Table Statistics

BASIC STATISTICAL PROCEDURES and TABLES Tenth Edition © 2001

P. Cabilio and J. Masaro

BASIC STATISTICAL PROCEDURES and TABLES

by Paul Cabilio and Joe Masaro Department of Mathematics and Statistics ACADIA UNIVERSITY

Tenth Edition 2001

i

ii

Basic Statistical Procedures and Tables

INTRODUCTION

This booklet is a summary of those fundamental statistical methods and associated tables that a student is likely to see in a general statistics course. While not a textbook, it may be used in conjunction with any textbook or set of notes that an instructor might select for such a course. In addition, it may serve as a useful reference handbook for any user of statistical methods. We have developed this handbook o ver many years of teaching introductory statistics to students in various disciplines and at various levels. In the experience of the authors it has proved to be a very efficient way of encapsulating the large variety of techniques available to the student. Every attempt has been made to make this booklet as convenient as possible for the user. On each page of the first part, headed Statistical Inference, the reader will find step-by -step instructions on how to conduct a particular statistical procedure. The layout in general is as follows.

• • • • • • • •

TITLE of the page names the procedure, and below it appears a table with various subheadings. ASSUMPTION lists the assumptions. TEST STATISTIC details the variable to be calculated. HYPOTHESIS gives the alternative (research) hypothesis and, when clarity requires it, the null hypothesis. The α-level rejection region and the P-value, if appropriate. In each case, the reader is referred to the appropriate table in the second part of the booklet. 100(1-α)% confidence interval, where appropriate. NOTES following each procedure provide additional useful information such as when a normal approximation can be used, to sample size requirements for confidence intervals. MINITAB commands when available for the procedure.

The second part of the booklet contains the tables. These were selected with “ease of use” in mind; and in order to increase their usefulness as a reference, tables not usually found in undergraduate textbooks have also been included.

Paul Cabilio Joe Masaro

iii


iv


CONTENTS INFERENCE CONCERNING THE LOCATION OF A POPULATION One Sample T-test for the Population Mean µ ........................................................................ One Sample Sign Test for the Population Median M............................................................... One Sample Wilcoxon Signed Rank Test (WSRT) for the Population Median M....................

3 4 5

INFERENCE CONCERNING SYMMETRY AND NORMALITY Sk-test for Asymmetry.............................................................................................................. Normal Scores Test for Non-Normality.................................................................................... Procedure for Determining the Proper Test of Location and/or the Associated Confidence Interval..............................................................................

6 6 7

INFERENCE CONCERNING THE LOCATION OF TWO POPULATIONS PAIRED SAMPLE DESIGN Paired T-test for the Population Mean µd................................................................................. Paired Sign Test for the Population Median Md....................................................................... Paired Wilcoxon Signed Rank Test (WSRT) for the Population Median Md............................

8 9 10

INDEPENDENT SAMPLE DESIGN Two Sample Pooled T-test for µX - µY...................................................................................... Two Sample Unpooled T-test for µX - µY.................................................................................. Two Sample Approximate Z-test for µX - µY............................................................................. Mann-Whitney-Wilcoxon Test (MWWT) for M X - MY................................................................

11 12 13 14

INFERENCE CONCERNING POPULATION VARIANCES 2

Test about a Population Varianceσ ...................................................................................... 2

2

Test Comparing Two Population Variances σ X and σ Y .......................................................

15 16

INFERENCE CONCERNING POPULATION PROPORTIONS Test about a Population Proportion p...................................................................................... Approximate Z-test Comparing Two Independent Population Proportions pX and pY.............. Fisher's Exact Test Comparing Two Independent Population Proportions pX and pY..............

17 18 19

ANALYSIS OF CATEGORICAL DATA Chi-Square Test for Goodness of Fit....................................................................................... Chi-Square Test for Independence.......................................................................................... Chi-Square Test for Homogeneity...........................................................................................

20 21 22

COMPLETELY RANDOMIZED DESIGN One-Way ANOVA: Fixed Effects, Normal Model..................................................................... Multiple Comparison Confidence Intervals for µi - µ j based on the One-Way ANOVA for a Completely Randomized Design, Fixed Effects Normal Model.......................... Newman-Keuls Multiple Range Procedure for the Comparison of k means µ1, µ2, ..., µk........ Kruskal-Wallis Test (KW-test).................................................................................................. Multiple Comparison Procedures based on the Kruskal-Wallis Test.......................................

v

23 24 25 26 27


CONTENTS (continued) RANDOMIZED COMPLETE BLOCK DESIGN Randomized Complete Block Design for the Comparison of k Treatments within b Blocks of Size k: Fixed Effects, Normal Model........................................................... 28 Multiple Comparison Confidence Intervals for µi. - µ j. based on the Two-Way ANOVA 29 for a Randomized Complete Block Design, Fixed Effects, Normal Model............................... Friedman Test.......................................................................................................................... 30 Multiple Comparison Procedures based on the Friedman Test............................................... 31 TWO-FACTOR FACTORIAL DESIGN: EQUAL NUMBERS PER CELL Two-Factor Factorial Design: Equal Numbers per Cell, Fixed Effects, Normal Model............ Multiple Pairwise Comparison Confidence Intervals based on the Two-Way ANOVA for a Two-Factor Factorial Design: Equal Numbers per Cell, Fixed Effects, Normal Model....

32 33

SIMPLE LINEAR REGRESSION AND CORRELATION Simple Linear Regression........................................................................................................ Test about the Population Correlation Coefficient: ρ Procedure for Testing H a: ρ < 0, ρ > 0, ρ ≠ 0........................................................................... Test about the Population Correlation Coefficient: ρ Procedure for Testing H a: ρ < ρ0, ρ > ρ0, ρ ≠ ρ0 where ρ0 ≠ 0.................................................. Kendall's Rank Correlation Coefficient: τ................................................................................. Spearman's Rank Correlation Coefficient: ρS..........................................................................

34 35 36 37 38

TABLES Table 1 Table 2 Table 3 Table 4 Table 5 Table 6 Table 7 Table 8 Table 9 Table 10 Table 11 Table 12 Table 13 Table 14 Table 15 Table 16 Table 17 Table 18 Table 19 Table 20 Table 21 Table 22 Table 23 Table 24 Table 25

Cumulative Binomial Probabilities: P(X ≤ x)............................................................ Cumulative Poisson Probabilities: P(X ≤ x)............................................................. Standard Normal: P(X ≤ x)...................................................................................... Upper Critical Values of Student's T-distribution..................................................... Exact Confidence Intervals for a Population Proportion p...................................... Distribution of the Sign Test Statistic: P(X ≤ x), X ~ Bin(n, .5)............................... Wilcoxon Signed-Rank Distribution: P(V ≤ v).......................................................... Mann-Whitney-Wilcoxon Distribution: P(W ≤ w)..................................................... Upper Critical Values of the Chi-Square Distribution.............................................. Upper Critical Values of the F-distribution with (ν1, ν2) df....................................... Critical Values for the Sk-test for Asymmetry......................................................... Critical Values for the Normal Scores Test for Non-Normality................................ Upper Critical Values for the Kruskal Wallis Test................................................... Upper Critical Values for the Friedman Test........................................................... Bonferroni Critical Values....................................................................................... Upper Critical Values for the Studentized Range: qα(r, ν)...................................... Critical Values dα for Multiple Comparisons based on the Kruskal-Wallis Test...... Critical Values eα for Multiple Comparisons based on the Friedman Test.............. ) Upper Critical Values for Kendall's Rank Correlation Coefficient τ ....................... Upper Critical Values for Spearman's Rank Correlation Coefficient RS.................. Sample Size for the T-test...................................................................................... Sample Size for the Pooled T-test.......................................................................... Sample Size for the One-Way ANOVA, Fixed Effects, Normal Model.................... Table of Pseudo-random Permutations of Size 9................................................... Table of Pseudo-random Digits..............................................................................

Acknowledgements..................................................................................................................

vi

41 47 48 52 53 56 57 61 65 66 70 70 71 73 74 77 80 81 82 83 84 86 88 90 91 93


STATISTICAL INFERENCE

1


2


One Sample T-test for the Population Mean

Assumption

A simple random sample of size n is chosen from a normal distribution; if n is large the normality assumption is not required.

T =

Test Statistic

X − µ 0 SE

, SE =

s n

; df = n − 1

t0 is the observed value of T

Ha

µ < µ0

µ > µ0

µ ≠ µ0

Reject H0 if (Table 4)

t0 ≤ -tα

t0 ≥ tα

|t0| ≥ tα/2

P(T ≤ t0)

P(T ≥ t0)

2P(T ≥ |t0|)

P-value

= P(T ≥ -t0)

100(1- )% CI for Notes:

X

±

t α / 2 SE

(a) For df > 100 use df = ∞, tα = zα and replace T by Z to calculate the approximate P-value from Table 3. (b) To estimate µ with Error Bound E and Confidence Level (Reliability) 100(1 - α)% use

 z σ ˆ  n =  α /2   E 

2

rounded up to the next whole number

where σˆ is a prior estimate of σ. Minitab:

Place the sample in column c1. For the T-test and/or CI, click Stat > Basic Statistics > 1-Sample t... Insert c1 in the Variables box then type in the appropriate Test mean ( µ0). Then click the Options... button and select the desired options.

3


One Sample Sign Test for the Population Median M

Assumption

Test Statistic

Ha

P-value (Table 6)

Notes:

A simple random sample is chosen from an arbitrary distribution

X x+ xn'

= = = = =

the number of xi > M0 the observed value of X the observed number of xi < M0 x+ + xn - (the observed number of xi = M0)

M < M0

M ≠ M0

M > M0

P(X ≤ x+)

P(X ≥ x+)

2P(X ≤ a)

= P(X ≤ x-)

a = min(x+, x-)

(a) For n' not in Table 6 (i.e. n' > 30) use the normal approximation to find the P-value.

  

P ( X ≤ x) ≈ P  Z ≤

x + .5 − n' (.5)  n' (.25)

 

(b) For values of x exceeding those in Table 6 use P ( X ≤ x) = 1 − P ( X ≤ n' − x − 1)

Minitab:

Place the sample in column c1. For the Sign Test and/or CI, click Stat > Nonparametrics > 1-Sample Sign... Insert c1 in the Variables box then type in the appropriate Test median (M 0) and select the Alternative. For a CI, select the Confidence interval option.

4


One Sample Wilcoxon Signed Rank Test (WSRT) for the Population Median M

Assumption

A simple random sample is chosen from a symmetric distribution

Let di = xi - M0 and n' = the number of non-zero di Test Statistic

Ha

V = the sum of the ranks of |di| for which d i > 0 v+ = the observed value of V v- = the sum of the ranks of |di| for which d i < 0 = n'(n'+1)/2 - v+

M < M0

P-value (Table 7)

Notes:

After throwing out all di = 0, let

M ≠ M0

M > M0

P(V ≤ v+)

P(V ≥ v+)

2P(V ≤ a)

= P(V ≤ v-)

a = min(v+, v-)

(a) Even for normal data when the T-test is superior, the performance of the WSRT is almost as good, and may be much better when the population is not normal. (b) For n' not in Table 7 (i.e. n' > 20) use the normal approximation to calculate the P-value.

   P (V ≤ v) ≈ P  Z ≤    (c)

   4  n' ( n' + 1)(2n' + 1)   24 

v + .5 −

For values of v exceeding those in Table 7 use

 

P (V ≤ v) = 1 - P  V ≤

Minitab:

n' (n' + 1)

n ′ ( n ′ + 1) 2

 

-v -1 

Place the sample in column c1. For the Wilcoxon Test and/or CI, click Stat > Nonparametrics > 1-Sample Wilcoxon... Insert c1 in the Variables box then type in the appropriate Test median (M0) and select the Alternative. For a CI, select the Confidence interval option.

5


Sk-test for Asymmetry

Assumption

Test Statistic


Sk =

X − m SE

, SE =

s n

sk0 is the observed value of Sk

H0: SCFSD (the Sample Comes From a Symmetric Distribution) Ha: SDNCFSD (the Sample Does NOT Come From a Symmetric Distribution)


|sk0| ≥ skα/2

P-value

2P(Sk ≥ |sk0|)

Note:

For n > 30, one may use critical values for n = ∞. In this case the test will be conservative i.e. the exact significance level will be lower than the nominal level.

Normal Scores Test for Non-Normality

Assumption


H0: SCFND (the Sample Comes From a Normal Distribution) Ha: SDNCFND (the Sample Does NOT Come From a Normal Distribution)

1. Place the sample in column c1 2. Click Stats > Basic Statistics > Normality Test... 3. Insert c1 in the Variable Box and under Tests for Normality click Ryan-Joiner 4. Click OK and use the P-value obtained from the output to conduct the test Alternatively, one may place the normal scores in column c2 (MTB > nsco c1 c2), calculate the correlation coefficient r (MTB > corr c1 c2), then Reject Ho if r r , where r α is obtained from Table 12.

6


Procedure for Determining the Proper Test of Location and/or the Associated Confidence Interval

T1: Sk-Test for Asymmetry [ H 0: SCFSD, H a: SDNCFSD;

α = .10 ]

∗

T2: Normal Scores Test for Non-Normality [ H0: SCFND, H a: SDNCFND;

α = .10 ]

STEP I NEITHER test is significant (i.e. neither rejects H 0 )

T1 rejects H0 OR T2 rejects H0

Inconclusive

The T-test is NOT appropriate

Examine the

Examine the

FIRST INDICATION

i) boxplot ii) histogram iii) stem and leaf plot

STEP II

Draw the appropriate conclusion from the choices below

CONCLUSION

SELECTED TEST/CI

i) boxplot ii) histogram iii) stem and leaf plot

Draw the appropriate conclusion from the choices below

SCF a Bell-Shaped probably Normal Distribution

SCF a Symmetric, not BellShaped Non-Normal Distribution

SCF a NonSymmetric Distribution

SCF a NonSymmetric Distribution

SCF a Symmetric Non-Normal Distribution (possibly BellShaped)

T-test / CI

WSRT / CI

Sign Test / CI

Sign Test / CI

WSRT / CI

∗

The nominal level of significance (in this case α = .10) for the Sk-Test for Asymmetry is approximate. The test maintains the nominal level of significance quite well over a wide selection of distributions including the normal distribution. However for some symmetric distributions (notably the Uniform, others in the Beta family, and the Cauchy) the nominal significance level underestimates the true significance level. For this reason, when the Sk-Test rejects H0, the procedure above views this as sufficient evidence to conclude the sample is not normally distributed ("FIRST INDICATION") but suspends judgement on the symmetry of the sample until "STEP II" has been completed.

7


Paired T-test for the Population Mean

Assumption

d

A simple random sample of paired observations (xi,yi), i = 1,2, . . . n is chosen whose differences di = xi - yi form a simple random sample from a normal distribution; if n is large the normality assumption is not required.

T =

d − D0 SE

Test Statistic

, SE =

s d

; df = n − 1

n

t0 is the observed value of T Ha

µd < D0

µd > D0

µd ≠ D0


t0 ≤ -tα

t0 ≥ tα

|t0| ≥ tα/2

P(T ≥ t0)

2 P(T ≥ |t0|)

P-value

P(T ≤ t0) = P(T ≥ -t0)

100(1- )% CI for d

d

±

t α / 2 SE

Note:

For df > 100 use df = ∞, tα = zα and replace T by Z to calculate the approximate P-value from Table 3.

Minitab:

Place the x-values in column c1 and the corresponding y-values in column c2. Then put the differences di = xi - yi in column c3: MTB > let c3 = c1 - c2 For the Paired T-test and/or CI, click Stat > Basic Statistics > 1-Sample t... Insert c3 in the Variables Variables box then type in the appropriate Test mean (D mean (D0). Then click the Options... button Options... button and select the desired options.

8


Paired Sign Test for the Population Population Median Md

Assumption

A simple random sample of paired observations (xi,yi), i = 1,2, . . . n is chosen whose differences di = xi - yi form a simple random sample from an arbitrary distribution Let di = xi - yi

Test Statistic

Ha

P-value (Table 6)

Notes:

X x+ xn'

= = = = =

the number of di > D0 the observed value of X the observed number of di < D0 x+ + xn - (the observed number number of di = D0)

Md < D0

Md > D0

Md ≠ D0

P(X ≤ x+)

P(X ≥ x+)

2P(X ≤ a)

= P(X ≤ x-)

a = min(x+, x-)

(a) For n' not in Table 6 (i.e. n' > 30) use the normal approximation to find the P-value.

  

P ( X ≤ x ) ≈ P  Z ≤

x + .5 − n' (.5)  n' (.25)

 

(b) For values of x exceeding those in Table 6 use P ( X ≤ x ) = 1 − P ( X ≤ n' − x − 1)

Minitab:

Place the x-values in column c1 and the corresponding y-values in column c2. Then put the differences di = xi - yi in column c3: MTB > let c3 = c1 - c2 For the Paired Sign Test and/or CI, click Stat > Basic Statistics > 1-Sample Sign... Insert c3 in the Variables Variables box then type in the appropriate Test median (D0) and select the Alternative. Alternative. For a CI, select the Confidence interval option. option.

9


Paired Wilcoxon Signed Rank Test (WSRT) for the Population Median Md

Assumption

A simple random sample of paired observations (xi,yi), i = 1,2, . . . n is chosen whose whose differences di = xi - yi form a simple random sample from a symmetric distribution Let di = (xi - yi) - D0 and n' = the number of non-zero di After throwing out all di = 0,

Test Statistic

V = the sum of the ranks of |di| for which d i > 0 v+ = the observed value of V v- = the sum of the ranks of |di| for which d i < 0 = n'(n'+1)/2 - v+

Ha

P-value (Table 7)

Notes:

Md < D0

Md > D0

P(V ≤ v+)

P(V ≥ v+)

2P(V ≤ a)

= P(V ≤ v-)

(a)

(b)

a = min(v+, v-)

For n' not in Table Table 7 (i.e. n' > 20) use the normal approximation to calculate the P-value.

   P (V ≤ v) ≈ P  Z ≤   

   4  n' ( n' + 1)(2n' + 1)   24 

v + .5 −

n' ( n' + 1)

For values of v exceeding those those in Table 7 use

 

P (V ≤ v) = 1 - P  V ≤

Minitab:

Md ≠ D0

n ′ ( n ′ + 1) 2

 

-v -1 

Place the x-values in column c1 and the corresponding y-values in column c2. Then put the differences di = xi - yi in column c3: MTB > let c3 = c1 - c2 For the Paired WSRT and/or CI, click Stat > Nonparametrics > 1-Sample Wilcoxon... Insert c3 in the Variables box Variables box then type in the appropriate Test median (D0) and select the Alternative. Alternative. For a CI, select the Confidence interval option. option.

10


Two Sample Pooled T-test for

Assumption

Y

Two independent simple random samples are chosen from normal distributions with equal variances σ 2X = σ 2Y

T =

Test Statistic

X -

s p2

( X − Y ) − D0

=

SE p

( n X

,

SE p

=



s p2 

2 − 1) s X + (nY − 1) sY 2 n X + nY − 2

1

 n X

;

+

1 



nY 

df = n X

+ nY − 2


µX - µY < D0

µX - µY > D0

µX - µY ≠ D0


t0 ≤ -tα

t0 ≥ tα

|t0| ≥ tα/2

P(T ≤ t0)

P(T ≥ t0)

2P(T ≥ |t0|)

P-value

= P(T ≥ -t0)

100(1 - )% CI for X - Y

Minitab:

X − Y

±

t α / 2 SE p

Place both the x- and y-samples in column c1 with subscripts in c2 OR place the x-sample in column c1 and the y-sample in column c2. For the Pooled T-test and/or CI, click Stat > Basic Statistics > 2-sample t... Select Samples in one column OR Samples in different columns whichever is appropriate and insert c1 and c2 in the appropriate boxes. Select the Assume equal variances option. Click the Options... button, type in the appropriate Test mean (D0) if conducting a test, and select the desired options.

11


Two Sample Unpooled T-test for

Assumption

Y

Two independent simple random samples are chosen from normal distributions (no assumptions regarding

T =

Test Statistic

X -

( X − Y ) − D0

df =

SE

(

2 s X / n X

(

2 s X / n X

n X

,

2 s X

SE =

n X

+ sY 2 / nY )

)

2

−1

2

( s +

2 Y

/ nY

nY

)

2

;

2 X and

σ

+

2 Y

σ

are required)

2 s Y

nY

Rounded DOWN to a whole number

−1


µX - µY < D0

µX - µY > D0

µX - µY ≠ D0


t0 ≤ -tα

t0 ≥ tα

|t0| ≥ tα/2

P(T ≤ t0)

P(T ≥ t0)

2P(T ≥ |t0|)

P-value (Table 4)

= P(T ≥ -t0)

100(1 - )% CI for X - Y

X − Y

±

t α / 2 SE

Note:

The nominal level of significance for this test is approximate.

Minitab:

Place both the x- and y-samples in column c1 with subscripts in c2 OR place the x-sample in column c1 and the y-sample in column c2. For the Unpooled T-test and/or CI, click Stat > Basic Statistics > 2-sample t... Select Samples in one column OR Samples in different columns whichever is appropriate and insert c1 and c2 in the appropriate boxes. Make sure the Assume equal variances option is disabled. Click the Options... button, type in the appropriate Test mean (D0) if conducting a test, and select the desired options.

12


Two Sample Approximate Z-test for

Assumption

X -

Y

Two independent simple random samples are chosen from arbitrary distributions and both nX > 30 and nY > 30

Z =

Test Statistic

( X − Y ) − D0 SE

,

SE =

2 s X

n X

+

2 s Y

nY

z0 is the observed value of Z Ha

µX - µY < D0

µX - µY > D0

µX - µY ≠ D0

Reject H0 if (Table 4 at df = )

z0 ≤ -zα

z0 ≥ zα

|z0| ≥ zα/2

P(Z ≤ z0)

P(Z ≥ z0)

2 P(Z ≤ -|z0|)

P-value (Table 3)

= 1 - P(Z ≤ z0)

100(1 - )% CI for X - Y Minitab:

[ X − Y ] ± z α / 2 SE

The same commands used for the Two Sample Unpooled T-test will give a reasonable approximation of the Two Sample Approximate Z-test. Place both the x- and y-samples in column c1 with subscripts in c2 OR place the x-sample in column c1 and the y-sample in column c2. For the Two Sample Approximate Z-test and/or CI, click Stat > Basic Statistics > 2-sample t... Select Samples in one column OR Samples in different columns whichever is appropriate and insert c1 and c2 in the appropriate boxes. Make sure the Assume equal variances option is disabled. Click the Options... button, type in the appropriate Test mean (D0) if conducting a test, and select the desired options.

13


Mann-Whitney-Wilcoxon Test (MWWT) for MX - M Y

Assumption

Two independent simple random samples are chosen from distributions of the same shape Let:

Test Statistic

(xi) be the sample with the fewest observations (yi) be the sample with the most observations y'i = yi + D0

Rank the combined x- and y'-samples and let W = the sum of the ranks of the x-sample wX = the observed value of W wX* = nX(nX + nY + 1) - w X

Ha

MX - MY < D0

P-value (Table 8)

Notes:

MX - MY ≠ D0

MX - MY > D0

P(W ≤ wX)

P(W ≥ wX) = P(W ≤ wX*)

2 P(W ≤ a), a = min(wX, wX*)

(a) Even for normal data when the Pooled T-test is superior, the performance of the MWWT is almost as good, and may be much better when the populations are not normal. (b) For nX and nY not in Table 8 and both nX and nY > 10 use the normal approximation to calculate the P-value.

   P (W ≤ w) ≈ P  Z ≤   

+ nY + 1)   2  n X nY ( n X + nY + 1)   12 

w + .5 −

n X ( n X

(c) For values of w exceeding those in Table 8 use P (W ≤ w) = 1 − P (W ≤ n X (n X

Minitab:

+ nY + 1) − w − 1)

Place the x-sample in column c1, the y-sample in column c2, and the y'-sample in c3: MTB > let c3 = c2 + D0 For the Mann-Whitney-Wilcoxon Test and/or CI, click Stat > Nonparametrics > Mann-Whitney... For the test, use columns c1 and c3 as the First and Second Samples; for the CI use columns c1 and c2. Then select the Alternative and Confidence level .

14


Test about a Population Variance

Assumption

σ

2

A simple random sample is chosen from a normal distribution

χ

Test Statistic

2

=

(n − 1) S 2 σ 02

df = n − 1

,

2

χ 0 is the observed value of χ

σ 2 < σ 02

Ha

σ 2 > σ 02

2

σ 2 ≠ σ 02 2


P-value

2 χ 0

≤

χ 12− α

2 χ 0

≥

χ 0

2 χ α

or χ 02

PL = P( χ 2

≤ χ 02 )

PU = P( χ 2 ≥ χ 02 )

 (n − 1)S 2   χ α 2 / 2 

Notes:

,

(n − 1) S 2  χ 12− α / 2

(a) To calculate P( χ 2 ≤ x) for 'a' df on Minitab use: MTB > cdf x; SUBC > chis a. (b) To calculate χ α 2 for 'a' df on Minitab use: MTB > invcdf 1- ; SUBC > chis a.

15

≥ χ α 2 / 2

2 min(PL, PU)

= 1 - PL

100(1 - )% CI for σ 2

≤ χ 12− α

 


Test Comparing Two Population Variances

Assumption

2 Y

σ

Two independent simple random samples are chosen from normal distributions

Test Statistic

F =

2

Ha

2 S X

S Y 2

F0 is the observed value of F

,

2

2

σ X < σ Y


2 X and

σ

F 0

≤

1 F α (nY

− 1, n X − 1)

σ X ≠ σ Y

2

2

σ X > σ Y

F 0 F 0

≥ F α ( n X − 1, nY − 1)

≤

1 F α / 2 (nY

P-value

PU = P(F ≥ F0)

df = (nX - 1, nY - 1)

= 1 - PL

− 1, n X − 1) or

F 0 ≥ F α / 2 (n X

PL = P(F ≤ F0)

2

− 1, nY − 1)

2 min(PL, PU)

df = (nX - 1, nY - 1)

100(1 - )% CI for

2 σX

Notes:

/

2 σ Y

2  S X 1   S 2 F α / 2 (n X − 1, nY − 1) ,  Y

2 S X 2

S Y

F α / 2 ( nY

 − 1, n X − 1)  

(a) To calculate P(F ≤ x) for df = (a,b) on Minitab use: MTB > cdf x; SUBC > f a b. (b) To calculate Fα , df = (a,b), on Minitab use: MTB > invcdf 1- ; SUBC > f a b.

Minitab:

Place both the x- and y-samples in c1 with subscripts in c2 OR place the x-sample in c1 and the y-sample in c2. For a two sided test, click Stat > Basic Statistics > 2 Variances..., select the appropriate data format. Then click the Options... button and select the desired options. Both the F-test and Levene's Test are conducted.

16


Test about a Population Proportion p

Assumption Test Statistic

X = the number of successes in the n trials x0 = the observed value of X p < p0

Ha

P-value (Table 1: n, po)

'Exact' 90%, 95%, 99% CI's for p

Approximate Large Sample 100(1 - )% CI for p Notes:

A binomial experiment with n trials and probability of success p (unknown) is performed

p ≠ p0

p > p0

PL = P(X ≤ x0)

PU = P(X ≥ x0)

2 min(PL, PU)

= 1 - P(X ≤ x0 -1)

Unless p0 = .5, this P-value is approximate

For 5 ≤ n ≤ 20, these can be found in Table 5

pˆ

± z α / 2 SE ;

SE =

pˆ qˆ n

, pˆ =

x 0 n

, qˆ = 1 − pˆ

(valid if both x0 ≥ 5 and n – x0 ≥ 5)

(a) For n or p0 not in Table 1 and such that both np0 ≥ 5 and nq0 ≥ 5, use the normal approximation to calculate the P-value.

  

P ( X ≤ x) ≈ P  Z ≤

x + .5 − np 0  np 0 q 0

 

(c) To estimate p with Error Bound E and Confidence Level (Reliability) 100(1 - α)% use 2 ~ if a prior estimate p of p  z α / 2  ~ ~ n = pq   is available  E 

 z  n = (.25)  α / 2   E 

2

if no prior estimate of p is available

The value obtained is rounded up to the next whole number.

Minitab:

Click Stat > Basic Statistics > 1 Proportion..., select the appropriate data format and insert the required information. Then click the Options... button and select the desired options.

17


Approximate Z-test Comparing Two Independent Population Proportions pX and p Y

Independent Bin(nX, pX) and Bin(nY, pY) experiments are performed with nX and nY large enough so that the number of successes and failures in each experiment is at least 5

Assumption

Z = pˆ X

Test Statistic

pˆ X

− pˆ Y

SE p

=

X

;

, pˆ Y

n X

SE p

=

Y nY

=



1

pˆ (1 − pˆ )

 n X

, pˆ =

+

1 



nY 

X + Y n X

+ nY

X and Y are the number of successes in each of the binomial experiments When pX = pY, pˆ is a "pooled" estimate for p X = pY z0 is the observed value of Z

Ha Reject H0 if (Table 4 at df =

)

P-value (Table 3)

pX - pY < 0

pX - pY > 0

pX - pY ≠ 0

z0 ≤ -zα

z0 ≥ zα

|z0| ≥ zα/2

P(Z ≤ z0)

P(Z ≥ z0)

2 P(Z ≤ -|z0|)

= 1 - P(Z ≤ z0)

Large Sample 100(1 - )% CI for pX - p Y Note:

( pˆ X − pˆ Y ) ± z α / 2 SE ;

SE =

pˆ X qˆ X n X

+

pˆ Y qˆ Y nY

One may also test: Ha : pX – pY < p0 , pX – pY > p0 or pX – pY ≠ p0 using the test statistic Z =

( pˆ X − pˆ Y ) − p 0 SE

;

SE =

pˆ X qˆ X n X

+

pˆ Y qˆ Y nY

and the same rejection rules as above. Minitab:

Click Stat > Basic Statistics > 2 Proportions..., select the appropriate data format and insert the required information. Then click the Options... button and select the desired options.

18


Fisher's Exact Test Comparing Two Independent Population Proportions pX and p Y

Assumption

Independent Bin(nX, pX) performed

and

Bin(nY, pY)

experiments are

x0 = the observed number of successes in the x-sample y0 = the observed number of successes in the y-sample n = nX,

M = x0 + y0,

N = nX + nY

Let H be the hypergeometric (n, M, N) random variable and use H to calculate the appropriate P-value as indicated below Conditional Test Statistic Remark : If X and Y are independent Bin(n X, p X) and Bin(nY, p Y) random variables then, under the null hypothesis pX = pY, the distribution of X given X + Y = M is hypergeometric (n, M, N). Thus P(H ≤ x0) and P(H ≥ x0) are the conditional (null) probabilities of observing outcomes as extreme or more extreme than what was actually observed (i.e. they are Pvalues). Ha

pX - pY < 0

pX - pY > 0

pX - pY ≠ 0

P-value

PL = P(H ≤ x0)

PU = P(H ≥ x0)

2 min(PL, PU) (approximately)

Calculation of the P-value using Minitab (a) PL = P(H x0) 1. Click Calc > Probability Distributions > Hypergeometric... 2. Choose Cumulative probability and enter the values for - Population size (N), - Successes in the population (M, which Minitab calls X), - Sample size (n) 3. Choose Input constant , enter the value of x0 and under Optional storage 4. Click OK and type the command MTB > prin k1 Then PL = k1.

(b) PU = P(H x0) Repeat steps 1. to 4. in (a) with x0 replaced by x0 - 1 in step 3. and obtain a new value for 'k1'. Then PU = 1 - k1.

19


Chi-Square Test for Goodness of Fit

Assumption

Test Statistic

A multinomial experiment is performed: a single random sample of size n is chosen from a single population and classified according to k categories

χ 2

=

∑

(Oi − E i )2 E i

, E i

= npio

2

2


p1 = p1o, p2 = p2o ,. . . . . . , p k = pko H0 [ p1o + p2o + . . . + p ko = 1 ] Ha

At least one value of pi in H0 is incorrect 2


χ 0

df = k - 1 - [the number of free parameters estimated] P( χ 2 ≥ χ 02 )

P-value

Notes:

≥ χ α 2

(a) The nominal level of significance for this test is approximate. The approximation will be adequate if all E i are ≥ 5. (b) If some Ei are < 5, categories (cells) can be combined (pooled) so that each of the new categories has E i ≥ 5.

20


Chi-Square Test for Independence

Assumption

Test Statistic

A multinomial experiment is performed: a simple random sample of size n is chosen from a single population and crossclassified according to 'r' row and 'c' column categories

χ

2

=

∑

(Oij − E ij )2 E ij

, E ij

2

=


n 2

H0

Row and column categories are independent

Ha

Row and column categories are dependent


2

χ 0

≥ χ α 2 , df = (r - 1)(c - 1) P( χ 2

P-value

Notes:

r i c j

≥ χ 02 )

(a) The nominal level of significance for this test is approximate. The approximation will be adequate if all Eij are ≥ 5. (b) If some Eij are < 5, categories (cells) can be combined (pooled) so that each of the new categories has E ij ≥ 5.

Minitab:

Place the column entries of the contingency table in columns c1, c2 ..., then click Stat > Tables > Chi-Square Test... Insert c1, c2, ... in the Columns containing the table box.

21


Chi-Square Test for Homogeneity

Assumption

Test Statistic

Several independent multinomial experiments are performed: simple random samples are chosen from each of 'c' independent populations and each sample is classified according to the same 'r' categories

χ

2

=

∑

(Oij − E ij )2 E ij

, E ij

2

=


n 2

H0

The populations are homogeneous with respect to each of the 'r' categories

Ha

The populations are not homogeneous with respect to the 'r' categories


2

χ 0

≥ χ α 2 , df = (r - 1)(c - 1) P( χ 2 ≥ χ 02 )

P-value

Notes:

r i c j

(a) The nominal level of significance for this test is approximate. The approximation will be adequate if all E ij are ≥ 5. (b) If some Eij are < 5, categories (cells) can be combined (pooled) so that each of the new categories has E ij ≥ 5.

Minitab:

Place the column entries of the contingency table in columns c1, c2 ..., then click Stat > Tables > Chi-Square Test... Insert c1, c2, ... in the Columns containing the table box.

22


One-Way ANOVA: Fixed Effects, Normal Model

The data arise as 'k' independent simple random samples from 'k' 2 normal distributions with equal variances σ Assumption

Yij =

µ + τi + εij = µi + εij th

th

Yij is the j observation from the i sample

µ is a constant , τi is the ith 'treatment effect', Σ τi = 0 µi = µ + τi is the mean of the ith distribution εij are independent N(0, σ2) , i = 1, 2, . . . , k; j = 1, 2, . . . , n i

Test Statistic

th

ni

= the size of the i sample

N

=

Σ ni th

y i. =

the mean of the i sample

y .. =

the grand mean

SSTr =

Σ ni ( y i. -

SSE

=

Σ (yij -

y i.)

SSTo =

Σ (yij -

y ..)

y ..)

2

2

2

Analysis of Variance Table Source of Variation

Null and Alternative Hypotheses

df

Sum of Squares

Treatments Error

k-1 N-k

SSTr SSE

Total

N-1

SSTo

H0:

Mean Square MS = SS/df MSTr MSE

F0 = MSTr/MSE

µ1 = µ2 = . . . = µk [or τ1 = τ2 = . . . = τk = 0]

Ha: not all

µi are equal [or not all τi = 0]


F0 ≥ Fα ; df = (k-1, N-k)

P-value

P(F ≥ F0)

Minitab:

F = MS/MSE

Place all the data values in column c1 and their corresponding treatment levels in column c2, then click Stat > ANOVA > Oneway... Insert c1 in the Response box and c2 in the Factor box.

23


Multiple Comparison Confidence Intervals for i - j based on the One-Way ANOVA for a Completely Randomized Design , Fixed Effects, Normal Model

Let: k = the number of groups (treatments) th ni = the sample size of the i group N = the combined sample size µi = the true ith treatment mean th

y i. = the sample mean for the i treatment

1 - α = the nominal family level of confidence

[ y i⋅ − y j⋅ ] ± B

 1  ni + 

MSE 

Bonferroni Procedure

1 

n j 

B = t α / 2 g , df = N − k , is obtained from Table 15 g = the number of pairwise comparisons

[ yi⋅ − y j⋅ ] ±

 1  ni + 

T MSE 

Tukey Procedure T = qα (k , N − k )

1 

n j 

2

qα (k , N − k ) is obtained from Table 16

Notes:

(a) If all k(k-1)/2 pairwise comparisons are of interest the Tukey CI's will be narrower than the Bonferroni CI's. If not all pairwise comparisons are of interest the Bonferroni CI's may be narrower. One may tell which is narrower by comparing the size of B and T. (b) "Data snooping" is allowed with the Tukey Procedure but not with the Bonferroni Procedure. (c) The true family level of confidence for the Tukey Procedure is exactly the nominal level if all sample sizes are equal and all pairwise comparisons are of interest; in any other case the true level of confidence is greater than the nominal level. The true family level of confidence for the Bonferroni Procedure is always at least the nominal level.

Minitab:

Place all the data values in column c1 and their corresponding treatment levels in column c2, then click Stat > ANOVA > Oneway... Insert c1 in the Response box and c2 in the Factor box and click the Comparisons button.

24


Newman-Keuls Multiple Range Procedure for the Comparison of k Means 1, 2,..., k

Assumption

The assumptions for the one way anova normal model hold and in addition each of the k samples have the same sample size n and the null hypothesis H0 has been rejected

1. For r = 2, 3, ..., k, tabulate Dα

Procedure (Table 16)

= qα (r , k (n − 1) )

r

2

3

.

.

.

.

k

Dα

-

-

-

-

-

-

-

MSE n

:

≤ y (2) ≤ . . . . . ≤ y (k )

2.

List the k ordered sample means : y (1)

3.

For each group of r consecutive ordered means r = k, k-1, . . ., 2, compare the group's Range to Dα (a) If Range < Dα

[no significance]

Draw a line under this group of sample means and stop testing subgroups of this group* (b) If Range ≥ Dα

[significance]

Draw no line and continue testing subgroups of this group

4.

Notes:

When step 3. is finished for all r = k, k-1, . . ., 2, only those population means whose estimates are not connected by a line underneath are judged to be significantly different

*(a) Whenever a group of means is underlined, there is no point in comparing the range of any subgroup. By the credo of multiple range tests no subgroup within this group can be significant, since it is contained in a larger nonsignificant group. (b) For the NK procedure the family Type I error rate is at most a

1 - (1 - α)

where 'a' is the greatest integer ≤ k/2.

25


Kruskal-Wallis Test (KW-test)

Assumption

The data arise as 'k' independent simple random samples chosen from distributions of the same shape Rank the combined samples Let:

Test Statistic

ni = N = Ri = Ri =

th

the size of the i sample the combined sample size th the sum of the ranks of the i sample th the average of the ranks for the i sample H =

=

12 N ( N + 1)

∑

N + 1   ni  Ri −  2   Ri2

12 N ( N + 1)

∑n

−

2

3( N + 1)

i

h0 is the observed value of H Null and Alternative Hypotheses

H0 : M1 = M2 = . . . = Mk


h0 ≥ hα

P-value

P(H ≥ h0)

Notes:

Ha : not all Mi are equal

(a) When the number of samples k = 2, the Kruskal-Wallis Test is equivalent to the two-sided Mann-Whitney Test. (b) When Table 13 is not applicable and k = 3 with all group sizes above 5 or k > 3 with all group sizes above 4, one may use the approximation: 2 hα ≈ χ α , df = k -1

Minitab:

Place all the data values in column c1 and their corresponding treatment levels in column c2 then click Stat > Nonparametrics > Kruskal-Wallis... Insert c1 in the Response box and c2 in the Factor box.

26


Multiple Comparison Procedures based on the Kruskal-Wallis Test Let: Mi k ni N Ri Ri

= = = = = =

th

the i treatment median the number of samples (treatments) th the sample size of the i sample the combined sample size th the sum of the ranks for the i sample th the average of the ranks for the i sample

α

= the nominal family type I error rate (or 1 - α = the nominal family level of confidence) Procedure 1:

Equal Sample Sizes, n i = n If Ri

− R j ≥ d α and Ri > R j

, conclude M i

> M j

If Ri

− R j ≥ d α and Ri < R j

, conclude M i

< M j

where d α is obtained from Table 17 Procedure 2:

Large Sample Approximation, Equal Sample Sizes ni = n, n Large Replace d α in procedure 1 by qα ( k , ∞)

nN ( N + 1) 12

where qα ( k , ∞) is obtained from Table 16 Procedure 3:

Conservative Procedure (always applicable)

If Ri

− R j ≥

hα

N ( N + 1)  1

If Ri

− R j ≥

hα

N ( N + 1)  1

12

12

 ni   ni 

+ +

1 

and Ri

> R j

, conclude M i

> M j

1 

and Ri

< R j

, conclude M i

< M j

n j  n j 

where hα is obtained from Table 13 Procedure 4:

Large Sample Approximation , Unequal Sample Sizes Replace

hα in procedure 3 by z α / 2 g where g =

27

k (k − 1) 2


Randomized Complete Block Design for the Comparison of k Treatments within b Blocks of Size k: Fixed Effects, Normal Model

Yij =

µ + τi + β j + εij , where th

th

Yij = the observation under the i treatment and the j block Assumption

τi = treatment effects , Σ τi = 0 ; β j = block effects , Σ β j = 0 εij are independent N(0, σ2) , i = 1, 2, . . ., k; j = 1, 2, . . ., b th

SSTr = b Σ ( y i. - y ..)

y . j = the j block mean

th

SSBl = k Σ ( y . j - y ..)

y .. = the grand mean

SSE

y i. = the i treatment mean

2

2

= SSTo - SSTr - SSBl

SSTo = Test Statistic

Analysis of Variance Table Source of Variation

Null and Alternative Hypotheses

Σ (yij - y ..)2

Sum of Squares

df

Treatments k -1 Blocks b –1 Error (k -1)(b -1)

SSTr SSBl SSE

Total

SSTo

kb - 1

Mean Square MS = SS/df F = MS/MSE MSTr MSBl MSE

H0: there are no treatment effects :

τ1 = τ2 = . . . = τk = 0

Ha: there is a treatment effect : not all τi are 0


F0 ≥ Fα ; df = [k-1, (k-1)(b-1)]

P-value

P(F ≥ F0)

Minitab:

F 0 = MSTr/MSE

Place all the data values in column c1, their corresponding treatment levels and block numbers in columns c2 and c3 respectively, then click Stat > ANOVA > Twoway... Insert c1 in the Response box, c2 in the Row factor box and c3 in the Column factor box.

28


Multiple Comparison Confidence Intervals for i. - j. based on the Two-Way ANOVA for a Randomized Complete Block Design, Fixed Effects, Normal Model

Let:

k = the number of treatments b = the number of blocks τi = the ith treatment effect µi. = µi. = µ + τi , the true ith treatment mean th

y i. = the sample mean for the i treatment

1 - α = the nominal family level of confidence

[ yi⋅ − y j⋅ ] ± B Bonferroni Procedure

2 MSE b

B = t α / 2 g is obtained from Table 15 df = (k − 1)(b − 1), g = the number of pairwise comparisons

[ y i⋅ − y j⋅ ] ± Tukey Procedure

T =

T

2 MSE b

qα (k , (k − 1)(b − 1) ) 2

qα (k , (k − 1)(b − 1) ) is obtained from Table 16 Notes:

(a)

µi. - µ j. = τi - τ j

(b) If all k(k-1)/2 pairwise comparisons are of interest the Tukey CI's will be narrower than the Bonferroni CI's. If not all pairwise comparisons are of interest the Bonferroni CI's may be narrower. One may tell which is narrower by comparing the size of B and T. (c) "Data snooping" is allowed with the Tukey Procedure but not with the Bonferroni Procedure.

29


Friedman Test

Assumption

The data arises from a randomized complete block design for the comparison of 'k' treatments within 'b' blocks of size 'k' Rank the observations within each block Let:

Ri = the sum of the ranks received by the th observations under the i treatment th

Test Statistic

Ri = the average of the ranks under the i treatment S =

=

12b k ( k + 1)

∑

k + 1    Ri −  2  

R bk (k + 1) ∑ 12

2 i

−

2

3b( k + 1)

s0 is the observed value of S Null and Alternative Hypotheses

H0: There is no difference between treatments


s0 ≥ sα

P-value

P(S ≥ s0)

Notes:

Ha: There is a difference between treatments

(a) When k and b fall outside the range of Table 14, the critical values 2 of S may be approximated by those of a χ distribution with df = k -1. This approximation is not always accurate. (b) Do not use the Friedman test when k = 2 as it is equivalent to the two-sided sign test in this case.

Minitab:

Place all the data values in column c1, their corresponding treatment levels and block numbers in columns c2 and c3 respectively, then click Stat > Nonparametrics > Friedman... Insert c1 in the Response box, c2 in the Treatment box and c3 in the Blocks box.

30


Multiple Comparison Procedures based on the Friedman Test

Let:

τi = the ith treatment effect k = the number of treatments b = the number of blocks th Ri = the sum of the ranks for the i treatment α = the nominal family type I error rate (or 1 - α = the nominal family level of confidence)

Procedure:

If Ri

− R j ≥ eα and Ri > R j

, conclude τ i

> τ j

If Ri

− R j ≥ eα and Ri < R j

, conclude τ i

< τ j

where eα is obtained from Table 18 Large Sample Approximation: For b large and beyond those values in Table 18,

Replace eα by qα (k , ∞)

bk ( k + 1) 12

is obtained from Table 16

31

, where qα (k , ∞)


Two-Factor Factorial Design: Equal Numbers per Cell, Fixed Effects, Normal Model

µ + αi + β j + (αβ)ij + εijk , where

Yijk

=

Yijk

= the k observation when factor A is at the i level

th

th

th

and factor B is at the j level Assumption

Σ αi = Σ β j = Σi (αβ)ij = Σ j (αβ)ij = 0 εijk are independent N(0, σ2) i = 1, 2, . . ., a; j = 1, 2, . . ., b; k = 1, 2, . . ., n SSA

= nb Σ ( y i.. - y ...)

2

SSE

SSB

= na Σ ( y . j. - y ...)

2

SSTo =

SSAB = n Σ ( y ij. - y i.. - y . j. + y ...)

Test Statistic

Notes:

=

Σ (yijk - y ij.)2 Σ (yijk - y ...)2

2

Analysis of Variance Table Source of Sum of Variation df

Mean Square Squares MS = SS/dfF F = MS/MSE

A B AB Error

a-1 b-1 (a-1)(b-1) ab(n-1)

SSA SSB SSAB SSE

Total

abn - 1

SSTo

MSA MSB MSAB MSE

MSA/MSE MSB/MSE MSAB/MSE

(a) If n = 1 (i.e. one observation per cell), one cannot test for interaction effects and so the interaction terms should be omitted from the model. (b) If n > 1, test for interaction effects first. (1) if interaction effects are present, transform the data (2) if interactions are still present return to the original data and examine the factor effects jointly (3) if no interactions are present test for the main effects and if they are present examine them separately

Minitab:

Place all the data values in column c1, their corresponding factor A levels and factor B levels in columns c2 and c3 respectively, then click Stat > ANOVA > Twoway... Insert c1 in the Response box, c2 in the Row factor box and c3 in the Column factor box.

32


Multiple Pairwise Comparison Confidence Intervals based on the Two-Way ANOVA for a Two-Factor Factorial Design: Equal Numbers per Cell, Fixed Effects, Normal Model

Let:

a b n

= = = µi. = µ. j = µij =

the number of levels for factor A the number of levels for factor B the number of observations per cell th the true mean for the i level of factor A th the true mean for the j level of factor B th the true mean for the (i, j) treatment th

y i.. = the sample mean for the i level of factor A th

y . j. = the sample mean for the j level of factor B th

y ij. = the sample mean for the (i,j) treatment

g

= the number of pairwise comparisons of interest

1-α

CI's for i . - k. (αi - αk)

CI's for . j - .l ( j - l)

CI's for ij - kl

Note:

= the nominal family level of confidence

Bonferroni :

[ y i⋅⋅ − y k ⋅⋅ ] ± B

2 MSE

Tukey :

[ y i⋅⋅ − y k ⋅⋅ ] ±

2 MSE

T

bn bn

Bonferroni :

[ y⋅ j⋅ − y⋅l ⋅ ] ± B

2 MSE

Tukey :

[ y⋅ j⋅ − y⋅l ⋅ ] ±

2 MSE

Bonferroni :

[ y ij⋅ − y kl ⋅ ] ± B

2 MSE

Tukey :

[ y ij⋅ − y kl ⋅ ] ±

2 MSE

T

T

an an

n n

, B = t α / 2 g , df = (n − 1)ab , T =

qα (a, ( n − 1) ab ) 2

, B = t α / 2 g , df = ( n − 1) ab , T =

qα (b, ( n − 1) ab ) 2

, B = t α / 2 g , df = ( n − 1) ab , T =

qα (ab, ( n − 1)ab ) 2

If it is desired to have a family level of confidence coefficient 1 - α for a joint set of pairwise comparisons involving both factor A and factor B, the Bonferroni method can be used directly, with g representing the total number of pairwise comparisons in the joint set.

33


Simple Linear Regression Yi = β0 +

β1xi + εi , where the εi are independent and normally

Assumption distributed with mean 0 and standard deviation 2

2

= ∑ x i - (∑ xi ) / n,

SSXX

SSyy =

SSXY = ∑ xiyi - (∑ xi )(∑ yi ) / n,

∑ y2i - (∑ yi )2 / n 2

SSE = SS YY - (SSXY) / SSXX

Estimate of Slope ( β1):

b1 = SSXY / SSXX

Estimate of Intercept ( β0):

b0 = y - b1 x

Estimate of

σ2:

MSE = SSE / (n - 2) 2

Estimated Variance of b1:

Basic Calculations

σ, i = 1, 2, . . ., n

s (b1) = MSE / SSXX

Estimated Standard s (b1 ) = s 2 (b1 )

Deviation of b1:

yˆ = b0 + b1 x

Least Squares Line:

2

Estimated Variance of yˆ :

2

s ( yˆ ) = MSE [1/n + (x - x ) / SSXX ]

Estimated Standard s(yˆ ) =

Deviation of yˆ :

To test the alternatives Ha: β1 <, >, or =/ Test and CI for the Slope

1

T =

b1

− β 10

s (b1 )

100(1 -

β10, use the t-statistic

, df = n −2 , and the usual rejection regions

)% CI for

α

100(1 - )% CI for E(Y | x) 100(1 - )% PI for Ynew(x)

s 2 (yˆ )

β1 : yˆ

ˆ y

±

b1

±

±

t α / 2 s (b1 ) , df = n − 2

t α / 2 s ( yˆ ) , df = n − 2

ˆ ) + MSE , df = n − 2 t α / 2 s 2 ( y

Note:

The above test for slope, with β10 = 0, can also be used as a test about the population correlation coefficient ρ: i.e. a test of Ha: ρ <, >, or =/ 0

Minitab:

Place the x-values in column c1 and corresponding y-values in column c2, then click Stat > Regression > Regression... Insert c2 in the Response box and c1 in the Predictors box. To obtain a CI for E(Y | x) and a PI for Ynew(x), click the Options button.

34


Test about the Population Correlation Coefficient: Procedure for Testing Ha: < 0, > 0, or 0

Assumption

The data consists of a simple random sample (x1, y1), (x2, y2), ..., (xn, yn), from a bivariate normal distribution

r =

∑ ( x − x )( y − y ) ∑ ( x − x ) ∑ ( y − y ) i

i

2

i

Test Statistic

T = r

n−2 1 − r 2

2

i

,

df = n − 2


ρ < 0

ρ > 0

ρ ≠ 0


t0 ≤ -tα

t0 ≥ tα

|t0| ≥ tα/2

P(T ≤ t0)

P(T ≥ t0)

2 P(T ≥ |t0|)

P-value

= P(T ≥ -t0)

Note:

Pearson's r measures the strength of the linear relationship between X and Y. It estimates the population correlation coefficient ρ.

Minitab:

Place the x-values in column c1 and corresponding y-values in column c2. To calculate r and obtain the P-value for testing Ha: ρ ≠ 0, click Stat > Basic Statistics > Correlation... Insert c1 and c2 in the Variables box and select the Display p-values box. To obtain the observed test statistic to, click Stat > Regression > Regression... Insert c2 in the Response box and c1 in the Predictors box.

35


Test about the Population Correlation Coefficient: Procedure for Testing Ha: < 0, > 0, or 0, where

0

0

The data consists of a simple random sample (x1, y1), (x2, y2), . . ., (xn, yn), from a bivariate normal distribution with n 'large' (n > 25)

Assumption

∑ ( x − x )( y − y ) ∑ ( x − x ) ∑ ( y − y ) i

r =

i

2

2

i

V =

Test Statistic

v0

=

1 2

 1 + r   , (" Fisher' s Z")  1 − r 

ln

1 2

Z =

i

 1 + ρ 0    1 − ρ 0 

ln

n − 3 (V − v 0 )

z0 is the observed value of Z Ha Reject H0 if (Table 4 at df =

P-value (Table 3)

Approximate 100(1 - )% CI for

Notes:

)

ρ < ρ0

ρ > ρ0

ρ ≠ ρ0

z0 ≤ -zα

z0 ≥ zα

|z0| ≥ zα/2

P(Z ≤ z0)

P(Z ≥ z0)

2 P(Z ≤ -|z0|)

= 1 - P(Z ≤ z0)

  tanh v −   

 , n − 3 

z α / 2

where tanh ( x ) =

e x e x

 tanh v + 

   n − 3  

z α / 2

− e − x + e − x

(a) One can use minitab to calculate r but then the rest of the test must be done by hand. (b) For n ≤ 25, 'exact' tests and confidence intervals for ρ can be obtained using "Tables of the Ordinates and Probability Integral of the Distribution of the Correlation Coefficient in Small Samples" by F. N. David (1938), Cambridge University Press, Cambridge, reprinted in "Biometrika rd Tablesfor Statisticians" (1966), Vol I, 3 Edition, Hartley, H. O. and Pearson, E. S.,Editors, Cambridge University Press, Cambridge.

36


Kendall's Rank Correlation Coefficient:

Assumption

The data consists of a bivariate sample (x1, y 1), (x2, y 2), . . . , (xn, yn) Let:

Nc = the number of concordant pairs Nd = the number of discordant pairs

Test Statistic

τ ˆ =

2( N c

− N d ) n(n − 1)

(estimator of τ )

τ ˆ0 is the observed value of τ ˆ

τ = 0 (there is neither concordance nor

H0

discordance between X and Y)

τ <0

Ha Reject H0 if (Table 19)

τ ˆ0

τ >0

≤ -τ α

τ ˆ0

P( τ ˆ ≤ τ ˆ0 ) P-value

Notes:

τ ≠ 0

≥τα

P( τ ˆ ≥ τ ˆ0 )

| τ ˆ0 | ≥ τ

α/2

2 P( τ ˆ ≥ | τ ˆ0 |)

= P( τ ˆ ≥ - τ ˆ0 )

(a) Kendall's coefficient τ is defined as Thus:

τ = 2 P[(Xi - X j)(Yi - Y j) > 0] - 1

τ > 0 implies P(Concordance) > .5 τ < 0 implies P(Concordance) < .5 τ = 0 implies P(Concordance) = .5

(b) Pairs of observations with ties between respective members are neither concordant nor discordant and therefore contribute nothing to the value of τ ˆ . (c) This test may also be used as a test for independence. (d) For values of n not in Table 19 (i.e. n > 60) one can conduct a large sample approximate test of the above hypotheses using the Z-Test Statistic Z = τ ˆ

9n( n − 1) 2( 2n + 5)

and the usual critical values and P-value formulas associated with any Z-test.

37


Spearman's Rank Correlation Coefficient:

Assumption

S

The data consists of a bivariate sample (x1, y1), (x2, y2), . . . ,(x n, yn) Rank the x-values: R(x i) and y-values: R(yi) separately

Test Statistic

RS = corr [R(Xi), R(Yi)] (estimator of r S

ρ S)

= the observed value of RS

ρS = 0

H0 Ha

ρS < 0

ρS > 0

ρS ≠ 0


r S ≤ -r S,α

r S ≥ r S,α

|r S| ≥ r S,α/2

P(RS ≤ r S)

P(RS ≥ r S)

2 P(RS ≥ |r S|)

P-value

Notes:

= P(RS ≥ -r S)

(a) The parameter ρS is the expected value of R S. Its precise interpretation is difficult, and is best left as another measure of similarity between the rankings of X and Y. (b) This test may also be used as a test for independence. (c) For values of n not in Table 20 (i.e. n > 60) one can conduct a large sample Z =

n − 1 R S

and the usual critical values and P-value formulas associated with any Z-test. Minitab:

Place the x-values in column c1 and the corresponding y-values in column c2. Rank the data in each column, placing the ranks in columns c3 and c4 respectively MTB > rank c1 c3 MTB > rank c2 c4 To calculate r S type MTB > corr c3 c4

38


TABLES

39


40


Table 1

Cumulative Binomial Probabilities: P(X x)

p n

x

.05

.10

.20

.30

.40

.50

2

0 1 2

0.9025 0.8100 0.6400 0.4900 0.3600 0.2500 0.1600 0.0900 0.0400 0.0100 0.0025 0.9975 0.9900 0.9600 0.9100 0.8400 0.7500 0.6400 0.5100 0.3600 0.1900 0.0975 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0 1 2

3

0 1 2 3

0.8574 0.9928 0.9999 1.0000

0.7290 0.9720 0.9990 1.0000

0.5120 0.8960 0.9920 1.0000

0.3430 0.7840 0.9730 1.0000

0.2160 0.6480 0.9360 1.0000

0.1250 0.5000 0.8750 1.0000

0.0640 0.3520 0.7840 1.0000

0.0270 0.2160 0.6570 1.0000

0.0080 0.1040 0.4880 1.0000

0.0010 0.0280 0.2710 1.0000

0.0001 0.0073 0.1426 1.0000

0 1 2 3

4

0 1 2 3 4

0.8145 0.9860 0.9995 1.0000 1.0000

0.6561 0.9477 0.9963 0.9999 1.0000

0.4096 0.8192 0.9728 0.9984 1.0000

0.2401 0.6517 0.9163 0.9919 1.0000

0.1296 0.4752 0.8208 0.9744 1.0000

0.0625 0.3125 0.6875 0.9375 1.0000

0.0256 0.1792 0.5248 0.8704 1.0000

0.0081 0.0837 0.3483 0.7599 1.0000

0.0016 0.0272 0.1808 0.5904 1.0000

0.0001 0.0037 0.0523 0.3439 1.0000

0.0000 0.0005 0.0140 0.1855 1.0000

0 1 2 3 4

5

0 1 2 3 4 5

0.7738 0.9774 0.9988 1.0000 1.0000 1.0000

0.5905 0.9185 0.9914 0.9995 1.0000 1.0000

0.3277 0.7373 0.9421 0.9933 0.9997 1.0000

0.1681 0.5282 0.8369 0.9692 0.9976 1.0000

0.0778 0.3370 0.6826 0.9130 0.9898 1.0000

0.0312 0.1875 0.5000 0.8125 0.9688 1.0000

0.0102 0.0870 0.3174 0.6630 0.9222 1.0000

0.0024 0.0308 0.1631 0.4718 0.8319 1.0000

0.0003 0.0067 0.0579 0.2627 0.6723 1.0000

0.0000 0.0005 0.0086 0.0815 0.4095 1.0000

0.0000 0.0000 0.0012 0.0226 0.2262 1.0000

0 1 2 3 4 5

6

0 1 2 3 4 5 6

0.7351 0.9672 0.9978 0.9999 1.0000 1.0000 1.0000

0.5314 0.8857 0.9842 0.9987 0.9999 1.0000 1.0000

0.2621 0.6554 0.9011 0.9830 0.9984 0.9999 1.0000

0.1176 0.4202 0.7443 0.9295 0.9891 0.9993 1.0000

0.0467 0.2333 0.5443 0.8208 0.9590 0.9959 1.0000

0.0156 0.1094 0.3438 0.6562 0.8906 0.9844 1.0000

0.0041 0.0410 0.1792 0.4557 0.7667 0.9533 1.0000

0.0007 0.0109 0.0705 0.2557 0.5798 0.8824 1.0000

0.0001 0.0016 0.0170 0.0989 0.3446 0.7379 1.0000

0.0000 0.0001 0.0013 0.0159 0.1143 0.4686 1.0000

0.0000 0.0000 0.0001 0.0022 0.0328 0.2649 1.0000

0 1 2 3 4 5 6

7

0 1 2 3 4 5 6 7

0.6983 0.9556 0.9962 0.9998 1.0000 1.0000 1.0000 1.0000

0.4783 0.8503 0.9743 0.9973 0.9998 1.0000 1.0000 1.0000

0.2097 0.5767 0.8520 0.9667 0.9953 0.9996 1.0000 1.0000

0.0824 0.3294 0.6471 0.8740 0.9712 0.9962 0.9998 1.0000

0.0280 0.1586 0.4199 0.7102 0.9037 0.9812 0.9984 1.0000

0.0078 0.0625 0.2266 0.5000 0.7734 0.9375 0.9922 1.0000

0.0016 0.0188 0.0963 0.2898 0.5801 0.8414 0.9720 1.0000

0.0002 0.0038 0.0288 0.1260 0.3529 0.6706 0.9176 1.0000

0.0000 0.0004 0.0047 0.0333 0.1480 0.4233 0.7903 1.0000

0.0000 0.0000 0.0002 0.0027 0.0257 0.1497 0.5217 1.0000

0.0000 0.0000 0.0000 0.0002 0.0038 0.0444 0.3017 1.0000

0 1 2 3 4 5 6 7

8

0 1 2 3 4 5 6 7 8

0.6634 0.9428 0.9942 0.9996 1.0000 1.0000 1.0000 1.0000 1.0000

0.4305 0.8131 0.9619 0.9950 0.9996 1.0000 1.0000 1.0000 1.0000

0.1678 0.5033 0.7969 0.9437 0.9896 0.9988 0.9999 1.0000 1.0000

0.0576 0.2553 0.5518 0.8059 0.9420 0.9887 0.9987 0.9999 1.0000

0.0168 0.1064 0.3154 0.5941 0.8263 0.9502 0.9915 0.9993 1.0000

0.0039 0.0352 0.1445 0.3633 0.6367 0.8555 0.9648 0.9961 1.0000

0.0007 0.0085 0.0498 0.1737 0.4059 0.6846 0.8936 0.9832 1.0000

0.0001 0.0013 0.0113 0.0580 0.1941 0.4482 0.7447 0.9424 1.0000

0.0000 0.0001 0.0012 0.0104 0.0563 0.2031 0.4967 0.8322 1.0000

0.0000 0.0000 0.0000 0.0004 0.0050 0.0381 0.1869 0.5695 1.0000

0.0000 0.0000 0.0000 0.0000 0.0004 0.0058 0.0572 0.3366 1.0000

0 1 2 3 4 5 6 7 8

9

0 1 2 3 4 5 6 7 8 9

0.6302 0.9288 0.9916 0.9994 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.3874 0.7748 0.9470 0.9917 0.9991 0.9999 1.0000 1.0000 1.0000 1.0000

0.1342 0.4362 0.7382 0.9144 0.9804 0.9969 0.9997 1.0000 1.0000 1.0000

0.0404 0.1960 0.4628 0.7297 0.9012 0.9747 0.9957 0.9996 1.0000 1.0000

0.0101 0.0705 0.2318 0.4826 0.7334 0.9006 0.9750 0.9962 0.9997 1.0000

0.0020 0.0195 0.0898 0.2539 0.5000 0.7461 0.9102 0.9805 0.9980 1.0000

0.0003 0.0038 0.0250 0.0994 0.2666 0.5174 0.7682 0.9295 0.9899 1.0000

0.0000 0.0004 0.0043 0.0253 0.0988 0.2703 0.5372 0.8040 0.9596 1.0000

0.0000 0.0000 0.0003 0.0031 0.0196 0.0856 0.2618 0.5638 0.8658 1.0000

0.0000 0.0000 0.0000 0.0001 0.0009 0.0083 0.0530 0.2252 0.6126 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0006 0.0084 0.0712 0.3698 1.0000

0 1 2 3 4 5 6 7 8 9

n

x

.05

.10

.20

.30

.40

.50

.60

.70

.80

.90

.95

X

41

.60

.70

.80

.90

.95

x


Table 1

Cumulative Binomial Probabilities: P(X x) (continued)

p n

x

.05

.10

.20

.30

.40

.50

.60

.70

.80

.90

.95

x

10

0 1 2 3 4 5 6 7 8 9 10

0.5987 0.9139 0.9885 0.9990 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.3487 0.7361 0.9298 0.9872 0.9984 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000

0.1074 0.3758 0.6778 0.8791 0.9672 0.9936 0.9991 0.9999 1.0000 1.0000 1.0000

0.0282 0.1493 0.3828 0.6496 0.8497 0.9527 0.9894 0.9984 0.9999 1.0000 1.0000

0.0060 0.0464 0.1673 0.3823 0.6331 0.8338 0.9452 0.9877 0.9983 0.9999 1.0000

0.0010 0.0107 0.0547 0.1719 0.3770 0.6230 0.8281 0.9453 0.9893 0.9990 1.0000

0.0001 0.0017 0.0123 0.0548 0.1662 0.3669 0.6177 0.8327 0.9536 0.9940 1.0000

0.0000 0.0001 0.0016 0.0106 0.0473 0.1503 0.3504 0.6172 0.8507 0.9718 1.0000

0.0000 0.0000 0.0001 0.0009 0.0064 0.0328 0.1209 0.3222 0.6242 0.8926 1.0000

0.0000 0.0000 0.0000 0.0000 0.0001 0.0016 0.0128 0.0702 0.2639 0.6513 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0010 0.0115 0.0861 0.4013 1.0000

0 1 2 3 4 5 6 7 8 9 10

11

0 1 2 3 4 5 6 7 8 9 10 11

0.5688 0.8981 0.9848 0.9984 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.3138 0.6974 0.9104 0.9815 0.9972 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0859 0.3221 0.6174 0.8389 0.9496 0.9883 0.9980 0.9998 1.0000 1.0000 1.0000 1.0000

0.0198 0.1130 0.3127 0.5696 0.7897 0.9218 0.9784 0.9957 0.9994 1.0000 1.0000 1.0000

0.0036 0.0302 0.1189 0.2963 0.5328 0.7535 0.9006 0.9707 0.9941 0.9993 1.0000 1.0000

0.0005 0.0059 0.0327 0.1133 0.2744 0.5000 0.7256 0.8867 0.9673 0.9941 0.9995 1.0000

0.0000 0.0007 0.0059 0.0293 0.0994 0.2465 0.4672 0.7037 0.8811 0.9698 0.9964 1.0000

0.0000 0.0000 0.0006 0.0043 0.0216 0.0782 0.2103 0.4304 0.6873 0.8870 0.9802 1.0000

0.0000 0.0000 0.0000 0.0002 0.0020 0.0117 0.0504 0.1611 0.3826 0.6779 0.9141 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0028 0.0185 0.0896 0.3026 0.6862 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0016 0.0152 0.1019 0.4312 1.0000

0 1 2 3 4 5 6 7 8 9 10 11

12

0 1 2 3 4 5 6 7 8 9 10 11 12

0.5404 0.8816 0.9804 0.9978 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.2824 0.6590 0.8891 0.9744 0.9957 0.9995 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0687 0.2749 0.5583 0.7946 0.9274 0.9806 0.9961 0.9994 0.9999 1.0000 1.0000 1.0000 1.0000

0.0138 0.0850 0.2528 0.4925 0.7237 0.8822 0.9614 0.9905 0.9983 0.9998 1.0000 1.0000 1.0000

0.0022 0.0196 0.0834 0.2253 0.4382 0.6652 0.8418 0.9427 0.9847 0.9972 0.9997 1.0000 1.0000

0.0002 0.0032 0.0193 0.0730 0.1938 0.3872 0.6128 0.8062 0.9270 0.9807 0.9968 0.9998 1.0000

0.0000 0.0003 0.0028 0.0153 0.0573 0.1582 0.3348 0.5618 0.7747 0.9166 0.9804 0.9978 1.0000

0.0000 0.0000 0.0002 0.0017 0.0095 0.0386 0.1178 0.2763 0.5075 0.7472 0.9150 0.9862 1.0000

0.0000 0.0000 0.0000 0.0001 0.0006 0.0039 0.0194 0.0726 0.2054 0.4417 0.7251 0.9313 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0005 0.0043 0.0256 0.1109 0.3410 0.7176 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0022 0.0196 0.1184 0.4596 1.0000

0 1 2 3 4 5 6 7 8 9 10 11 12

13

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0.5133 0.8646 0.9755 0.9969 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.2542 0.6213 0.8661 0.9658 0.9935 0.9991 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0550 0.2336 0.5017 0.7473 0.9009 0.9700 0.9930 0.9988 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000

0.0097 0.0637 0.2025 0.4206 0.6543 0.8346 0.9376 0.9818 0.9960 0.9993 0.9999 1.0000 1.0000 1.0000

0.0013 0.0126 0.0579 0.1686 0.3530 0.5744 0.7712 0.9023 0.9679 0.9922 0.9987 0.9999 1.0000 1.0000

0.0001 0.0017 0.0112 0.0461 0.1334 0.2905 0.5000 0.7095 0.8666 0.9539 0.9888 0.9983 0.9999 1.0000

0.0000 0.0001 0.0013 0.0078 0.0321 0.0977 0.2288 0.4256 0.6470 0.8314 0.9421 0.9874 0.9987 1.0000

0.0000 0.0000 0.0001 0.0007 0.0040 0.0182 0.0624 0.1654 0.3457 0.5794 0.7975 0.9363 0.9903 1.0000

0.0000 0.0000 0.0000 0.0000 0.0002 0.0012 0.0070 0.0300 0.0991 0.2527 0.4983 0.7664 0.9450 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0009 0.0065 0.0342 0.1339 0.3787 0.7458 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0031 0.0245 0.1354 0.4867 1.0000

0 1 2 3 4 5 6 7 8 9 10 11 12 13

n

x

.05

.10

.20

.30

.40

.50

.60

.70

.80

.90

.95

x

42


Table 1


p n

x

.05

.10

.20

.30

.40

.50

.60

.70

.80

.90

.95

x

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0.4877 0.8470 0.9699 0.9958 0.9996 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.2288 0.5846 0.8416 0.9559 0.9908 0.9985 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0440 0.1979 0.4481 0.6982 0.8702 0.9561 0.9884 0.9976 0.9996 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0068 0.0475 0.1608 0.3552 0.5842 0.7805 0.9067 0.9685 0.9917 0.9983 0.9998 1.0000 1.0000 1.0000 1.0000

0.0008 0.0081 0.0398 0.1243 0.2793 0.4859 0.6925 0.8499 0.9417 0.9825 0.9961 0.9994 0.9999 1.0000 1.0000

0.0001 0.0009 0.0065 0.0287 0.0898 0.2120 0.3953 0.6047 0.7880 0.9102 0.9713 0.9935 0.9991 0.9999 1.0000

0.0000 0.0001 0.0006 0.0039 0.0175 0.0583 0.1501 0.3075 0.5141 0.7207 0.8757 0.9602 0.9919 0.9992 1.0000

0.0000 0.0000 0.0000 0.0002 0.0017 0.0083 0.0315 0.0933 0.2195 0.4158 0.6448 0.8392 0.9525 0.9932 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0004 0.0024 0.0116 0.0439 0.1298 0.3018 0.5519 0.8021 0.9560 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0015 0.0092 0.0441 0.1584 0.4154 0.7712 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0004 0.0042 0.0301 0.1530 0.5123 1.0000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.4633 0.8290 0.9638 0.9945 0.9994 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.2059 0.5490 0.8159 0.9444 0.9873 0.9978 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0352 0.1671 0.3980 0.6482 0.8358 0.9389 0.9819 0.9958 0.9992 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0047 0.0353 0.1268 0.2969 0.5155 0.7216 0.8689 0.9500 0.9848 0.9963 0.9993 0.9999 1.0000 1.0000 1.0000 1.0000

0.0005 0.0052 0.0271 0.0905 0.2173 0.4032 0.6098 0.7869 0.9050 0.9662 0.9907 0.9981 0.9997 1.0000 1.0000 1.0000

0.0000 0.0005 0.0037 0.0176 0.0592 0.1509 0.3036 0.5000 0.6964 0.8491 0.9408 0.9824 0.9963 0.9995 1.0000 1.0000

0.0000 0.0000 0.0003 0.0019 0.0093 0.0338 0.0950 0.2131 0.3902 0.5968 0.7827 0.9095 0.9729 0.9948 0.9995 1.0000

0.0000 0.0000 0.0000 0.0001 0.0007 0.0037 0.0152 0.0500 0.1311 0.2784 0.4845 0.7031 0.8732 0.9647 0.9953 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0008 0.0042 0.0181 0.0611 0.1642 0.3518 0.6020 0.8329 0.9648 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0022 0.0127 0.0556 0.1841 0.4510 0.7941 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0006 0.0055 0.0362 0.1710 0.5367 1.0000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

16

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.4401 0.8108 0.9571 0.9930 0.9991 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.1853 0.5147 0.7892 0.9316 0.9830 0.9967 0.9995 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0281 0.1407 0.3518 0.5981 0.7982 0.9183 0.9733 0.9930 0.9985 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0033 0.0261 0.0994 0.2459 0.4499 0.6598 0.8247 0.9256 0.9743 0.9929 0.9984 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000

0.0003 0.0033 0.0183 0.0651 0.1666 0.3288 0.5272 0.7161 0.8577 0.9417 0.9809 0.9951 0.9991 0.9999 1.0000 1.0000 1.0000

0.0000 0.0003 0.0021 0.0106 0.0384 0.1051 0.2272 0.4018 0.5982 0.7728 0.8949 0.9616 0.9894 0.9979 0.9997 1.0000 1.0000

0.0000 0.0000 0.0001 0.0009 0.0049 0.0191 0.0583 0.1423 0.2839 0.4728 0.6712 0.8334 0.9349 0.9817 0.9967 0.9997 1.0000

0.0000 0.0000 0.0000 0.0000 0.0003 0.0016 0.0071 0.0257 0.0744 0.1753 0.3402 0.5501 0.7541 0.9006 0.9739 0.9967 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0015 0.0070 0.0267 0.0817 0.2018 0.4019 0.6482 0.8593 0.9719 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0005 0.0033 0.0170 0.0684 0.2108 0.4853 0.8147 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0009 0.0070 0.0429 0.1892 0.5599 1.0000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

n

x

.05

.10

.20

.30

.40

.50

.60

.70

.80

.90

.95

x

43


Table 1


p n

x

.05

.10

.20

.30

.40

.50

.60

.70

.80

.90

.95

x

17

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0.4181 0.7922 0.9497 0.9912 0.9988 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.1668 0.4818 0.7618 0.9174 0.9779 0.9953 0.9992 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0225 0.1182 0.3096 0.5489 0.7582 0.8943 0.9623 0.9891 0.9974 0.9995 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0023 0.0193 0.0774 0.2019 0.3887 0.5968 0.7752 0.8954 0.9597 0.9873 0.9968 0.9993 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000

0.0002 0.0021 0.0123 0.0464 0.1260 0.2639 0.4478 0.6405 0.8011 0.9081 0.9652 0.9894 0.9975 0.9995 0.9999 1.0000 1.0000 1.0000

0.0000 0.0001 0.0012 0.0064 0.0245 0.0717 0.1662 0.3145 0.5000 0.6855 0.8338 0.9283 0.9755 0.9936 0.9988 0.9999 1.0000 1.0000

0.0000 0.0000 0.0001 0.0005 0.0025 0.0106 0.0348 0.0919 0.1989 0.3595 0.5522 0.7361 0.8740 0.9536 0.9877 0.9979 0.9998 1.0000

0.0000 0.0000 0.0000 0.0000 0.0001 0.0007 0.0032 0.0127 0.0403 0.1046 0.2248 0.4032 0.6113 0.7981 0.9226 0.9807 0.9977 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0005 0.0026 0.0109 0.0377 0.1057 0.2418 0.4511 0.6904 0.8818 0.9775 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0008 0.0047 0.0221 0.0826 0.2382 0.5182 0.8332 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0012 0.0088 0.0503 0.2078 0.5819 1.0000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

18

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.3972 0.7735 0.9419 0.9891 0.9985 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.1501 0.4503 0.7338 0.9018 0.9718 0.9936 0.9988 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0180 0.0991 0.2713 0.5010 0.7164 0.8671 0.9487 0.9837 0.9957 0.9991 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0016 0.0142 0.0600 0.1646 0.3327 0.5344 0.7217 0.8593 0.9404 0.9790 0.9939 0.9986 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0001 0.0013 0.0082 0.0328 0.0942 0.2088 0.3743 0.5634 0.7368 0.8653 0.9424 0.9797 0.9942 0.9987 0.9998 1.0000 1.0000 1.0000 1.0000

0.0000 0.0001 0.0007 0.0038 0.0154 0.0481 0.1189 0.2403 0.4073 0.5927 0.7597 0.8811 0.9519 0.9846 0.9962 0.9993 0.9999 1.0000 1.0000

0.0000 0.0000 0.0000 0.0002 0.0013 0.0058 0.0203 0.0576 0.1347 0.2632 0.4366 0.6257 0.7912 0.9058 0.9672 0.9918 0.9987 0.9999 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0014 0.0061 0.0210 0.0596 0.1407 0.2783 0.4656 0.6673 0.8354 0.9400 0.9858 0.9984 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0009 0.0043 0.0163 0.0513 0.1329 0.2836 0.4990 0.7287 0.9009 0.9820 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0012 0.0064 0.0282 0.0982 0.2662 0.5497 0.8499 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0015 0.0109 0.0581 0.2265 0.6028 1.0000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

19

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

0.3774 0.7547 0.9335 0.9868 0.9980 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.1351 0.4203 0.7054 0.8850 0.9648 0.9914 0.9983 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0144 0.0829 0.2369 0.4551 0.6733 0.8369 0.9324 0.9767 0.9933 0.9984 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0011 0.0104 0.0462 0.1332 0.2822 0.4739 0.6655 0.8180 0.9161 0.9674 0.9895 0.9972 0.9994 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0001 0.0008 0.0055 0.0230 0.0696 0.1629 0.3081 0.4878 0.6675 0.8139 0.9115 0.9648 0.9884 0.9969 0.9994 0.9999 1.0000 1.0000 1.0000 1.0000

0.0000 0.0000 0.0004 0.0022 0.0096 0.0318 0.0835 0.1796 0.3238 0.5000 0.6762 0.8204 0.9165 0.9682 0.9904 0.9978 0.9996 1.0000 1.0000 1.0000

0.0000 0.0000 0.0000 0.0001 0.0006 0.0031 0.0116 0.0352 0.0885 0.1861 0.3325 0.5122 0.6919 0.8371 0.9304 0.9770 0.9945 0.9992 0.9999 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0006 0.0028 0.0105 0.0326 0.0839 0.1820 0.3345 0.5261 0.7178 0.8668 0.9538 0.9896 0.9989 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0016 0.0067 0.0233 0.0676 0.1631 0.3267 0.5449 0.7631 0.9171 0.9856 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0017 0.0086 0.0352 0.1150 0.2946 0.5797 0.8649 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0020 0.0132 0.0665 0.2453 0.6226 1.0000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

n

x

.05

.10

.20

.30

.40

.50

.60

.70

.80

.90

.95

x

44


Table 1


p n

x

.05

.10

.20

.30

.40

.50

.60

.70

.80

.90

.95

x

20

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.3585 0.7358 0.9245 0.9841 0.9974 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.1216 0.3917 0.6769 0.8670 0.9568 0.9887 0.9976 0.9996 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0115 0.0692 0.2061 0.4114 0.6296 0.8042 0.9133 0.9679 0.9900 0.9974 0.9994 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0008 0.0076 0.0355 0.1071 0.2375 0.4164 0.6080 0.7723 0.8867 0.9520 0.9829 0.9949 0.9987 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0000 0.0005 0.0036 0.0160 0.0510 0.1256 0.2500 0.4159 0.5956 0.7553 0.8725 0.9435 0.9790 0.9935 0.9984 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000

0.0000 0.0000 0.0002 0.0013 0.0059 0.0207 0.0577 0.1316 0.2517 0.4119 0.5881 0.7483 0.8684 0.9423 0.9793 0.9941 0.9987 0.9998 1.0000 1.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0003 0.0016 0.0065 0.0210 0.0565 0.1275 0.2447 0.4044 0.5841 0.7500 0.8744 0.9490 0.9840 0.9964 0.9995 1.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0013 0.0051 0.0171 0.0480 0.1133 0.2277 0.3920 0.5836 0.7625 0.8929 0.9645 0.9924 0.9992 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0006 0.0026 0.0100 0.0321 0.0867 0.1958 0.3704 0.5886 0.7939 0.9308 0.9885 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0004 0.0024 0.0113 0.0432 0.1330 0.3231 0.6083 0.8784 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0026 0.0159 0.0755 0.2642 0.6415 1.0000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

25

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.2774 0.6424 0.8729 0.9659 0.9928 0.9988 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0718 0.2712 0.5371 0.7636 0.9020 0.9666 0.9905 0.9977 0.9995 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0038 0.0274 0.0982 0.2340 0.4207 0.6167 0.7800 0.8909 0.9532 0.9827 0.9944 0.9985 0.9996 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0001 0.0016 0.0090 0.0332 0.0905 0.1935 0.3407 0.5118 0.6769 0.8106 0.9022 0.9558 0.9825 0.9940 0.9982 0.9995 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0000 0.0001 0.0004 0.0024 0.0095 0.0294 0.0736 0.1536 0.2735 0.4246 0.5858 0.7323 0.8462 0.9222 0.9656 0.9868 0.9957 0.9988 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0000 0.0000 0.0000 0.0001 0.0005 0.0020 0.0073 0.0216 0.0539 0.1148 0.2122 0.3450 0.5000 0.6550 0.7878 0.8852 0.9461 0.9784 0.9927 0.9980 0.9995 0.9999 1.0000 1.0000 1.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0003 0.0012 0.0043 0.0132 0.0344 0.0778 0.1538 0.2677 0.4142 0.5754 0.7265 0.8464 0.9264 0.9706 0.9905 0.9976 0.9996 0.9999 1.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0005 0.0018 0.0060 0.0175 0.0442 0.0978 0.1894 0.3231 0.4882 0.6593 0.8065 0.9095 0.9668 0.9910 0.9984 0.9999 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0004 0.0015 0.0056 0.0173 0.0468 0.1091 0.2200 0.3833 0.5793 0.7660 0.9018 0.9726 0.9962 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0005 0.0023 0.0095 0.0334 0.0980 0.2364 0.4629 0.7288 0.9282 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0012 0.0072 0.0341 0.1271 0.3576 0.7226 1.0000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

n

x

.05

.10

.20

.30

.40

.50

.60

.70

.80

.90

.95

x

45


Table 1


p n

x

.05

.10

.20

.30

.40

.50

.60

.70

.80

.90

.95

x

30

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0.2146 0.5535 0.8122 0.9392 0.9984 0.9967 0.9994 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0424 0.1837 0.4114 0.6474 0.8245 0.9268 0.9742 0.9922 0.9980 0.9995 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0012 0.0105 0.0442 0.1227 0.2552 0.4275 0.6070 0.7608 0.8713 0.9389 0.9744 0.9905 0.9969 0.9991 0.9998 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0000 0.0003 0.0021 0.0093 0.0302 0.0766 0.1595 0.2814 0.4315 0.5888 0.7304 0.8407 0.9155 0.9599 0.9831 0.9936 0.9979 0.9994 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0000 0.0000 0.0000 0.0003 0.0015 0.0057 0.0172 0.0435 0.0940 0.1763 0.2915 0.4311 0.5785 0.7145 0.8246 0.9029 0.9519 0.9788 0.9917 0.9971 0.9991 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0007 0.0026 0.0081 0.0214 0.0494 0.1002 0.1808 0.2923 0.4278 0.5722 0.7077 0.8192 0.8998 0.9506 0.9786 0.9919 0.9974 0.9993 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0009 0.0029 0.0083 0.0212 0.0481 0.0971 0.1754 0.2855 0.4215 0.5689 0.7085 0.8237 0.9060 0.9565 0.9828 0.9943 0.9985 0.9997 1.0000 1.0000 1.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0006 0.0021 0.0064 0.0169 0.0401 0.0845 0.1593 0.2696 0.4112 0.5685 0.7186 0.8405 0.9234 0.9698 0.9907 0.9979 0.9997 1.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0009 0.0031 0.0095 0.0256 0.0611 0.1287 0.2392 0.3930 0.5725 0.7448 0.8773 0.9558 0.9895 0.9988 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0005 0.0020 0.0078 0.0258 0.0732 0.1755 0.3526 0.5886 0.8163 0.9576 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0006 0.0033 0.0156 0.0608 0.1878 0.4465 0.7854 1.0000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

n

x

.05

.10

.20

.30

.40

.50

.60

.70

.80

.90

.95

x

46


Table 2

Cumulative Poisson Probabilities: P(X x)

Value of x

.1

.2

.3

.4

.5

.6

.7

.8

.9

1.0

x

0 1 2 3 4 5 6 7 8

0.9048 0.9953 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.8187 0.9825 0.9989 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000

0.7408 0.9631 0.9964 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000

0.6703 0.9384 0.9921 0.9992 0.9999 1.0000 1.0000 1.0000 1.0000

0.6065 0.9098 0.9856 0.9982 0.9998 1.0000 1.0000 1.0000 1.0000

0.5488 0.8781 0.9769 0.9966 0.9996 1.0000 1.0000 1.0000 1.0000

0.4966 0.8442 0.9659 0.9942 0.9992 0.9999 1.0000 1.0000 1.0000

0.4493 0.8088 0.9526 0.9909 0.9986 0.9998 1.0000 1.0000 1.0000

0.4066 0.7725 0.9371 0.9865 0.9977 0.9997 1.0000 1.0000 1.0000

0.3679 0.7358 0.9197 0.9810 0.9963 0.9994 0.9999 1.0000 1.0000

0 1 2 3 4 5 6 7 8

Value of x

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

15.0

x

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

0.1353 0.4060 0.6767 0.8571 0.9473 0.9834 0.9955 0.9989 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0498 0.1991 0.4232 0.6472 0.8153 0.9161 0.9665 0.9881 0.9962 0.9989 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0183 0.0916 0.2381 0.4335 0.6288 0.7851 0.8893 0.9489 0.9786 0.9919 0.9972 0.9991 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0067 0.0404 0.1247 0.2650 0.4405 0.6160 0.7622 0.8666 0.9319 0.9682 0.9863 0.9945 0.9980 0.9993 0.9998 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0025 0.0174 0.0620 0.1512 0.2851 0.4457 0.6063 0.7440 0.8472 0.9161 0.9574 0.9799 0.9912 0.9964 0.9986 0.9995 0.9998 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0009 0.0073 0.0296 0.0818 0.1730 0.3007 0.4497 0.5987 0.7291 0.8305 0.9015 0.9467 0.9730 0.9872 0.9943 0.9976 0.9990 0.9996 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0003 0.0030 0.0138 0.0424 0.0996 0.1912 0.3134 0.4530 0.5925 0.7166 0.8159 0.8881 0.9362 0.9658 0.9827 0.9918 0.9963 0.9984 0.9993 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0001 0.0012 0.0062 0.0212 0.0550 0.1157 0.2068 0.3239 0.4557 0.5874 0.7060 0.8030 0.8758 0.9261 0.9585 0.9780 0.9889 0.9947 0.9976 0.9989 0.9996 0.9998 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0000 0.0005 0.0028 0.0103 0.0293 0.0671 0.1301 0.2202 0.3328 0.4579 0.5830 0.6968 0.7916 0.8645 0.9165 0.9513 0.9730 0.9857 0.9928 0.9965 0.9984 0.9993 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0000 0.0000 0.0000 0.0002 0.0009 0.0028 0.0076 0.0180 0.0374 0.0699 0.1185 0.1848 0.2676 0.3632 0.4657 0.5681 0.6641 0.7489 0.8195 0.8752 0.9170 0.9469 0.9673 0.9805 0.9888 0.9938 0.9967 0.9983 0.9991 0.9996 0.9998 0.9999 1.0000 1.0000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

47


Table 3

Standard Normal Probabilities

z

0

Area = P(Z ≤ z) z

P(Z ≤ z)

z

P(Z ≤ z)

z

P(Z ≤ z)

z

P(Z ≤ z)

-4.00 -3.99 -3.98 -3.97 -3.96 -3.95 -3.94 -3.93 -3.92 -3.91 -3.90 -3.89 -3.88 -3.87 -3.86 -3.85 -3.84 -3.83 -3.82 -3.81 -3.80 -3.79 -3.78 -3.77 -3.76 -3.75 -3.74 -3.73 -3.72 -3.71 -3.70 -3.69 -3.68 -3.67 -3.66 -3.65 -3.64 -3.63 -3.62 -3.61 -3.60 -3.59 -3.58 -3.57 -3.56 -3.55 -3.54 -3.53 -3.52 -3.51

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002

-3.50 -3.49 -3.48 -3.47 -3.46 -3.45 -3.44 -3.43 -3.42 -3.41 -3.40 -3.39 -3.38 -3.37 -3.36 -3.35 -3.34 -3.33 -3.32 -3.31 -3.30 -3.29 -3.28 -3.27 -3.26 -3.25 -3.24 -3.23 -3.22 -3.21 -3.20 -3.19 -3.18 -3.17 -3.16 -3.15 -3.14 -3.13 -3.12 -3.11 -3.10 -3.09 -3.08 -3.07 -3.06 -3.05 -3.04 -3.03 -3.02 -3.01

0.0002 0.0002 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0006 0.0006 0.0006 0.0006 0.0006 0.0007 0.0007 0.0007 0.0007 0.0008 0.0008 0.0008 0.0008 0.0009 0.0009 0.0009 0.0010 0.0010 0.0010 0.0011 0.0011 0.0011 0.0012 0.0012 0.0013 0.0013

-3.00 -2.99 -2.98 -2.97 -2.96 -2.95 -2.94 -2.93 -2.92 -2.91 -2.90 -2.89 -2.88 -2.87 -2.86 -2.85 -2.84 -2.83 -2.82 -2.81 -2.80 -2.79 -2.78 -2.77 -2.76 -2.75 -2.74 -2.73 -2.72 -2.71 -2.70 -2.69 -2.68 -2.67 -2.66 -2.65 -2.64 -2.63 -2.62 -2.61 -2.60 -2.59 -2.58 -2.57 -2.56 -2.55 -2.54 -2.53 -2.52 -2.51

0.0013 0.0014 0.0014 0.0015 0.0015 0.0016 0.0016 0.0017 0.0018 0.0018 0.0019 0.0019 0.0020 0.0021 0.0021 0.0022 0.0023 0.0023 0.0024 0.0025 0.0026 0.0026 0.0027 0.0028 0.0029 0.0030 0.0031 0.0032 0.0033 0.0034 0.0035 0.0036 0.0037 0.0038 0.0039 0.0040 0.0041 0.0043 0.0044 0.0045 0.0047 0.0048 0.0049 0.0051 0.0052 0.0054 0.0055 0.0057 0.0059 0.0060

-2.50 -2.49 -2.48 -2.47 -2.46 -2.45 -2.44 -2.43 -2.42 -2.41 -2.40 -2.39 -2.38 -2.37 -2.36 -2.35 -2.34 -2.33 -2.32 -2.31 -2.30 -2.29 -2.28 -2.27 -2.26 -2.25 -2.24 -2.23 -2.22 -2.21 -2.20 -2.19 -2.18 -2.17 -2.16 -2.15 -2.14 -2.13 -2.12 -2.11 -2.10 -2.09 -2.08 -2.07 -2.06 -2.05 -2.04 -2.03 -2.02 -2.01

0.0062 0.0064 0.0066 0.0068 0.0069 0.0071 0.0073 0.0075 0.0078 0.0080 0.0082 0.0084 0.0087 0.0089 0.0091 0.0094 0.0096 0.0099 0.0102 0.0104 0.0107 0.0110 0.0113 0.0116 0.0119 0.0122 0.0125 0.0129 0.0132 0.0136 0.0139 0.0143 0.0146 0.0150 0.0154 0.0158 0.0162 0.0166 0.0170 0.0174 0.0179 0.0183 0.0188 0.0192 0.0197 0.0202 0.0207 0.0212 0.0217 0.0222

48


Table 3

Standard Normal Probabilities (continued)

z

0

Area = P(Z ≤ z) z

P(Z ≤ z)

z

P(Z ≤ z)

z

P(Z ≤ z)

z

P(Z ≤ z)

-2.00 -1.99 -1.98 -1.97 -1.96 -1.95 -1.94 -1.93 -1.92 -1.91 -1.90 -1.89 -1.88 -1.87 -1.86 -1.85 -1.84 -1.83 -1.82 -1.81 -1.80 -1.79 -1.78 -1.77 -1.76 -1.75 -1.74 -1.73 -1.72 -1.71 -1.70 -1.69 -1.68 -1.67 -1.66 -1.65 -1 64 -1.63 -1.62 -1.61 -1.60 -1.59 -1.58 -1.57 -1.56 -1.55 -1.54 -1.53 -1.52 -1.51

0.0228 0.0233 0.0239 0.0244 0.0250 0.0256 0.0262 0.0268 0.0274 0.0281 0.0287 0.0294 0.0301 0.0307 0.0314 0.0322 0.0329 0.0336 0.0344 0.0351 0.0359 0.0367 0.0375 0.0384 0.0392 0.0401 0.0409 0.0418 0.0427 0.0436 0.0446 0.0455 0.0465 0.0475 0.0485 0.0495 0.0505 0.0516 0.0526 0.0537 0.0548 0.0559 0.0571 0.0582 0.0594 0.0606 0.0618 0.0630 0.0643 0.0655

-1.50 -1.49 -1.48 -1.47 -1.46 -1.45 -1.44 -1.43 -1.42 -1.41 -1.40 -1.39 -1.38 -1.37 -1.36 -1.35 -1.34 -1.33 -1.32 -1.31 -1.30 -1.29 -1.28 -1.27 -1.26 -1.25 -1.24 -1.23 -1.22 -1.21 -1.20 -1.19 -1.18 -1.17 -1.16 -1.15 -1.14 -1.13 -1.12 -1.11 -1.10 -1.09 -1.08 -1.07 -1.06 -1.05 -1.04 -1.03 -1.02 -1.01

0.0668 0.0681 0.0694 0.0708 0.0721 0.0735 0.0749 0.0764 0.0778 0.0793 0.0808 0.0823 0.0838 0.0853 0.0869 0.0885 0.0901 0.0918 0.0934 0.0951 0.0968 0.0985 0.1003 0.1020 0.1038 0.1056 0.1075 0.1093 0.1112 0.1131 0.1151 0.1170 0.1190 0.1210 0.1230 0.1251 0.1271 0.1292 0.1314 0.1335 0.1357 0.1379 0.1401 0.1423 0.1446 0.1469 0.1492 0.1515 0.1539 0.1562

-1.00 -0.99 -0 98 -0.97 -0.96 -0.95 -0.94 -0.93 -0.92 -0.91 -0.90 -0.89 -0.88 -0.87 -0.86 -0.85 -0.84 -0.83 -0.82 -0.81 -0.80 -0.79 -0.78 -0.77 -0.76 -0.75 -0.74 -0.73 -0.72 -0.71 -0.70 -0.69 -0.68 -0.67 -0.66 -0.65 -0.64 -0.63 -0.62 -0.61 -0.60 -0.59 -0.58 -0.57 -0.56 -0.55 -0.54 -0.53 -0.52 -0.51

0.1587 0.1611 0.1635 0.1660 0.1685 0.1711 0.1736 0.1762 0.1788 0.1814 0.1841 0.1867 0.1894 0.1922 0.1949 0.1977 0.2005 0.2033 0.2061 0.2090 0.2119 0.2148 0.2177 0.2206 0.2236 0.2266 0.2296 0.2327 0.2358 0.2389 0.2420 0.2451 0.2483 0.2514 0.2546 0.2578 0.2611 0.2643 0.2676 0.2709 0.2743 0.2776 0.2810 0.2843 0.2877 0.2912 0.2946 0.2981 0.3015 0.3050

-0.50 -0.49 -0.48 -0.47 -0.46 -0.45 -0.44 -0.43 -0.42 -0.41 -0.40 -0.39 -0.38 -0.37 -0.36 -0.35 -0.34 -0.33 -0.32 -0.31 -0.30 -0.29 -0.28 -0.27 -0.26 -0.25 -0.24 -0.23 -0.22 -0.21 -0.20 -0.19 -0.18 -0.17 -0.16 -0.15 -0.14 -0.13 -0.12 -0.11 -0.10 -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01

0.3085 0.3121 0.3156 0.3192 0.3228 0.3264 0.3300 0.3336 0.3372 0.3409 0.3446 0.3483 0.3520 0.3557 0.3594 0.3632 0.3669 0.3707 0.3745 0.3783 0.3821 0.3859 0.3897 0.3936 0.3974 0.4013 0.4052 0.4090 0.4129 0.4168 0.4207 0.4247 0.4286 0.4325 0.4364 0.4404 0.4443 0.4483 0.4522 0.4562 0.4602 0.4641 0.4681 0.4721 0.4761 0.4801 0.4840 0.4880 0.4920 0.4960

49


Table 3


0

z

Area = P(Z ≤ z) z

P(Z ≤ z)

z

P(Z ≤ z)

z

P(Z ≤ z)

z

P(Z ≤ z)

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49

0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879

0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99

0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389

1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49

0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319

1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99

0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767

50


Table 3


0

z

Area = P(Z ≤ z) z

P(Z ≤ z)

z

P(Z ≤ z)

z

P(Z ≤ z)

z

P(Z ≤ z)

2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.40 2.41 2.42 2.43 2.44 2.45 2.46 2.47 2.48 2.49

0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936

2.50 2.51 2.52 2.53 2.54 2.55 2.56 2.57 2.58 2.59 2.60 2.61 2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69 2.70 2.71 2.72 2.73 2.74 2.75 2.76 2.77 2.78 2.79 2.80 2.81 2.82 2.83 2.84 2.85 2.86 2.87 2.88 2.89 2.90 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99

0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986

3.00 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 3.40 3.41 3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49

0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998

3.50 3.51 3.52 3.53 3.54 3.55 3.56 3.57 3.58 3.59 3.60 3.61 3.62 3.63 3.64 3.65 3.66 3.67 3.68 3.69 3.70 3.71 3.72 3.73 3.74 3.75 3.76 3.77 3.78 3.79 3.80 3.81 3.82 3.83 3.84 3.85 3.86 3.87 3.88 3.89 3.90 3.91 3.92 3.93 3.94 3.95 3.96 3.97 3.98 3.99 4.00

0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

51


Table 4

Upper Critical Values of Student's T-distribution

t α

0

Tail Area

Tail Area

df

.10

.05

.025

.01

.005

df

.10

.05

.025

.01

.005

1 2 3 4 5

3.0777 1.8856 1.6377 1.5332 1.4759

6.3138 2.9200 2.3534 2.1318 2.0150

12.706 4.3027 3.1824 2.7764 2.5706

31.821 6.9646 4.5407 3.7469 3.3649

63.657 9.9248 5.8409 4.6041 4.0321

51 52 53 54 55

1.2984 1.2980 1.2977 1.2974 1.2971

1.6753 1.6747 1.6741 1.6736 1.6730

2.0076 2.0066 2.0057 2.0049 2.0040

2.4017 2.4002 2.3988 2.3974 2.3961

2.6757 2.6737 2.6718 2.6700 2.6682

6 7 8 9 10

1.4398 1.4149 1.3968 1.3830 1.3722

1.9432 1.8946 1.8595 1.8331 1.8125

2.4469 2.3646 2.3060 2.2622 2.2281

3.1427 2.9980 2.8965 2.8214 2.7638

3.7074 3.4995 3.3554 3.2498 3.1693

56 57 58 59 60

1.2969 1.2966 1.2963 1.2961 1.2958

1.6725 1.6720 1.6716 1.6711 1.6706

2.0032 2.0025 2.0017 2.0010 2.0003

2.3948 2.3936 2.3924 2.3912 2.3901

2.6665 2.6649 2.6633 2.6618 2.6603

11 12 13 14 15

1.3634 1.3562 1.3502 1.3450 1.3406

1.7959 1.7823 1.7709 1.7613 1.7531

2.2010 2.1788 2.1604 2.1448 2.1314

2.7181 2.6810 2.6503 2.6245 2.6025

3.1058 3.0545 3.0123 2.9768 2.9467

61 62 63 64 65

1.2956 1.2954 1.2951 1.2949 1.2947

1.6702 1.6698 1.6694 1.6690 1.6686

1.9996 1.9990 1.9983 1.9977 1.9971

2.3890 2.3880 2.3870 2.3860 2.3851

2.6589 2.6575 2.6561 2.6549 2.6536

16 17 18 19 20

1.3368 1.3334 1.3304 1.3277 1.3253

1.7459 1.7396 1.7341 1.7291 1.7247

2.1199 2.1098 2.1009 2.0930 2.0860

2.5835 2.5669 2.5524 2.5395 2.5280

2.9208 2.8982 2.8784 2.8609 2.8453

66 67 68 69 70

1.2945 1.2943 1.2941 1.2939 1.2938

1.6683 1.6679 1.6676 1.6672 1.6669

1.9966 1.9960 1.9955 1.9949 1.9944

2.3842 2.3833 2.3824 2.3816 2.3808

2.6524 2.6512 2.6501 2.6490 2.6479

21 22 23 24 25

1.3232 1.3212 1.3195 1.3178 1.3163

1.7207 1.7171 1.7139 1.7109 1.7081

2.0796 2.0739 2.0687 2.0639 2.0595

2.5176 2.5083 2.4999 2.4922 2.4851

2.8314 2.8188 2.8073 2.7969 2.7874

71 72 73 74 75

1.2936 1.2934 1.2933 1.2931 1.2929

1.6666 1.6663 1.6660 1.6657 1.6654

1.9939 1.9935 1.9930 1.9925 1.9921

2.3800 2.3793 2.3785 2.3778 2.3771

2.6469 2.6459 2.6449 2.6439 2.6430

26 27 28 29 30

1.3150 1.3137 1.3125 1.3114 1.3104

1.7056 1.7033 1.7011 1.6991 1.6973

2.0555 2.0518 2.0484 2.0452 2.0423

2.4786 2.4727 2.4671 2.4620 2.4573

2.7787 2.7707 2.7633 2.7564 2.7500

76 77 78 79 80

1.2928 1.2926 1.2925 1.2924 1.2922

1.6652 1.6649 1.6646 1.6644 1.6641

1.9917 1.9913 1.9908 1.9905 1.9901

2.3764 2.3758 2.3751 2.3745 2.3739

2.6421 2.6412 2.6403 2.6395 2.6387

31 32 33 34 35

1.3095 1.3086 1.3077 1.3070 1.3062

1.6955 1.6939 1.6924 1.6909 1.6896

2.0395 2.0369 2.0345 2.0322 2.0301

2.4528 2.4487 2.4448 2.4411 2.4377

2.7440 2.7385 2.7333 2.7284 2.7238

81 82 83 84 85

1.2921 1.2920 1.2918 1.2917 1.2916

1.6639 1.6636 1.6634 1.6632 1.6630

1.9897 1.9893 1.9890 1.9886 1.9883

2.3733 2.3727 2.3721 2.3716 2.3710

2.6379 2.6371 2.6364 2.6356 2.6349

36 37 38 39 40

1.3055 1.3049 1.3042 1.3036 1.3031

1.6883 1.6871 1.6860 1.6849 1.6839

2.0281 2.0262 2.0244 2.0227 2.0211

2.4345 2.4314 2.4286 2.4258 2.4233

2.7195 2.7154 2.7116 2.7079 2.7045

86 87 88 89 90

1.2915 1.2914 1.2912 1.2911 1.2910

1.6628 1.6626 1.6624 1.6622 1.6620

1.9879 1.9876 1.9873 1.9870 1.9867

2.3705 2.3700 2.3695 2.3690 2.3685

2.6342 2.6335 2.6329 2.6322 2.6316

41 42 43 44 45

1.3025 1.3020 1.3016 1.3011 1.3006

1.6829 1.6820 1.6811 1.6802 1.6794

2.0195 2.0181 2.0167 2.0154 2.0141

2.4208 2.4185 2.4163 2.4141 2.4121

2.7012 2.6981 2.6951 2.6923 2.6896

91 92 93 94 95

1.2909 1.2908 1.2907 1.2906 1.2905

1.6618 1.6616 1.6614 1.6612 1.6611

1.9864 1.9861 1.9858 1.9855 1.9853

2.3680 2.3676 2.3671 2.3667 2.3662

2.6309 2.6303 2.6297 2.6291 2.6286

46 47 48 49 50

1.3002 1.2998 1.2994 1.2991 1.2987

1.6787 1.6779 1.6772 1.6766 1.6759

2.0129 2.0117 2.0106 2.0096 2.0086

2.4102 2.4083 2.4066 2.4049 2.4033

2.6870 2.6846 2.6822 2.6800 2.6778

96 97 98 99 100

1.2904 1.2903 1.2902 1.2902 1.2901

1.6609 1.6607 1.6606 1.6604 1.6602

1.9850 1.9847 1.9845 1.9842 1.9840

2.3658 2.3654 2.3650 2.3646 2.3642

2.6280 2.6275 2.6269 2.6264 2.6259

1.2816

1.6449

1.9600

2.3263

2.5758

52


Table 5

Exact Confidence Intervals for a Population Proportion p

Exact 90% Confidence Interval for a Population Proportion p x 0 1 2 3 4 5 6 7 8

n=5 0.000 0.010 0.076 0.189 0.343 0.549

x 0 1 2 3 4 5 6 7 8 9 10 11 12 x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

n=6 0.451 0.657 0.811 0.924 0.990 1.000

n=9 0.000 0.006 0.041 0.098 0.169 0.251 0.345 0.450 0.571 0.717

0.283 0.429 0.550 0.655 0.749 0.831 0.902 0.959 0.994 1.000

0.206 0.316 0.410 0.495 0.573 0.645 0.713 0.776 0.834 0.887 0.934 0.972 0.996 1.000

n = 17 0.000 0.003 0.021 0.050 0.085 0.124 0.166 0.212 0.260 0.311 0.364 0.420 0.478 0.539 0.604 0.674 0.750 0.838

n=7 0.393 0.582 0.729 0.847 0.937 0.991 1.000

n = 10

n = 13 0.000 0.004 0.028 0.066 0.113 0.166 0.224 0.287 0.355 0.427 0.505 0.590 0.684 0.794

0.000 0.009 0.063 0.153 0.271 0.418 0.607

0.162 0.250 0.326 0.396 0.461 0.522 0.580 0.636 0.689 0.740 0.788 0.834 0.876 0.915 0.950 0.979 0.997 1.000

0.000 0.005 0.037 0.087 0.150 0.222 0.304 0.393 0.493 0.606 0.741

0.259 0.394 0.507 0.607 0.696 0.778 0.850 0.913 0.963 0.995 1.000

0.000 0.005 0.033 0.079 0.135 0.200 0.271 0.350 0.436 0.530 0.636 0.762

0.238 0.364 0.470 0.564 0.650 0.729 0.800 0.865 0.921 0.967 0.995 1.000

n = 15

0.193 0.297 0.385 0.466 0.540 0.610 0.675 0.736 0.794 0.847 0.896 0.939 0.974 0.996 1.000

n = 18 0.000 0.003 0.020 0.047 0.080 0.116 0.156 0.199 0.244 0.291 0.341 0.392 0.446 0.502 0.561 0.623 0.690 0.762 0.847

n=8 0.348 0.521 0.659 0.775 0.871 0.947 0.993 1.000

n = 11

n = 14 0.000 0.004 0.026 0.061 0.104 0.153 0.206 0.264 0.325 0.390 0.460 0.534 0.615 0.703 0.807

0.000 0.007 0.053 0.129 0.225 0.341 0.479 0.652

0.000 0.003 0.024 0.057 0.097 0.142 0.191 0.244 0.300 0.360 0.423 0.489 0.560 0.637 0.721 0.819

0.181 0.279 0.363 0.440 0.511 0.577 0.640 0.700 0.756 0.809 0.858 0.903 0.943 0.976 0.997 1.000

n = 19

0.153 0.238 0.310 0.377 0.439 0.498 0.554 0.608 0.659 0.709 0.756 0.801 0.844 0.884 0.920 0.953 0.980 0.997 1.000

53

0.000 0.003 0.019 0.044 0.075 0.110 0.147 0.188 0.230 0.274 0.320 0.368 0.418 0.470 0.524 0.581 0.641 0.704 0.774 0.854

0.146 0.226 0.296 0.359 0.419 0.476 0.530 0.582 0.632 0.680 0.726 0.770 0.812 0.853 0.890 0.925 0.956 0.981 0.997 1.000

0.000 0.006 0.046 0.111 0.193 0.289 0.400 0.529 0.688

0.312 0.471 0.600 0.711 0.807 0.889 0.954 0.994 1.000

n = 12 0.000 0.004 0.030 0.072 0.123 0.181 0.245 0.315 0.391 0.473 0.562 0.661 0.779

0.221 0.339 0.438 0.527 0.609 0.685 0.755 0.819 0.877 0.928 0.970 0.996 1.000

n = 16 0.000 0.003 0.023 0.053 0.090 0.132 0.178 0.227 0.279 0.333 0.391 0.452 0.516 0.583 0.656 0.736 0.829

0.171 0.264 0.344 0.417 0.484 0.548 0.609 0.667 0.721 0.773 0.822 0.868 0.910 0.947 0.977 0.997 1.000

n = 20 0.000 0.003 0.018 0.042 0.071 0.104 0.140 0.177 0.217 0.259 0.302 0.347 0.394 0.442 0.492 0.544 0.599 0.656 0.717 0.784 0.861

0.139 0.216 0.283 0.344 0.401 0.456 0.508 0.558 0.606 0.653 0.698 0.741 0.783 0.823 0.860 0.896 0.929 0.958 0.982 0.997 1.000


Table 5 (continued)


n=5 0.000 0.005 0.053 0.147 0.284 0.478

x 0 1 2 3 4 5 6 7 8 9 10 11 12 x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

n=6 0.522 0.716 0.853 0.947 0.995 1.000

n=9 0.000 0.003 0.028 0.075 0.137 0.212 0.299 0.400 0.518 0.664

0.336 0.482 0.600 0.701 0.788 0.863 0.925 0.972 0.997 1.000

0.247 0.360 0.454 0.538 0.614 0.684 0.749 0.808 0.861 0.909 0.950 0.981 0.998 1.000

n = 17 0.000 0.001 0.015 0.038 0.068 0.103 0.142 0.184 0.230 0.278 0.329 0.383 0.440 0.501 0.566 0.636 0.713 0.805

n=7 0.459 0.641 0.777 0.882 0.957 0.996 1.000

n = 10

n = 13 0.000 0.002 0.019 0.050 0.091 0.139 0.192 0.251 0.316 0.386 0.462 0.546 0.640 0.753

0.000 0.004 0.043 0.118 0.223 0.359 0.541

0.195 0.287 0.364 0.434 0.499 0.560 0.617 0.671 0.722 0.770 0.816 0.858 0.897 0.932 0.962 0.985 0.999 1.000

0.000 0.003 0.025 0.067 0.122 0.187 0.262 0.348 0.444 0.555 0.692

0.308 0.445 0.556 0.652 0.738 0.813 0.878 0.933 0.975 0.997 1.000

0.000 0.002 0.023 0.060 0.109 0.167 0.234 0.308 0.390 0.482 0.587 0.715

0.285 0.413 0.518 0.610 0.692 0.766 0.833 0.891 0.940 0.977 0.998 1.000

n = 15

0.232 0.339 0.428 0.508 0.581 0.649 0.711 0.770 0.823 0.872 0.916 0.953 0.982 0.998 1.000

n = 18 0.000 0.001 0.014 0.036 0.064 0.097 0.133 0.173 0.215 0.260 0.308 0.357 0.410 0.465 0.524 0.586 0.653 0.727 0.815

n=8 0.410 0.579 0.710 0.816 0.901 0.963 0.996 1.000

n = 11

n = 14 0.000 0.002 0.018 0.047 0.084 0.128 0.177 0.230 0.289 0.351 0.419 0.492 0.572 0.661 0.768

0.000 0.004 0.037 0.099 0.184 0.290 0.421 0.590

0.000 0.002 0.017 0.043 0.078 0.118 0.163 0.213 0.266 0.323 0.384 0.449 0.519 0.595 0.681 0.782

0.218 0.319 0.405 0.481 0.551 0.616 0.677 0.734 0.787 0.837 0.882 0.922 0.957 0.983 0.998 1.000

n = 19

0.185 0.273 0.347 0.414 0.476 0.535 0.590 0.643 0.692 0.740 0.785 0.827 0.867 0.903 0.936 0.964 0.986 0.999 1.000

54

0.000 0.001 0.013 0.034 0.061 0.091 0.126 0.163 0.203 0.244 0.289 0.335 0.384 0.434 0.488 0.544 0.604 0.669 0.740 0.824

0.176 0.260 0.331 0.396 0.456 0.512 0.566 0.616 0.665 0.711 0.756 0.797 0.837 0.874 0.909 0.939 0.966 0.987 0.999 1.000

0.000 0.003 0.032 0.085 0.157 0.245 0.349 0.473 0.631

0.369 0.527 0.651 0.755 0.843 0.915 0.968 0.997 1.000

n = 12 0.000 0.002 0.021 0.055 0.099 0.152 0.211 0.277 0.349 0.428 0.516 0.615 0.735

0.265 0.385 0.484 0.572 0.651 0.723 0.789 0.848 0.901 0.945 0.979 0.998 1.000

n = 16 0.000 0.002 0.016 0.040 0.073 0.110 0.152 0.198 0.247 0.299 0.354 0.413 0.476 0.544 0.617 0.698 0.794

0.206 0.302 0.383 0.456 0.524 0.587 0.646 0.701 0.753 0.802 0.848 0.890 0.927 0.960 0.984 0.998 1.000

n = 20 0.000 0.001 0.012 0.032 0.057 0.087 0.119 0.154 0.191 0.231 0.272 0.315 0.361 0.408 0.457 0.509 0.563 0.621 0.683 0.751 0.832

0.168 0.249 0.317 0.379 0.437 0.491 0.543 0.592 0.639 0.685 0.728 0.769 0.809 0.846 0.881 0.913 0.943 0.968 0.988 0.999 1.000


Table 5 (continued)


n=5 0.000 0.001 0.023 0.083 0.185 0.347

x 0 1 2 3 4 5 6 7 8 9 10 11 12 x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

n=6 0.653 0.815 0.917 0.977 0.999 1.000

n=9 0.000 0.001 0.012 0.042 0.087 0.146 0.219 0.307 0.415 0.555

0.445 0.585 0.693 0.781 0.854 0.913 0.958 0.988 0.999 1.000

0.335 0.449 0.541 0.621 0.691 0.755 0.811 0.862 0.906 0.943 0.972 0.992 1.000 1.000

n = 17 0.000 0.000 0.006 0.021 0.043 0.070 0.101 0.137 0.176 0.219 0.266 0.315 0.369 0.427 0.490 0.559 0.637 0.732

n=7 0.586 0.746 0.856 0.934 0.981 0.999 1.000

n = 10

n = 13 0.000 0.000 0.008 0.028 0.057 0.094 0.138 0.189 0.245 0.309 0.379 0.459 0.551 0.665

0.000 0.001 0.019 0.066 0.144 0.254 0.414

0.268 0.363 0.441 0.510 0.573 0.631 0.685 0.734 0.781 0.824 0.863 0.899 0.930 0.957 0.979 0.994 1.000 1.000

0.000 0.001 0.011 0.037 0.077 0.128 0.191 0.265 0.352 0.456 0.589

0.411 0.544 0.648 0.735 0.809 0.872 0.923 0.963 0.989 0.999 1.000

0.000 0.000 0.010 0.033 0.069 0.114 0.169 0.233 0.307 0.392 0.491 0.618

0.382 0.509 0.608 0.693 0.767 0.831 0.886 0.931 0.967 0.990 1.000 1.000

n = 15

0.315 0.424 0.512 0.589 0.658 0.720 0.777 0.828 0.873 0.913 0.947 0.974 0.992 1.000 1.000

n = 18 0.000 0.000 0.006 0.020 0.040 0.065 0.095 0.128 0.165 0.205 0.247 0.293 0.342 0.395 0.451 0.512 0.578 0.654 0.745

n=8 0.531 0.685 0.797 0.882 0.945 0.984 0.999 1.000

n = 11

n = 14 0.000 0.000 0.008 0.026 0.053 0.087 0.127 0.172 0.223 0.280 0.342 0.411 0.488 0.576 0.685

0.000 0.001 0.016 0.055 0.118 0.203 0.315 0.469

0.000 0.000 0.007 0.024 0.049 0.080 0.117 0.159 0.205 0.256 0.312 0.373 0.439 0.514 0.598 0.702

0.298 0.402 0.486 0.561 0.627 0.688 0.744 0.795 0.841 0.883 0.920 0.951 0.976 0.993 1.000 1.000

n = 19

0.255 0.346 0.422 0.488 0.549 0.605 0.658 0.707 0.753 0.795 0.835 0.872 0.905 0.935 0.960 0.980 0.994 1.000 1.000

55

0.000 0.000 0.006 0.019 0.038 0.062 0.090 0.121 0.155 0.192 0.232 0.274 0.319 0.367 0.418 0.473 0.532 0.596 0.669 0.757

0.243 0.331 0.404 0.468 0.527 0.582 0.633 0.681 0.726 0.768 0.808 0.845 0.879 0.910 0.938 0.962 0.981 0.994 1.000 1.000

0.000 0.001 0.014 0.047 0.100 0.170 0.258 0.368 0.516

0.484 0.632 0.742 0.830 0.900 0.953 0.986 0.999 1.000

n = 12 0.000 0.000 0.009 0.030 0.062 0.103 0.152 0.209 0.272 0.345 0.427 0.523 0.643

0.357 0.477 0.573 0.655 0.728 0.791 0.848 0.897 0.938 0.970 0.991 1.000 1.000

n = 16 0.000 0.000 0.007 0.022 0.045 0.075 0.109 0.147 0.190 0.236 0.287 0.342 0.401 0.466 0.537 0.619 0.718

0.282 0.381 0.463 0.534 0.599 0.658 0.713 0.764 0.810 0.853 0.891 0.925 0.955 0.978 0.993 1.000 1.000

n = 20 0.000 0.000 0.005 0.018 0.036 0.058 0.085 0.114 0.146 0.181 0.218 0.257 0.299 0.343 0.390 0.440 0.493 0.551 0.613 0.683 0.767

0.233 0.317 0.387 0.449 0.507 0.560 0.610 0.657 0.701 0.743 0.782 0.819 0.854 0.886 0.915 0.942 0.964 0.982 0.995 1.000 1.000


Table 6

Distribution of The Sign-Test Statistic: P(X x), X ~ Bin(n, .5)

n x

3

4

5

6

7

8

9

0 1 2 3 4

0.1250 0.5000 0.8750 1.0000

0.0625 0.3125 0.6875 0.9375 1.0000

0.0313 0.1875 0.5000 0.8125 0.9687

0.0156 0.1094 0.3438 0.6562 0.8906

0.0078 0.0625 0.2266 0.5000 0.7734

0.0039 0.0352 0.1445 0.3633 0.6367

0.0020 0.0195 0.0898 0.2539 0.5000

n x

10

11

12

13

14

15

16

0 1 2 3 4 5 6 7 8

0.0001 0.0107 0.0547 0.1719 0.3770 0.6230 0.8281 0.9453 0.9893

0.0005 0.0059 0.0327 0.1133 0.2744 0.5000 0.7256 0.8867 0.9673

0.0002 0.0032 0.0193 0.0730 0.1938 0.3872 0.6128 0.8062 0.9270

0.0001 0.0017 0.0112 0.0461 0.1334 0.2905 0.5000 0.7095 0.8666

0.0001 0.0009 0.0065 0.0287 0.0898 0.2120 0.3953 0.6047 0.7880

0.0000 0.0005 0.0037 0.0176 0.0592 0.1509 0.3036 0.5000 0.6964

0.0000 0.0003 0.0021 0.0106 0.0384 0.1051 0.2272 0.4018 0.5982

n x

17

18

19

20

21

22

23

0 1 2 3 4 5 6 7 8 9 10 11

0.0000 0.0001 0.0012 0.0064 0.0245 0.0717 0.1662 0.3145 0.5000 0.6855 0.8338 0.9283

0.0000 0.0001 0.0007 0.0038 0.0154 0.0481 0.1189 0.2403 0.4073 0.5927 0.7597 0.8811

0.0000 0.0000 0.0004 0.0022 0.0096 0.0318 0.0835 0.1796 0.3238 0.5000 0.6762 0.8204

0.0000 0.0000 0.0002 0.0013 0.0059 0.0207 0.0577 0.1316 0.2517 0.4119 0.5881 0.7483

0.0000 0.0000 0.0001 0.0007 0.0036 0.0133 0.0392 0.0946 0.1917 0.3318 0.5000 0.6682

0.0000 0.0000 0.0001 0.0004 0.0022 0.0085 0.0262 0.0669 0.1431 0.2617 0.4159 0.5841

0.0000 0.0000 0.0000 0.0002 0.0013 0.0053 0.0173 0.0466 0.1050 0.2024 0.3388 0.5000

n x

24

25

26

27

28

29

30

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.0000 0.0000 0.0000 0.0001 0.0008 0.0033 0.0113 0.0320 0.0758 0.1537 0.2706 0.4194 0.5806 0.7294 0.8463 0.9242

0.0000 0.0000 0.0000 0.0001 0.0005 0.0020 0.0073 0.0216 0.0539 0.1148 0.2122 0.3450 0.5000 0.6550 0.7878 0.8852

0.0000 0.0000 0.0000 0.0000 0.0003 0.0012 0.0047 0.0145 0.0378 0.0843 0.1635 0.2786 0.4225 0.5775 0.7214 0.8365

0.0000 0.0000 0.0000 0.0000 0.0002 0.0008 0.0030 0.0096 0.0261 0.0610 0.1239 0.2210 0.3506 0.5000 0.6494 0.7790

0.0000 0.0000 0.0000 0.0000 0.0001 0.0005 0.0019 0.0063 0.0178 0.0436 0.0925 0.1725 0.2858 0.4253 0.5747 0.7142

0.0000 0.0000 0.0000 0.0000 0.0001 0.0003 0.0012 0.0041 0.0121 0.0307 0.0680 0.1325 0.2291 0.3555 0.5000 0.6445

0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0007 0.0026 0.0081 0.0214 0.0494 0.1002 0.1808 0.2923 0.4278 0.5722

56


Table 7

Wilcoxon Signed-Rank Distribution: P(V

v)

n v

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

0.5000 1.0000

0.2500 0.5000 0.7500 1.0000

0.1250 0.2500 0.3750 0.6250 0.7500 0.8750 1.0000

0.0625 0.1250 0.1875 0.3125 0.4375 0.5625 0.6875 0.8125 0.8750 0.9375 1.0000

0.0313 0.0625 0.0938 0.1563 0.2188 0.3125 0.4063 0.5000 0.5937 0.6875 0.7812 0.8437 0.9062 0.9375 0.9687 1.0000

0.0156 0.0313 0.0469 0.0781 0.1094 0.1563 0.2188 0.2813 0.3438 0.4219 0.5000 0.5781 0.6562 0.7187 0.7812 0.8437 0.8906 0.9219 0.9531 0.9687 0.9844 1.0000

0.0078 0.0156 0.0234 0.0391 0.0547 0.0781 0.1094 0.1484 0.1875 0.2344 0.2891 0.3438 0.4063 0.4688 0.5312 0.5937 0.6562 0.7109 0.7656 0.8125 0.8516 0.8906 0.9219 0.9453 0.9609 0.9766 0.9844 0.9922 1.0000

57


Table 7

Wilcoxon Signed-Rank Distribution: P(V v) (continued)

n v

8

9

10

11

12

13

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

0.0039 0.0078 0.0117 0.0195 0.0273 0.0391 0.0547 0.0742 0.0977 0.1250 0.1563 0.1914 0.2305 0.2734 0.3203 0.3711 0.4219 0.4727 0.5273 0.5781 0.6289 0.6797 0.7266 0.7695 0.8086 0.8437 0.8750 0.9023 0.9258 0.9453 0.9609 0.9727 0.9805 0.9883 0.9922 0.9961 1.0000

0.0020 0.0039 0.0059 0.0098 0.0137 0.0195 0.0273 0.0371 0.0488 0.0645 0.0820 0.1016 0.1250 0.1504 0.1797 0.2129 0.2480 0.2852 0.3262 0.3672 0.4102 0.4551 0.5000 0.5449 0.5898 0.6328 0.6738 0.7148 0.7520 0.7871 0.8203 0.8496 0.8750 0.8984 0.9180 0.9355 0.9512 0.9629 0.9727 0.9805 0.9863 0.9902 0.9941 0.9961 0.9980 1.0000

0.0010 0.0020 0.0029 0.0049 0.0068 0.0098 0.0137 0.0186 0.0244 0.0322 0.0420 0.0527 0.0654 0.0801 0.0967 0.1162 0.1377 0.1611 0.1875 0.2158 0.2461 0.2783 0.3125 0.3477 0.3848 0.4229 0.4609 0.5000 0.5391 0.5771 0.6152 0.6523 0.6875 0.7217 0.7539 0.7842 0.8125 0.8389 0.8623 0.8838 0.9033 0.9199 0.9346 0.9473 0.9580 0.9678 0.9756 0.9814 0.9863 0.9902 0.9932 0.9951 0.9971

0.0005 0.0010 0.0015 0.0024 0.0034 0.0049 0.0068 0.0093 0.0122 0.0161 0.0210 0.0269 0.0337 0.0415 0.0508 0.0615 0.0737 0.0874 0.1030 0.1201 0.1392 0.1602 0.1826 0.2065 0.2324 0.2598 0.2886 0.3188 0.3501 0.3823 0.4155 0.4492 0.4829 0.5171 0.5508 0.5845 0.6177 0.6499 0.6812 0.7114 0.7402 0.7676 0.7935 0.8174 0.8398 0.8608 0.8799 0.8970 0.9126 0.9263 0.9385 0.9492 0.9585

0.0002 0.0005 0.0007 0.0012 0.0017 0.0024 0.0034 0.0046 0.0061 0.0081 0.0105 0.0134 0.0171 0.0212 0.0261 0.0320 0.0386 0.0461 0.0549 0.0647 0.0757 0.0881 0.1018 0.1167 0.1331 0.1506 0.1697 0.1902 0.2119 0.2349 0.2593 0.2847 0.3110 0.3386 0.3667 0.3955 0.4250 0.4548 0.4849 0.5151 0.5452 0.5750 0.6045 0.6333 0.6614 0.6890 0.7153 0.7407 0.7651 0.7881 0.8098 0.8303 0.8494

0.0001 0.0002 0.0004 0.0006 0.0009 0.0012 0.0017 0.0023 0.0031 0.0040 0.0052 0.0067 0.0085 0.0107 0.0133 0.0164 0.0199 0.0239 0.0287 0.0341 0.0402 0.0471 0.0549 0.0636 0.0732 0.0839 0.0955 0.1082 0.1219 0.1367 0.1527 0.1698 0.1879 0.2072 0.2274 0.2487 0.2709 0.2939 0.3177 0.3424 0.3677 0.3934 0.4197 0.4463 0.4730 0.5000 0.5270 0.5537 0.5803 0.6066 0.6323 0.6576 0.6823

0.0001 0.0001 0.0002 0.0003 0.0004 0.0006 0.0009 0.0012 0.0015 0.0020 0.0026 0.0034 0.0043 0.0054 0.0067 0.0083 0.0101 0.0123 0.0148 0.0176 0.0209 0.0247 0.0290 0.0338 0.0392 0.0453 0.0520 0.0594 0.0676 0.0765 0.0863 0.0969 0.1083 0.1206 0.1338 0.1479 0.1629 0.1788 0.1955 0.2131 0.2316 0.2508 0.2708 0.2915 0.3129 0.3349 0.3574 0.3804 0.4039 0.4276 0.4516 0.4758 0.5000

58


Table 7


n v 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

15

16

17

18

19

20

0.0000 0.0001 0.0001 0.0002 0.0002 0.0003 0.0004 0.0006 0.0008 0.0010 0.0013 0.0017 0.0021 0.0027 0.0034 0.0042 0.0051 0.0062 0.0075 0.0090 0.0108 0.0128 0.0151 0.0177 0.0206 0.0240 0.0277 0.0319 0.0365 0.0416 0.0473 0.0535 0.0603 0.0677 0.0757 0.0844 0.0938 0.1039 0.1147 0.1262 0.1384 0.1514 0.1651 0.1796 0.1947 0.2106 0.2271 0.2444 0.2622 0.2807 0.2997

0.0000 0.0000 0.0000 0.0001 0.0001 0.0002 0.0002 0.0003 0.0004 0.0005 0.0007 0.0008 0.0011 0.0013 0.0017 0.0021 0.0026 0.0031 0.0038 0.0046 0.0055 0.0065 0.0078 0.0091 0.0107 0.0125 0.0145 0.0168 0.0193 0.0222 0.0253 0.0288 0.0327 0.0370 0.0416 0.0467 0.0523 0.0583 0.0649 0.0719 0.0795 0.0877 0.0964 0.1057 0.1156 0.1261 0.1372 0.1489 0.1613 0.1742 0.1877

0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001 0.0002 0.0003 0.0003 0.0004 0.0005 0.0007 0.0008 0.0010 0.0013 0.0016 0.0019 0.0023 0.0028 0.0033 0.0040 0.0047 0.0055 0.0064 0.0075 0.0087 0.0101 0.0116 0.0133 0.0153 0.0174 0.0198 0.0224 0.0253 0.0284 0.0319 0.0357 0.0398 0.0443 0.0492 0.0544 0.0601 0.0662 0.0727 0.0797 0.0871 0.0950 0.1034 0.1123

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001 0.0002 0.0002 0.0003 0.0003 0.0004 0.0005 0.0006 0.0008 0.0010 0.0012 0.0014 0.0017 0.0020 0.0024 0.0028 0.0033 0.0038 0.0045 0.0052 0.0060 0.0069 0.0080 0.0091 0.0104 0.0118 0.0134 0.0152 0.0171 0.0192 0.0216 0.0241 0.0269 0.0300 0.0333 0.0368 0.0407 0.0449 0.0494 0.0542 0.0594 0.0649

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001 0.0002 0.0002 0.0003 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0010 0.0012 0.0014 0.0017 0.0020 0.0023 0.0027 0.0031 0.0036 0.0041 0.0047 0.0054 0.0062 0.0070 0.0080 0.0090 0.0102 0.0115 0.0129 0.0145 0.0162 0.0180 0.0201 0.0223 0.0247 0.0273 0.0301 0.0331 0.0364

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0002 0.0002 0.0003 0.0004 0.0004 0.0005 0.0006 0.0007 0.0008 0.0010 0.0012 0.0014 0.0016 0.0018 0.0021 0.0024 0.0028 0.0032 0.0036 0.0042 0.0047 0.0053 0.0060 0.0068 0.0077 0.0086 0.0096 0.0107 0.0120 0.0133 0.0148 0.0164 0.0181 0.0200

59


Table 7


n v 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105

15

16

17

18

19

20

0.3193 0.3394 0.3599 0.3808 0.4020 0.4235 0.4452 0.4670 0.4890 0.5110 0.5330 0.5548 0.5765 0.5980 0.6192 0.6401 0.6606 0.6807 0.7003 0.7193 0.7378 0.7556 0.7729 0.7894 0.8053 0.8204 0.8349 0.8486 0.8616 0.8738 0.8853 0.8961 0.9062 0.9156 0.9243 0.9323 0.9397 0.9465 0.9527 0.9584 0.9635 0.9681 0.9723 0.9760 0.9794 0.9823 0.9849 0.9872 0.9892 0.9910 0.9925 0.9938 0.9949 0.9958 0.9966

0.2019 0.2166 0.2319 0.2477 0.2641 0.2809 0.2983 0.3161 0.3343 0.3529 0.3718 0.3910 0.4104 0.4301 0.4500 0.4699 0.4900 0.5100 0.5301 0.5500 0.5699 0.5896 0.6090 0.6282 0.6471 0.6657 0.6839 0.7017 0.7191 0.7359 0.7523 0.7681 0.7834 0.7981 0.8123 0.8258 0.8387 0.8511 0.8628 0.8739 0.8844 0.8943 0.9036 0.9123 0.9205 0.9281 0.9351 0.9417 0.9477 0.9533 0.9584 0.9630 0.9673 0.9712 0.9747

0.1217 0.1317 0.1421 0.1530 0.1645 0.1764 0.1889 0.2019 0.2153 0.2293 0.2437 0.2585 0.2738 0.2895 0.3056 0.3221 0.3389 0.3559 0.3733 0.3910 0.4088 0.4268 0.4450 0.4633 0.4816 0.5000 0.5184 0.5367 0.5550 0.5732 0.5912 0.6090 0.6267 0.6441 0.6611 0.6779 0.6944 0.7105 0.7262 0.7415 0.7563 0.7707 0.7847 0.7981 0.8111 0.8236 0.8355 0.8470 0.8579 0.8683 0.8783 0.8877 0.8966 0.9050 0.9129

0.0708 0.0770 0.0837 0.0907 0.0982 0.1061 0.1144 0.1231 0.1323 0.1419 0.1519 0.1624 0.1733 0.1846 0.1964 0.2086 0.2211 0.2341 0.2475 0.2613 0.2754 0.2899 0.3047 0.3198 0.3353 0.3509 0.3669 0.3830 0.3994 0.4159 0.4325 0.4493 0.4661 0.4831 0.5000 0.5169 0.5339 0.5507 0.5675 0.5841 0.6006 0.6170 0.6331 0.6491 0.6647 0.6802 0.6953 0.7101 0.7246 0.7387 0.7525 0.7659 0.7789 0.7914 0.8036

0.0399 0.0437 0.0478 0.0521 0.0567 0.0616 0.0668 0.0723 0.0782 0.0844 0.0909 0.0978 0.1051 0.1127 0.1206 0.1290 0.1377 0.1467 0.1562 0.1660 0.1762 0.1868 0.1977 0.2090 0.2207 0.2327 0.2450 0.2576 0.2706 0.2839 0.2974 0.3113 0.3254 0.3397 0.3543 0.3690 0.3840 0.3991 0.4144 0.4298 0.4453 0.4609 0.4765 0.4922 0.5078 0.5235 0.5391 0.5547 0.5702 0.5856 0.6009 0.6160 0.6310 0.6457 0.6603

0.0220 0.0242 0.0266 0.0291 0.0319 0.0348 0.0379 0.0413 0.0448 0.0487 0.0527 0.0570 0.0615 0.0664 0.0715 0.0768 0.0825 0.0884 0.0947 0.1012 0.1081 0.1153 0.1227 0.1305 0.1387 0.1471 0.1559 0.1650 0.1744 0.1841 0.1942 0.2045 0.2152 0.2262 0.2375 0.2490 0.2608 0.2729 0.2853 0.2979 0.3108 0.3238 0.3371 0.3506 0.3643 0.3781 0.3921 0.4062 0.4204 0.4347 0.4492 0.4636 0.4782 0.4927 0.5073

60


Table 8

Mann-Whitney-Wilcoxon Distribution: P(W

w)

Note: nX is the smaller sample size, n Y the larger sample size

nX = 2 n Y w

2

3

4

5

6

7

8

9

10

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

0.1667 0.3333 0.6667 0.8333 1.0000

0.1000 0.2000 0.4000 0.6000 0.8000 0.9000 1.0000

0.0667 0.1333 0.2667 0.4000 0.6000 0.7333 0.8667 0.9333 1.0000

0.0476 0.0952 0.1905 0.2857 0.4286 0.5714 0.7143 0.8095 0.9048 0.9524 1.0000

0.0357 0.0714 0.1429 0.2143 0.3214 0.4286 0.5714 0.6786 0.7857 0.8571 0.9286 0.9643 1.0000

0.0278 0.0556 0.1111 0.1667 0.2500 0.3333 0.4444 0.5556 0.6667 0.7500 0.8333 0.8889 0.9444 0.9722 1.0000

0.0222 0.0444 0.0889 0.1333 0.2000 0.2667 0.3556 0.4444 0.5556 0.6444 0.7333 0.8000 0.8667 0.9111 0.9556 0.9778 1.0000

0.0182 0.0364 0.0727 0.1091 0.1636 0.2182 0.2909 0.3636 0.4545 0.5455 0.6364 0.7091 0.7818 0.8364 0.8909 0.9273 0.9636 0.9818 1.0000

0.0152 0.0303 0.0606 0.0909 0.1364 0.1818 0.2424 0.3030 0.3788 0.4545 0.5455 0.6212 0.6970 0.7576 0.8182 0.8636 0.9091 0.9394 0.9697 0.9848 1.0000

nX = 3 n Y w

3

4

5

6

7

8

9

10

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

0.0500 0.1000 0.2000 0.3500 0.5000 0.6500 0.8000 0.9000 0.9500 1.0000

0.0286 0.0571 0.1143 0.2000 0.3143 0.4286 0.5714 0.6857 0.8000 0.8857 0.9429 0.9714 1.0000

0.0179 0.0357 0.0714 0.1250 0.1964 0.2857 0.3929 0.5000 0.6071 0.7143 0.8036 0.8750 0.9286 0.9643 0.9821 1.0000

0.0119 0.0238 0.0476 0.0833 0.1310 0.1905 0.2738 0.3571 0.4524 0.5476 0.6429 0.7262 0.8095 0.8690 0.9167 0.9524

0.0083 0.0167 0.0333 0.0583 0.0917 0.1333 0.1917 0.2583 0.3333 0.4167 0.5000 0.5833 0.6667 0.7417 0.8083 0.8667

0.0061 0.0121 0.0242 0.0424 0.0667 0.0970 0.1394 0.1879 0.2485 0.3152 0.3879 0.4606 0.5394 0.6121 0.6848 0.7515

0.0045 0.0091 0.0182 0.0318 0.0500 0.0727 0.1045 0.1409 0.1864 0.2409 0.3000 0.3636 0.4318 0.5000 0.5682 0.6364

0.0035 0.0070 0.0140 0.0245 0.0385 0.0559 0.0804 0.1084 0.1434 0.1853 0.2343 0.2867 0.3462 0.4056 0.4685 0.5315

61


Table 8

Mann-Whitney-Wilcoxon Distribution: P(W

w) (continued)

Note: nX is the smaller sample size, n Y the larger sample size

nX = 4 n Y w

4

5

6

7

8

9

10

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0.0143 0.0286 0.0571 0.1000 0.1714 0.2429 0.3429 0.4429 0.5571 0.6571 0.7571 0.8286 0.9000 0.9429 0.9714 0.9857 1.0000

0.0079 0.0159 0.0317 0.0556 0.0952 0.1429 0.2063 0.2778 0.3651 0.4524 0.5476 0.6349 0.7222 0.7937 0.8571 0.9048 0.9444 0.9683 0.9841 0.9921 1.0000

0.0048 0.0095 0.0190 0.0333 0.0571 0.0857 0.1286 0.1762 0.2381 0.3048 0.3810 0.4571 0.5429 0.6190 0.6952 0.7619 0.8238 0.8714 0.9143 0.9429 0.9667

0.0030 0.0061 0.0121 0.0212 0.0364 0.0545 0.0818 0.1152 0.1576 0.2061 0.2636 0.3242 0.3939 0.4636 0.5364 0.6061 0.6758 0.7364 0.7939 0.8424 0.8848

0.0020 0.0040 0.0081 0.0141 0.0242 0.0364 0.0545 0.0768 0.1071 0.1414 0.1838 0.2303 0.2848 0.3414 0.4040 0.4667 0.5333 0.5960 0.6586 0.7152 0.7697

0.0014 0.0028 0.0056 0.0098 0.0168 0.0252 0.0378 0.0531 0.0741 0.0993 0.1301 0.1650 0.2070 0.2517 0.3021 0.3552 0.4126 0.4699 0.5301 0.5874 0.6448

0.0010 0.0020 0.0040 0.0070 0.0120 0.0180 0.0270 0.0380 0.0529 0.0709 0.0939 0.1199 0.1518 0.1868 0.2268 0.2697 0.3177 0.3666 0.4196 0.4725 0.5275

nX = 5 n Y w

5

6

7

8

9

10

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

0.0040 0.0079 0.0159 0.0278 0.0476 0.0754 0.1111 0.1548 0.2103 0.2738 0.3452 0.4206 0.5000 0.5794 0.6548 0.7262 0.7897 0.8452 0.8889 0.9246 0.9524 0.9722 0.9841 0.9921 0.9960 1.0000

0.0022 0.0043 0.0087 0.0152 0.0260 0.0411 0.0628 0.0887 0.1234 0.1645 0.2143 0.2684 0.3312 0.3961 0.4654 0.5346 0.6039 0.6688 0.7316 0.7857 0.8355 0.8766 0.9113 0.9372 0.9589 0.9740

0.0013 0.0025 0.0051 0.0088 0.0152 0.0240 0.0366 0.0530 0.0745 0.1010 0.1338 0.1717 0.2159 0.2652 0.3194 0.3775 0.4381 0.5000 0.5619 0.6225 0.6806 0.7348 0.7841 0.8283 0.8662 0.8990

0.0008 0.0016 0.0031 0.0054 0.0093 0.0148 0.0225 0.0326 0.0466 0.0637 0.0855 0.1111 0.1422 0.1772 0.2176 0.2618 0.3108 0.3621 0.4165 0.4716 0.5284 0.5835 0.6379 0.6892 0.7382 0.7824

0.0005 0.0010 0.0020 0.0035 0.0060 0.0095 0.0145 0.0210 0.0300 0.0415 0.0559 0.0734 0.0949 0.1199 0.1489 0.1818 0.2188 0.2592 0.3032 0.3497 0.3986 0.4491 0.5000 0.5509 0.6014 0.6503

0.0003 0.0007 0.0013 0.0023 0.0040 0.0063 0.0097 0.0140 0.0200 0.0276 0.0376 0.0496 0.0646 0.0823 0.1032 0.1272 0.1548 0.1855 0.2198 0.2567 0.2970 0.3393 0.3839 0.4296 0.4765 0.5235

62


Table 8

Mann-Whitney-Wilcoxon Distribution: P(W w) (continued) Note: nX is the smaller sample size, n Y the larger sample size

nX = 6

nX = 7

n Y

n Y

w

6

7

8

9

10

w

7

8

9

10

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

0.0011 0.0022 0.0043 0.0076 0.0130 0.0206 0.0325 0.0465 0.0660 0.0898 0.1201 0.1548 0.1970 0.2424 0.2944 0.3496 0.4091 0.4686 0.5314 0.5909 0.6504 0.7056 0.7576 0.8030 0.8452 0.8799 0.9102 0.9340 0.9535 0.9675 0.9794 0.9870 0.9924 0.9957 0.9978 0.9989 1.0000

0.0006 0.0012 0.0023 0.0041 0.0070 0.0111 0.0175 0.0256 0.0367 0.0507 0.0688 0.0903 0.1171 0.1474 0.1830 0.2226 0.2669 0.3141 0.3654 0.4178 0.4726 0.5274 0.5822 0.6346 0.6859 0.7331 0.7774 0.8170 0.8526 0.8829 0.9097 0.9312 0.9493 0.9633 0.9744 0.9825 0.9889 0.9930 0.9959 0.9977 0.9988 0.9994 1.0000

0.0003 0.0007 0.0013 0.0023 0.0040 0.0063 0.0100 0.0147 0.0213 0.0296 0.0406 0.0539 0.0709 0.0906 0.1142 0.1412 0.1725 0.2068 0.2454 0.2864 0.3310 0.3773 0.4259 0.4749 0.5251 0.5741 0.6227 0.6690 0.7136 0.7546 0.7932 0.8275 0.8588 0.8858 0.9094 0.9291 0.9461 0.9594 0.9704 0.9787 0.9853 0.9900 0.9937 0.9960 0.9977 0.9987 0.9993 0.9997 1.0000

0.0002 0.0004 0.0008 0.0014 0.0024 0.0038 0.0060 0.0088 0.0128 0.0180 0.0248 0.0332 0.0440 0.0567 0.0723 0.0905 0.1119 0.1361 0.1638 0.1942 0.2280 0.2643 0.3035 0.3445 0.3878 0.4320 0.4773 0.5227 0.5680 0.6122 0.6555 0.6965 0.7357 0.7720 0.8058 0.8362 0.8639 0.8881 0.9095 0.9277 0.9433 0.9560 0.9668 0.9752 0.9820 0.9872 0.9912 0.9940 0.9962 0.9976 0.9986

0.0001 0.0002 0.0005 0.0009 0.0015 0.0024 0.0037 0.0055 0.0080 0.0112 0.0156 0.0210 0.0280 0.0363 0.0467 0.0589 0.0736 0.0903 0.1099 0.1317 0.1566 0.1838 0.2139 0.2461 0.2811 0.3177 0.3564 0.3962 0.4374 0.4789 0.5211 0.5626 0.6038 0.6436 0.6823 0.7189 0.7539 0.7861 0.8162 0.8434 0.8683 0.8901 0.9097 0.9264 0.9411 0.9533 0.9637 0.9720 0.9790 0.9844 0.9888

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

0.0003 0.0006 0.0012 0.0020 0.0035 0.0055 0.0087 0.0131 0.0189 0.0265 0.0364 0.0487 0.0641 0.0825 0.1043 0.1297 0.1588 0.1914 0.2279 0.2675 0.3100 0.3552 0.4024 0.4508 0.5000 0.5492 0.5976 0.6448 0.6900 0.7325 0.7721 0.8086 0.8412 0.8703 0.8957 0.9175 0.9359 0.9513 0.9636 0.9735 0.9811 0.9869 0.9913 0.9945 0.9965 0.9980 0.9988 0.9994 0.9997 1.0000

0.0002 0.0003 0.0006 0.0011 0.0019 0.0030 0.0047 0.0070 0.0103 0.0145 0.0200 0.0270 0.0361 0.0469 0.0603 0.0760 0.0946 0.1159 0.1405 0.1678 0.1984 0.2317 0.2679 0.3063 0.3472 0.3894 0.4333 0.4775 0.5225 0.5667 0.6106 0.6528 0.6937 0.7321 0.7683 0.8016 0.8322 0.8595 0.8841 0.9054 0.9240 0.9397 0.9531 0.9639 0.9730 0.9800 0.9855 0.9897 0.9930 0.9953 0.9970

0.0001 0.0002 0.0003 0.0006 0.0010 0.0017 0.0026 0.0039 0.0058 0.0082 0.0115 0.0156 0.0209 0.0274 0.0356 0.0454 0.0571 0.0708 0.0869 0.1052 0.1261 0.1496 0.1755 0.2039 0.2349 0.2680 0.3032 0.3403 0.3788 0.4185 0.4591 0.5000 0.5409 0.5815 0.6212 0.6597 0.6968 0.7320 0.7651 0.7961 0.8245 0.8504 0.8739 0.8948 0.9131 0.9292 0.9429 0.9546 0.9644 0.9726 0.9791

0.0001 0.0001 0.0002 0.0004 0.0006 0.0010 0.0015 0.0023 0.0034 0.0048 0.0068 0.0093 0.0125 0.0165 0.0215 0.0277 0.0351 0.0439 0.0544 0.0665 0.0806 0.0966 0.1148 0.1349 0.1574 0.1819 0.2087 0.2374 0.2681 0.3004 0.3345 0.3698 0.4063 0.4434 0.4811 0.5189 0.5566 0.5937 0.6302 0.6655 0.6996 0.7319 0.7626 0.7913 0.8181 0.8426 0.8651 0.8852 0.9034 0.9194 0.9335

63


Table 8

Mann-Whitney-Wilcoxon Distribution: P(W w) (continued) Note: nX is the smaller sample size, n Y the larger sample size

nX = 8

nX = 9

nX = 10

n Y

n Y

n Y

w

8

9

10

w

9

10

36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86

0.0001 0.0002 0.0003 0.0005 0.0009 0.0015 0.0023 0.0035 0.0052 0.0074 0.0103 0.0141 0.0190 0.0249 0.0325 0.0415 0.0524 0.0652 0.0803 0.0974 0.1172 0.1393 0.1641 0.1911 0.2209 0.2527 0.2869 0.3227 0.3605 0.3992 0.4392 0.4796 0.5204 0.5608 0.6008 0.6395 0.6773 0.7131 0.7473 0.7791 0.8089 0.8359 0.8607 0.8828 0.9026 0.9197 0.9348 0.9476 0.9585 0.9675 0.9751

0.0000 0.0001 0.0002 0.0003 0.0005 0.0008 0.0012 0.0019 0.0028 0.0039 0.0056 0.0076 0.0103 0.0137 0.0180 0.0232 0.0296 0.0372 0.0464 0.0570 0.0694 0.0836 0.0998 0.1179 0.1383 0.1606 0.1852 0.2117 0.2404 0.2707 0.3029 0.3365 0.3715 0.4074 0.4442 0.4813 0.5187 0.5558 0.5926 0.6285 0.6635 0.6971 0.7293 0.7596 0.7883 0.8148 0.8394 0.8617 0.8821 0.9002 0.9164

0.0000 0.0000 0.0001 0.0002 0.0003 0.0004 0.0007 0.0010 0.0015 0.0022 0.0031 0.0043 0.0058 0.0078 0.0103 0.0133 0.0171 0.0217 0.0273 0.0338 0.0416 0.0506 0.0610 0.0729 0.0864 0.1015 0.1185 0.1371 0.1577 0.1800 0.2041 0.2299 0.2574 0.2863 0.3167 0.3482 0.3809 0.4143 0.4484 0.4827 0.5173 0.5516 0.5857 0.6191 0.6518 0.6833 0.7137 0.7426 0.7701 0.7959 0.8200

45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95

0.0000 0.0000 0.0001 0.0001 0.0002 0.0004 0.0006 0.0009 0.0014 0.0020 0.0028 0.0039 0.0053 0.0071 0.0094 0.0122 0.0157 0.0200 0.0252 0.0313 0.0385 0.0470 0.0567 0.0680 0.0807 0.0951 0.1112 0.1290 0.1487 0.1701 0.1933 0.2181 0.2447 0.2729 0.3024 0.3332 0.3652 0.3981 0.4317 0.4657 0.5000 0.5343 0.5683 0.6019 0.6348 0.6668 0.6976 0.7271 0.7553 0.7819 0.8067

0.0000 0.0000 0.0000 0.0001 0.0001 0.0002 0.0003 0.0005 0.0007 0.0011 0.0015 0.0021 0.0028 0.0038 0.0051 0.0066 0.0086 0.0110 0.0140 0.0175 0.0217 0.0267 0.0326 0.0394 0.0474 0.0564 0.0667 0.0782 0.0912 0.1055 0.1214 0.1388 0.1577 0.1781 0.2001 0.2235 0.2483 0.2745 0.3019 0.3304 0.3598 0.3901 0.4211 0.4524 0.4841 0.5159 0.5476 0.5789 0.6099 0.6402 0.6696

64

w 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105

10 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0002 0.0002 0.0004 0.0005 0.0008 0.0010 0.0014 0.0019 0.0026 0.0034 0.0045 0.0057 0.0073 0.0093 0.0116 0.0144 0.0177 0.0216 0.0262 0.0315 0.0376 0.0446 0.0526 0.0615 0.0716 0.0827 0.0952 0.1088 0.1237 0.1399 0.1575 0.1763 0.1965 0.2179 0.2406 0.2644 0.2894 0.3153 0.3421 0.3697 0.3980 0.4267 0.4559 0.4853 0.5147


Table 9

0

Upper Critical Values of the Chi-Square Distribution

χ 12−α

χα2

0 Tail Area 1 -

Tail Area

df

.995

.99

.975

.95

.90

df

1 2 3 4 5

0.000 0.010 0.072 0.207 0.412

0.000 0.020 0.115 0.297 0.554

0.001 0.051 0.216 0.484 0.831

0.004 0.103 0.352 0.711 1.145

0.016 0.211 0.584 1.064 1.610

1 2 3 4 5

6 7 8 9 10

0.676 0.989 1.344 1.735 2.156

0.872 1.239 1.646 2.088 2.558

1.237 1.690 2.180 2.700 3.247

1.635 2.167 2.733 3.325 3.940

2.204 2.833 3.490 4.168 4.865

11 12 13 14 15

2.603 3.074 3.565 4.075 4.601

3.053 3.571 4.107 4.660 5.229

3.816 4.404 5.009 5.629 6.262

4.575 5.226 5.892 6.571 7.261

16 17 18 19 20

5.142 5.697 6.265 6.844 7.434

5.812 6.408 7.015 7.633 8.260

6.908 7.564 8.231 8.907 9.591

21 22 23 24 25

8.034 8.643 9.260 9.886 10.520

8.897 9.542 10.196 10.856 11.524

26 27 28 29 30

11.160 11.808 12.461 13.121 13.787

31 32 33 34 35

.05

.025

.01

.005

2.706 4.605 6.251 7.779 9.236

3.841 5.991 7.815 9.488 11.070

5.024 7.378 9.348 11.143 12.833

6.635 9.210 11.345 13.277 15.086

7.879 10.597 12.838 14.860 16.750

6 7 8 9 10

10.645 12.017 13.362 14.684 15.987

12.592 14.067 15.507 16.919 18.307

14.449 16.013 17.535 19.023 20.483

16.812 18.475 20.090 21.666 23.209

18.548 20.278 21.955 23.589 25.188

5.578 6.304 7.042 7.790 8.547

11 12 13 14 15

17.275 18.549 19.812 21.064 22.307

19.675 21.026 22.362 23.685 24.996

21.920 23.337 24.736 26.119 27.488

24.725 26.217 27.688 29.141 30.578

26.757 28.300 29.819 31.319 32.801

7.962 8.672 9.390 10.117 10.851

9.312 10.085 10.865 11.651 12.443

16 17 18 19 20

23.542 24.769 25.989 27.204 28.412

26.296 27.587 28.869 30.144 31.410

28.845 30.191 31.526 32.852 34.170

32.000 33.409 34.805 36.191 37.566

34.267 35.718 37.156 38.582 39.997

10.283 10.982 11.689 12.401 13.120

11.591 12.338 13.091 13.848 14.611

13.240 14.041 14.848 15.659 16.473

21 22 23 24 25

29.615 30.813 32.007 33.196 34.382

32.671 33.924 35.172 36.415 37.652

35.479 36.781 38.076 39.364 40.646

38.932 40.289 41.638 42.980 44.314

41.401 42.796 44.181 45.559 46.928

12.198 12.879 13.565 14.256 14.953

13.844 14.573 15.308 16.047 16.791

15.379 16.151 16.928 17.708 18.493

17.292 18.114 18.939 19.768 20.599

26 27 28 29 30

35.563 36.741 37.916 39.087 40.256

38.885 40.113 41.337 42.557 43.773

41.923 43.195 44.461 45.772 46.979

45.642 46.963 48.278 49.588 50.892

48.290 49.645 50.993 52.336 53.672

14.458 15.134 15.815 16.501 17.192

15.655 13.362 17.073 17.789 18.509

17.539 18.291 19.047 19.806 20.569

19.281 20.072 20.867 21.664 22.465

21.434 22.271 23.110 23.952 24.797

31 32 33 34 35

41.422 42.585 43.745 44.903 46.059

44.985 46.194 47.400 48.602 49.802

48.232 49.480 50.725 51.966 53.203

52.191 53.486 54.776 56.061 57.342

55.003 56.328 57.648 58.964 60.275

36 37 38 39 40

17.887 18.586 19.289 19.996 20.707

19.233 19.960 20.691 21.426 22.164

21.336 22.106 22.878 23.654 24.433

23.269 24.075 24.884 25.695 26.509

25.643 26.492 27.343 28.196 29.051

36 37 38 39 40

47.212 48.363 49.513 50.660 51.805

50.998 52.192 53.384 54.572 55.758

54.437 55.668 56.895 58.120 59.342

58.619 59.892 61.162 62.428 63.691

61.581 62.883 64.181 65.476 66.766

41 42 43 44 45

21.421 22.138 22.859 23.584 24.311

22.906 23.650 24.398 25.148 25.901

25.215 25.999 26.785 27.575 28.366

27.326 28.144 28.965 29.787 30.612

29.907 30.765 31.625 32.487 33.350

41 42 43 44 45

52.949 54.090 55.230 56.369 57.505

56.942 58.124 59.304 60.481 61.656

60.561 61.777 62.990 64.201 65.410

64.950 66.206 67.459 68.709 69.957

68.053 69.336 70.616 71.893 73.166

46 47 48 49 50

25.041 25.775 26.511 27.249 27.991

26.657 27.416 28.177 28.941 29.707

29.160 29.956 30.755 31.555 32.357

31.439 32.268 33.098 33.930 34.764

34.215 35.081 35.949 36.818 37.689

46 47 48 49 50

58.641 59.774 60.907 62.038 63.167

62.830 64.001 65.171 66.339 67.505

66.617 67.821 69.023 70.222 71.420

71.201 72.443 73.683 74.919 76.154

74.437 75.704 76.969 78.231 79.490

65

.10


Table 10

Upper Critical Values of the F-distribution with ( 1,

0

2)

df

Tail Area = α

Fα 1

2

1

2

3

4

5

6

7

8

9

1

.10 .05 .025 .01 .005

39.86 161.4 647.8 4052 16211

49.50 199.5 799.5 5000 20000

53.59 215.7 864.2 5403 20615

55.83 224.6 899.6 5625 22500

57.24 230.2 921.8 5764 23056

58.20 234.0 937.1 5859 23437

58.91 236.8 948.2 5928 23715

59.44 238.9 956.7 5981 23925

59.86 240.5 963.3 6022 24091

2

.10 .05 .025 .01 .005

8.53 18.51 38.51 98.50 198.5

9.00 19.00 39.00 99.00 199.0

9.16 19.16 39.17 99.17 199.2

9.24 19.25 39.25 99.25 199.2

9.29 19.30 39.30 99.30 199.3

9.33 19.33 39.33 99.33 199.3

9.35 19.35 39.36 99.36 199.4

9.37 19.37 39.37 99.37 199.4

9.38 19.38 39.39 99.39 199.4

3

.10 .05 .025 .01 .005

5.54 10.13 17.44 34.12 55.55

5.46 9.55 16.04 30.82 49.80

5.39 9.28 15.44 29.46 47.47

5.34 9.12 15.10 28.71 46.19

5.31 9.01 14.88 28.24 45.39

5.28 8.94 14.73 27.91 44.84

5.27 8.89 14.62 27.67 44.43

5.25 8.85 14.54 27.49 44.13

5.24 8.81 14.47 27.35 43.88

4

.10 .05 .025 .01 .005

4.54 7.71 12.22 21.20 31.33

4.32 6.94 10.65 18.00 26.28

4.19 6.59 9.98 16.69 24.26

4.11 6.39 9.60 15.98 23.15

4.05 6.26 9.36 15.52 22.46

4.01 6.16 9.20 15.21 21.97

3.98 6.09 9.07 14.98 21.62

3.95 6.04 8.98 14.80 21.35

3.94 6.00 8.90 14.66 21.14

5

.10 .05 .025 .01 .005

4.06 6.61 10.01 16.26 22.78

3.78 5.79 8.43 13.27 18.31

3.62 5.41 7.76 12.06 16.53

3.52 5.19 7.39 11.39 15.56

3.45 5.05 7.15 10.97 14.94

3.40 4.95 6.98 10.67 14.51

3.37 4.88 6.85 10.46 14.20

3.34 4.82 6.76 10.29 13.96

3.32 4.77 6.68 10.16 13.77

6

.10 .05 .025 .01 .005

3.78 5.99 8.81 13.75 18.63

3.46 5.14 7.26 10.92 14.54

3.29 4.76 6.60 9.78 12.92

3.18 4.53 6.23 9.15 12.03

3.11 4.39 5.99 8.75 11.46

3.05 4.28 5.82 8.47 11.07

3.01 4.21 5.70 8.26 10.79

2.98 4.15 5.60 8.10 10.57

2.96 4.10 5.52 7.98 10.39

7

.10 .05 .025 .01 .005

3.59 5.59 8.07 12.25 16.24

3.26 4.74 6.54 9.55 12.40

3.07 4.35 5.89 8.45 10.88

2.96 4.12 5.52 7.85 10.05

2.88 3.97 5.29 7.46 9.52

2.83 3.87 5.12 7.19 9.16

2.78 3.79 4.99 6.99 8.89

2.75 3.73 4.90 6.84 8.68

2.72 3.68 4.82 6.72 8.51

8

.10 .05 .025 .01 .005

3.46 5.32 7.57 11.26 14.69

3.11 4.46 6.06 8.65 11.04

2.92 4.07 5.42 7.59 9.60

2.81 3.84 5.05 7.01 8.81

2.73 3.69 4.82 6.63 8.30

2.67 3.58 4.65 6.37 7.95

2.62 3.50 4.53 6.18 7.69

2.59 3.44 4.43 6.03 7.50

2.56 3.39 4.36 5.91 7.34

9

.10 .05 .025 .01 .005

3.36 5.12 7.21 10.56 13.61

3.01 4.26 5.71 8.02 10.11

2.81 3.86 5.08 6.99 8.72

2.69 3.63 4.72 6.42 7.96

2.61 3.48 4.48 6.06 7.47

2.55 3.37 4.32 5.80 7.13

2.51 3.29 4.20 5.61 6.88

2.47 3.23 4.10 5.47 6.69

2.44 3.18 4.03 5.35 6.54

66


Table 10


0

2)

df (continued)

Tail Area = α

Fα 1

2

10

12

15

20

24

30

60

120

1

.10 .05 .025 .01 .005

60.19 241.9 968.6 6056 24224

60.71 243.9 976.7 6106 24426

61.22 245.9 984.9 6157 24630

61.74 248.0 993.1 6209 24836

62.00 249.1 997.2 6235 24940

62.26 250.1 1001 6261 25044

62.79 252.2 1010 6313 25253

63.06 253.3 1014 6339 25359

63.33 254.3 1018 6366 25465

2

.10 .05 .025 .01 .005

9.39 19.40 39.40 99.40 199.4

9.41 19.41 39.41 99.42 199.4

9.42 19.43 39.43 99.43 199.4

9.44 19.45 39.45 99.45 199.4

9.45 19.45 39.46 99.46 199.5

9.46 19.46 39.46 99.47 199.5

9.47 19.48 39.48 99.48 199.5

9.48 19.49 39.49 99.49 199.5

9.49 19.50 39.50 99.50 199.5

3

.10 .05 .025 .01 .005

5.23 8.79 14.42 27.23 43.69

5.22 8.74 14.34 27.05 43.39

5.20 8.70 14.25 26.87 43.08

5.18 8.66 14.17 26.69 42.78

5.18 8.64 14.12 26.60 42.62

5.17 8.62 14.08 26.50 42.47

5.15 8.57 13.99 26.32 42.15

5.14 8.55 13.95 26.22 41.99

5.13 8.53 13.90 26.13 41.83

4

.10 .05 .025 .01 .005

3.92 5.96 8.84 14.55 20.97

3.90 5.91 8.75 14.37 20.70

3.87 5.86 8.66 14.20 20.44

3.84 5.80 8.56 14.02 20.17

3.83 5.77 8.51 13.93 20.03

3.82 5.75 8.46 13.84 19.89

3.79 5.69 8.36 13.65 19.61

3.78 5.66 8.31 13.56 19.47

3.76 5.63 8.26 13.46 19.32

5

.10 .05 .025 .01 .005

3.30 4.74 6.62 10.05 13.62

3.27 4.68 6.52 9.89 13.38

3.24 4.62 6.43 9.72 13.15

3.21 4.56 6.33 9.55 12.90

3.19 4.53 6.28 9.47 12.78

3.17 4.50 6.23 9.38 12.66

3.14 4.43 6.12 9.20 12.40

3.12 4.40 6.07 9.11 12.27

3.10 4.36 6.02 9.02 12.14

6

.10 .05 .025 .01 .005

2.94 4.06 5.46 7.87 10.25

2.90 4.00 5.37 7.72 10.03

2.87 3.94 5.27 7.56 9.81

2.84 3.87 5.17 7.40 9.59

2.82 3.84 5.12 7.31 9.47

2.80 3.81 5.07 7.23 9.36

2.76 3.74 4.96 7.06 9.12

2.74 3.70 4.90 6.97 9.00

2.72 3.67 4.85 6.88 8.88

7

.10 .05 .025 .01 .005

2.70 3.64 4.76 6.62 8.38

2.67 3.57 4.67 6.47 8.18

2.63 3.51 4.57 6.31 7.97

2.59 3.44 4.47 6.16 7.75

2.58 3.41 4.42 6.07 7.65

2.56 3.38 4.36 5.99 7.53

2.51 3.30 4.25 5.82 7.31

2.49 3.27 4.20 5.74 7.19

2.47 3.23 4.14 5.65 7.08

8

.10 .05 .025 .01 .005

2.54 3.35 4.30 5.81 7.21

2.50 3.28 4.20 5.67 7.01

2.46 3.22 4.10 5.52 6.81

2.42 3.15 4.00 5.36 6.61

2.40 3.12 3.95 5.28 6.50

2.38 3.08 3.89 5.20 6.40

2.34 3.01 3.78 5.03 6.18

2.32 2.97 3.73 4.95 6.06

2.29 2.93 3.67 4.86 5.95

9

.10 .05 .025 .01 .005

2.42 3.14 3.96 5.26 6.42

2.38 3.07 3.87 5.11 6.23

2.34 3.01 3.77 4.96 6.03

2.30 2.94 3.67 4.81 5.83

2.28 2.90 3.61 4.73 5.73

2.25 2.86 3.56 4.65 5.62

2.21 2.79 3.45 4.48 5.41

2.18 2.75 3.39 4.40 5.30

2.16 2.71 3.33 4.31 5.19

67


Table 10


0

2)

df (continued)

Tail Area = α

Fα 1

2

1

2

3

4

5

6

7

8

9

10

.10 .05 .025 .01 .005

3.29 4.96 6.94 10.0 12.8

2.92 4.10 5.46 7.56 9.43

2.73 3.71 4.83 6.55 8.08

2.61 3.48 4.47 5.99 7.34

2.52 3.33 4.24 5.64 6.87

2.46 3.22 4.07 5.39 6.54

2.41 3.14 3.95 5.20 6.30

2.38 3.07 3.85 5.06 6.12

2.35 3.02 3.78 4.94 5.97

12

.10 .05 .025 .01 .005

3.18 4.75 6.55 9.33 11.8

2.81 3.89 5.10 6.93 8.51

2.61 3.49 4.47 5.95 7.23

2.48 3.26 4.12 5.41 6.52

2.39 3.11 3.89 5.06 6.07

2.33 3.00 3.73 4.82 5.76

2.28 2.91 3.61 4.64 5.52

2.24 2.85 3.51 4.50 5.35

2.21 2.80 3.44 4.39 5.20

15

.10 .05 .025 .01 .005

3.07 4.54 6.20 8.68 10.8

2.70 3.68 4.77 6.36 7.70

2.49 3.29 4.15 5.42 6.48

2.36 3.06 3.80 4.89 5.80

2.27 2.90 3.58 4.56 5.37

2.21 2.79 3.41 4.32 5.07

2.16 2.71 3.29 4.14 4.85

2.12 2.64 3.20 4.00 4.67

2.09 2.59 3.12 3.89 4.54

20

.10 .05 .025 .01 .005

2.97 4.35 5.87 8.10 9.94

2.59 3.49 4.46 5.85 6.99

2.38 3.10 3.86 4.94 5.82

2.25 2.87 3.51 4.43 5.17

2.16 2.71 3.29 4.10 4.76

2.09 2.60 3.13 3.87 4.47

2.04 2.51 3.01 3.70 4.26

2.00 2.45 2.91 3.56 4.09

1.96 2.39 2.84 3.46 3.96

24

.10 .05 .025 .01 .005

2.93 4.26 5.72 7.82 9.55

2.54 3.40 4.32 5.61 6.66

2.33 3.01 3.72 4.72 5.52

2.19 2.78 3.38 4.22 4.89

2.10 2.62 3.15 3.90 4.49

2.04 2.51 2.99 3.67 4.20

1.98 2.42 2.87 3.50 3.99

1.94 2.36 2.78 3.36 3.83

1.91 2.30 2.70 3.26 3.69

30

.10 .05 .025 .01 .005

2.88 4.17 5.57 7.56 9.18

2.49 3.32 4.18 5.39 6.35

2.28 2.92 3.59 4.51 5.24

2.14 2.69 3.25 4.02 4.62

2.05 2.53 3.03 3.70 4.23

1.98 2.42 2.87 3.47 3.95

1.93 2.33 2.75 3.30 3.74

1.88 2.27 2.65 3.17 3.58

1.85 2.21 2.57 3.07 3.45

60

.10 .05 .025 .01 .005

2.79 4.00 5.29 7.08 8.49

2.39 3.15 3.93 4.98 5.79

2.18 2.76 3.34 4.13 4.73

2.04 2.53 3.01 3.65 4.14

1.95 2.37 2.79 3.34 3.76

1.87 2.25 2.63 3.12 3.49

1.82 2.17 2.51 2.95 3.29

1.77 2.10 2.41 2.82 3.13

1.74 2.04 2.33 2.72 3.01

120

.10 .05 .025 .01 .005

2.75 3.92 5.15 6.85 8.18

2.35 3.07 3.80 4.79 5.54

2.13 2.68 3.23 3.95 4.50

1.99 2.45 2.89 3.48 3.92

1.90 2.29 2.67 3.17 3.55

1.82 2.18 2.52 2.96 3.28

1.77 2.09 2.39 2.79 3.09

1.72 2.02 2.30 2.66 2.93

1.68 1.96 2.22 2.56 2.81

.10 .05 .025 .01 .005

2.71 3.84 5.02 6.63 7.88

2.30 3.00 3.69 4.61 5.30

2.08 2.60 3.12 3.78 4.28

1.94 2.37 2.79 3.32 3.72

1.85 2.21 2.57 3.02 3.35

1.77 2.10 2.41 2.80 3.09

1.72 2.01 2.29 2.64 2.90

1.67 1.94 2.19 2.51 2.74

1.63 1.88 2.11 2.41 2.62

68


Table 10


0

2)

df (continued)

Tail Area = α

Fα 1

2

10

12

15

20

24

30

60

120

10

.10 .05 .025 .01 .005

2.32 2.98 3.72 4.85 5.85

2.28 2.91 3.62 4.71 5.66

2.24 2.85 3.52 4.56 5.47

2.20 2.77 3.42 4.41 5.27

2.18 2.74 3.37 4.33 5.17

2.16 2.70 3.31 4.25 5.07

2.11 2.62 3.20 4.08 4.86

2.08 2.58 3.14 4.00 4.75

2.06 2.54 3.08 3.91 4.64

12

.10 .05 .025 .01 .005

2.19 2.75 3.37 4.30 5.09

2.15 2.69 3.28 4.16 4.91

2.10 2.62 3.18 4.01 4.72

2.06 2.54 3.07 3.86 4.53

2.04 2.51 3.02 3.78 4.43

2.01 2.47 2.96 3.70 4.33

1.96 2.38 2.85 3.54 4.12

1.93 2.34 2.79 3.45 4.01

1.90 2.30 2.72 3.36 3.90

15

.10 .05 .025 .01 .005

2.06 2.54 3.06 3.80 4.42

2.02 2.48 2.96 3.67 4.25

1.97 2.40 2.86 3.52 4.07

1.92 2.33 2.76 3.37 3.88

1.90 2.29 2.70 3.29 3.79

1.87 2.25 2.64 3.21 3.69

1.82 2.16 2.52 3.05 3.48

1.79 2.11 2.46 2.96 3.37

1.76 2.07 2.40 2.87 3.26

20

.10 .05 .025 .01 .005

1.94 2.35 2.77 3.37 3.85

1.89 2.28 2.68 3.23 3.68

1.84 2.20 2.57 3.09 3.50

1.79 2.12 2.46 2.94 3.32

1.77 2.08 2.41 2.86 3.22

1.74 2.04 2.35 2.78 3.12

1.68 1.95 2.22 2.61 2.92

1.64 1.90 2.16 2.52 2.81

1.61 1.84 2.09 2.42 2.69

24

.10 .05 .025 .01 .005

1.88 2.25 2.64 3.17 3.59

1.83 2.18 2.54 3.03 3.42

1.78 2.11 2.44 2.89 3.25

1.73 2.03 2.33 2.74 3.06

1.70 1.98 2.27 2.66 2.97

1.67 1.94 2.21 2.58 2.87

1.61 1.84 2.08 2.40 2.66

1.57 1.79 2.01 2.31 2.55

1.53 1.73 1.94 2.21 2.43

30

.10 .05 .025 .01 .005

1.82 2.16 2.51 2.98 3.34

1.77 2.09 2.41 2.84 3.18

1.72 2.01 2.31 2.70 3.01

1.67 1.93 2.20 2.55 2.82

1.64 1.89 2.14 2.47 2.73

1.61 1.84 2.07 2.39 2.63

1.54 1.74 1.94 2.21 2.42

1.50 1.68 1.87 2.11 2.30

1.46 1.62 1.79 2.01 2.18

60

.10 .05 .025 .01 .005

1.71 1.99 2.27 2.63 2.90

1.66 1.92 2.17 2.50 2.74

1.60 1.84 2.06 2.35 2.57

1.54 1.75 1.94 2.20 2.39

1.51 1.70 1.88 2.12 2.29

1.48 1.65 1.82 2.03 2.19

1.40 1.53 1.67 1.84 1.96

1.35 1.47 1.58 1.73 1.83

1.29 1.39 1.48 1.60 1.69

120

.10 .05 .025 .01 .005

1.65 1.91 2.16 2.47 2.71

1.60 1.83 2.05 2.34 2.54

1.55 1.75 1.94 2.19 2.37

1.48 1.66 1.82 2.03 2.19

1.45 1.61 1.76 1.95 2.09

1.41 1.55 1.69 1.86 1.98

1.32 1.43 1.53 1.66 1.75

1.26 1.35 1.43 1.53 1.61

1.19 1.25 1.31 1.38 1.43

.10 .05 .025 .01 .005

1.60 1.83 2.05 2.32 2.52

1.55 1.75 1.94 2.18 2.36

1.49 1.67 1.83 2.04 2.19

1.42 1.57 1.71 1.88 2.00

1.38 1.52 1.64 1.79 1.90

1.34 1.46 1.57 1.70 1.79

1.24 1.32 1.39 1.47 1.53

1.17 1.22 1.27 1.32 1.36

1.00 1.00 1.00 1.00 1.00

69


Table 11

Critical Values for the Sk-test for Asymmetry

n

.10

.05

.025

n

.10

.05

.025

5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.88 0.71 0.91 0.77 0.92 0.81 0.93 0.83 0.93 0.85 0.94 0.86 0.94 0.87

1.07 0.88 1.13 0.96 1.15 1.01 1.17 1.05 1.18 1.08 1.19 1.10 1.19 1.11

1.21 1.02 1.30 1.13 1.35 1.19 1.37 1.24 1.39 1.27 1.41 1.30 1.41 1.32

19 20 21 22 23 24 25 26 27 28 29 30

0.94 0.88 0.95 0.89 0.95 0.89 0.95 0.90 0.95 0.90 0.95 0.91

1.20 1.12 1.20 1.13 1.21 1.14 1.21 1.15 1.21 1.15 1.21 1.16

1.42 1.33 1.43 1.35 1.43 1.36 1.44 1.37 1.44 1.37 1.44 1.38

0.97

1.24

1.48

Table 12

n

Critical Values for the Normal Scores Test for Non-Normality

.10

.05

.01

n

.10

.05

.01

0.972 0.972 0.973 0.974 0.974

0.965 0.966 0.967 0.968 0.969

0.950 0.950 0.951 0.953 0.954

3 4 5

0.891 0.894 0.903

0.879 0.868 0.880

0.869 0.824 0.826

31 32 33 34 35

6 7 8 9 10

0.910 0.918 0.924 0.930 0.934

0.888 0.898 0.906 0.912 0.918

0.838 0.850 0.861 0.871 0.879

36 37 38 39 40

0.975 0.976 0.976 0.977 0.977

0.969 0.970 0.971 0.971 0.972

0.955 0.956 0.957 0.958 0.959

11 12 13 14 15

0.938 0.942 0.945 0.948 0.951

0.923 0.928 0.932 0.935 0.939

0.886 0.892 0.899 0.905 0.910

41 42 43 44 45

0.977 0.978 0.978 0.979 0.979

0.973 0.973 0.974 0.974 0.974

0.960 0.961 0.961 0.962 0.963

16 17 18 19 20

0.953 0.954 0.957 0.958 0.960

0.941 0.944 0.946 0.949 0.951

0.913 0.917 0.920 0.924 0.926

46 47 48 49 50

0.980 0.980 0.980 0.981 0.981

0.975 0.976 0.976 0.976 0.977

0.963 0.965 0.965 0.966 0.966

21 22 23 24 25

0.961 0.963 0.964 0.965 0.966

0.952 0.954 0.956 0.957 0.959

0.930 0.933 0.935 0.937 0.939

55 60 65 70 75

0.982 0.984 0.985 0.986 0.987

0.979 0.980 0.981 0.983 0.984

0.969 0.971 0.973 0.975 0.976

26 27 28 29 30

0.967 0.968 0.969 0.970 0.971

0.960 0.961 0.962 0.963 0.964

0.941 0.943 0.944 0.946 0.947

80 85 90 95 100

0.987 0.988 0.988 0.989 0.989

0.985 0.985 0.986 0.987 0.987

0.978 0.979 0.980 0.981 0.982

70


Table 13

Upper Critical Values for the Kruskal-Wallis Test (k samples)

Notes: 1. In the table below, the critical values give significance levels as close as possible to, but not exceeding the nominal . The actual levels of significance are in brackets. 2. When the table below is not applicable and k = 3 with the three group sizes above 5 or k > 3 with all group sizes above 4, use the following approximation: h

2

≈ χ α , df = k – 1 Nominal

Group Sizes

.10

.05

.025

.01

222 321 322 331 332 333 421 422 431 432 433 441 442 443 444 521 522 531 532 533 541 542 543 544 551 552 553 554 555 611 621 622 631 632 633 641 642 643 644 651 652 653 654 655 661 662 663 664 665 666 777 888

4.571 (.06667) 4.286 (.10000) 4.500 (.06667) 4.571 (.10000) 4.556 (.10000) 4.622 (.10000) 4.500 (.07619) 4.458 (.10000) 4.056 (.09286) 4.511 (.09841) 4.709 (.09238) 4.167 (.08254) 4.555 (.09778) 4.545 (.09905) 4.654 (.09662) 4.200 (.09524) 4.373 (.08995) 4.018 (.09524) 4.651 (.09127) 4.533 (.09697) 3.987 (.09841) 4.541 (.09841) 4.549 (.09892) 4.668 (.09817) 4.109 (.08586) 4.623 (.09704) 4.545 (.09965) 4.523 (.09935) 4.560 (.09952) 4.200 (.09524) 4.545 (.08889) 3.909 (.09524) 4.682 (.08528) 4.590 (.09773) 4.038 (.09437) 4.494 (.09986) 4.604 (.09997) 4.595 (.09847) 4.128 (.09271) 4.596 (.09807) 4.535 (.09932) 4.522 (.09974) 4.547 (.09835) 4.000 (.09774) 4.438 (.09824) 4.558 (.09948) 4.548 (.09982) 4.542 (.09987) 4.643 (.09874) 4.594 (.09933) 4.595 (.09933)

4.714 (.04762) 5.143 (.04286) 5.361 (.03214) 5.600 (.05000) 5.333 (.03333) 5.208 (.05000) 5.444 (.04603) 5.791 (.04571) 4.967 (.04762) 5.455 (.04571) 5.598 (.04866) 5.692 (.04866) 5.000 (.04762) 5.160 (.03439) 4.960 (.04762) 5.251 (.04921) 5.648 (.04892) 4.985 (.04444) 5.273 (.04877) 5.656 (.04863) 5.657 (.04906) 5.127 (.04618) 5.338 (.04726) 5.705 (.04612) 5.666 (.04931) 5.780 (.04878) 4.822 (.04762) 5.345 (.03810) 4.855 (.05000) 5.348 (.04632) 5.615 (.04968) 4.947 (.04675) 5.340 (.04906) 5.610 (.04862) 5.681 (.04881) 4.990 (.04726) 5.338 (.04729) 5.602 (.04956) 5.661 (.04991) 5.729 (.04973) 4.945 (.04779) 5.410 (.04993) 5.625 (.04999) 5.724 (.04950) 5.765 (.04993) 5.801 (.04905) 5.819 (.04911) 5.805 (.04973)

5.556 (.02500) 5.956 (.02500) 5.500 (.02381) 5.833 (.02143) 6.000 (.02381) 6.155 (.02476) 6.167 (.02222) 6.327 (.02413) 6.394 (.02476) 6.615 (.02424) 6.000 (.01852) 6.044 (.01984) 6.004 (.02460) 6.315 (.02121) 5.858 (.02381) 6.068 (.02482) 6.410 (.02496) 6.673 (.02429) 6.000 (.02165) 6.346 (.02489) 6.549 (.02436) 6.760 (.02490) 6.740 (.02475) 5.600 (.02381) 5.745 (.02063) 5.945 (.02143) 6.136 (.02294) 6.436 (.02229) 5.856 (.02424) 6.186 (.02453) 6.538 (.02498) 6.667 (.02495) 5.951 (.02453) 6.196 (.02481) 6.667 (.02452) 6.750 (.02473) 6.788 (.02484) 5.923 (.02381) 6.210 (.02443) 6.725 (.02462) 6.812 (.02458) 6.848 (.02489) 6.889 (.02493) 6.954 (.02446) 6.995 (.02485)

7.200 (.00357) 6.444 (.00794) 6.745 (.01000) 6.667 (.00952) 7.036 (.00571) 7.144 (.00970) 7.654 (.00762) 6.533 (.00794) 6.909 (.00873) 7.079 (.00866) 6.955 (.00794) 7.205 (.00895) 7.445 (.00974) 7.760 (.00946) 7.309 (.00938) 7.338 (.00962) 7.578 (.00968) 7.823 (.00978) 8.000 (.00946) 6.655 (.00794) 6.873 (.00714) 6.970 (.00909) 7.410 (.00779) 7.106 (.00866) 7.340 (.00967) 7.500 (.00966) 7.795 (.00990) 7.182 (.00974) 7.376 (.00982) 7.590 (.00999) 7.936 (.00998) 8.028 (.00988) 7.121 (.00932) 7.467 (.00982) 7.725 (.00985) 8.000 (.00998) 8.124 (.00990) 8.222 (.00994) 8.378 (.00992) 8.465 (.00991)

71


Table 13

Upper Critical Values for the Kruskal-Wallis Test (continued)

Nominal Group Sizes 2211 2221 2222 3111 3211 3221 3222 3311 3321 3322 3331 3332 3333 4111 4211 4221 4222 4311 4321 4322 4331 4332 4333 4411 4421 4422 4431 4432 4433 4441 4442 4443 4444 22111 22211 22221 22222 31111 32111 32211 32221 32222 33111 33211 33221 33222 33311 33321 33322 33331 33332 33333

.10

.05

.025

5.357 (.06667) 5.667 (.07619) 5.143 (.08571) 5.556 (.07143) 5.644 (.10000) 5.333 (.09643) 5.689 (.08571) 5.745 (.09921) 5.655 (.09786) 5.879 (.09974) 6.026 (.09779) 5.250 (.09048) 5.533 (.09788) 5.755 (.09302) 5.067 (.09524) 5.591 (.09857) 5.750 (.09980) 5.689 (.09602) 5.872 (.09929) 6.016 (.09779) 5.182 (.09968) 5.568 (.09980) 5.808 (.09882) 5.692 (.09853) 5.901 (.09950) 6.019 (.09948) 5.654 (.09801) 5.914 (.09940) 6.042 (.09980) 6.088 (.09900) 5.786 (.09524) 6.250 (.08810) 6.600 (.08889) 6.982 (.09101) 6.139 (.10000) 6.511 (.10000) 6.709 (.09873) 6.955 (.09922) 6.311 (.09286) 6.600 (.09929) 6.788 (.09892) 7.026 (.09897) 6.788 (.09779) 6.910 (.09916) 7.121 (.09979) 7.077 (.09836) 7.210 (.09965) 7.333 (.09922)

5.679 (.03810) 6.167 (.03810) 5.833 (.04286) 6.333 (.04762) 6.333 (.02143) 6.244 (.04246) 6.527 (.04921) 6.600 (.04929) 6.727 (.04948) 7.000 (.04351) 5.833 (.04286) 6.133 (.04180) 6.545 (.04921) 6.178 (.04921) 6.309 (.04937) 6.621 (.04949) 6.545 (.04952) 6.795 (.04925) 6.984 (.04897) 5.945 (.04952) 6.386 (.04981) 6.731 (.04872) 6.635 (.04978) 6.874 (.04983) 7.038 (.04990) 6.725 (.04979) 6.957 (.04960) 7.142 (.04954) 7.235 (.04922) 6.750 (.02381) 7.133 (.04127) 7.418 (.04868) 6.583 (.03571) 6.800 (.04921) 7.309 (.04889) 7.682 (.04745) 7.111 (.04048) 7.200 (.05000) 7.591 (.04919) 7.910 (.04934) 7.576 (.04545) 7.769 (.04885) 8.044 (.04915) 8.000 (.04792) 8.200 (.04940) 8.333 (.04955)

72

6.667 (.00952) 6.250 (.02143) 6.978 (.01746) 6.333 (.02143) 6.689 (.01786) 7.055 (.02317) 7.036 (.02429) 7.515 (.02390) 7.667 (.02338) 6.533 (.02063) 7.064 (.02222) 6.711 (.01905) 6.955 (.02317) 7.326 (.02496) 7.326 (.02329) 7.564 (.02494) 7.775 (.02437) 6.955 (.02349) 7.159 (.02459) 7.538 (.02453) 7.500 (.02462) 7.747 (.02500) 7.929 (.02487) 7.648 (.02470) 7.914 (.02499) 8.079 (.02494) 8.228 (.02476) 6.750 (.02381) 7.333 (.02222) 7.964 (.02222) 7.200 (.02460) 7.745 (.02317) 8.182 (.02384) 7.467 (.01190) 7.618 (.02452) 8.121 (.02437) 8.538 (.02408) 8.061 (.02325) 8.449 (.02471) 8.813 (.02472) 8.703 (.02396) 9.038 (.02452) 9.200 (.02500)

.01 6.667 (.00952) 7.133 (.00794) 7.200 (.00595) 7.636 (.01000) 7.400 (.00857) 8.015 (.00961) 8.538 (.00838) 7.000 (.00952) 7.391 (.00889) 7.067 (.00952) 7.455 (.00984) 7.871 (.00999) 7.758 (.00974) 8.333 (.00985) 8.659 (.00990) 7.909 (.00381) 7.909 (.00906) 8.346 (.00941) 8.231 (.00955) 8.621 (.00999) 8.876 (.00974) 8.588 (.00986) 8.871 (.00987) 9.075 (.01000) 9.287 (.00999) 7.533 (.00952) 8.291 (.00952) 7.600 (.00794) 8.127 (.00937) 8.682 (.00958) 8.073 (.00738) 8.576 (.00984) 9.115 (.00996) 8.424 (.00909) 9.051 (.00976) 9.505 (.00999) 9.451 (.00997) 9.876 (.00966) 10.200 (.00986)


Table 14

Upper Critical Values for the Friedman Test (k treatments and b blocks)

Notes 1. In the table below, the critical values give significance levels as close as possible to, but not exceeding the nominal . 2. For values of k and b beyond the range of the table below, various approximations are a vailable. k=3

k=4

k=5

b

.05

.01

.05

.01

.05

2 3 4 5

6.000 6.500 6.400

8.000 8.400

6.000 7.400 7.800 7.800

9.000 9.600 9.960

7.600 8.533 8.800 8.960

6 7 8 9 10

7.000 7.143 6.250 6.222 6.200

9.000 8.857 9.000 9.556 9.600

7.600 7.800 7.650 7.667 7.680

10.200 10.543 10.500 10.73 10.68

9.067 9.143 9.200 9.244

11 12 13 14 15

6.545 6.500 6.615 6.143 6.400

9.455 9.500 9.385 9.143 8.933

7.691 7.700 7.800 7.714 7.720

10.75 10.80 10.85 10.89 10.92

16 17 18 19 20

6.500 6.118 6.333 6.421 6.300

9.375 9.294 9.000 9.579 9.300

7.800 7.800 7.733 7.863 7.800

10.95 11.05 10.93 11.02 11.10

21 22 23 24 25

6.095 6.091 6.348 6.250 6.080

9.238 9.091 9.391 9.250 8.960

7.800 7.800

11.06 11.07

26 27 28 29 30

6.077 6.000 6.500 6.276 6.200

9.308 9.407 9.214 9.172 9.267

31 32 33 34 35

6.000 6.063 6.061 6.059 6.171

9.290 9.250 9.152 9.176 9.314

36 37 38 39 40

6.167 6.054 6.158 6.000 6.050

9.389 9.243 9.053 9.282 9.150

41 42 43 44 45

6.195 6.143 6.186 6.318 6.178

9.366 9.190 9.256 9.136 9.244

46 47 48 49 50

6.043 6.128 6.167 6.041 6.040

9.435 9.319 9.125 9.184 9.160

73

k=6 .01

.05

.01

8.000 10.13 11.20 11.68

9.143 9.857 10.286 10.486

9.714 11.762 12.714 13.229

11.867 12.114 12.300 12.44

10.571

13.619


Table 15

Bonferroni Critical Values

Critical Values B = t /2g for Simultaneous 90% Confidence Intervals

Number of Intervals ( = g ) df

2

3

4

5

6

7

8

9

10

15

2 3 4 5

4.3027 3.1824 2.7764 2.5706

5.3393 3.7405 3.1863 2.9117

6.2053 4.1765 3.4954 3.1634

6.9646 4.5407 3.7469 3.3649

7.6488 4.8567 3.9608 3.5341

8.2767 5.1377 4.1478 3.6805

8.8602 5.3919 4.3147 3.8100

9.4076 5.6251 4.4657 3.9263

9.9248 5.8409 4.6041 4.0321

12.186 6.7411 5.1668 4.4558

6 7 8 9 10

2.4469 2.3646 2.3060 2.2622 2.2281

2.7491 2.6419 2.5660 2.5096 2.4660

2.9687 2.8412 2.7515 2.6850 2.6338

3.1427 2.9980 2.8965 2.8214 2.7638

3.2875 3.1276 3.0158 2.9333 2.8701

3.4119 3.2383 3.1174 3.0283 2.9601

3.5212 3.3353 3.2060 3.1109 3.0382

3.6190 3.4216 3.2846 3.1841 3.1073

3.7074 3.4995 3.3554 3.2498 3.1693

4.0579 3.8055 3.6319 3.5054 3.4093

11 12 13 14 15

2.2010 2.1788 2.1604 2.1448 2.1314

2.4313 2.4030 2.3796 2.3598 2.3429

2.5931 2.5600 2.5326 2.5096 2.4899

2.7181 2.6810 2.6503 2.6245 2.6025

2.8200 2.7795 2.7459 2.7178 2.6937

2.9062 2.8626 2.8265 2.7962 2.7705

2.9809 2.9345 2.8961 2.8640 2.8366

3.0468 2.9978 2.9575 2.9236 2.8948

3.1058 3.0545 3.0123 2.9768 2.9467

3.3338 3.2729 3.2229 3.1811 3.1456

16 17 18 19 20

2.1199 2.1098 2.1009 2.0930 2.0860

2.3283 2.3156 2.3043 2.2944 2.2855

2.4729 2.4581 2.4450 2.4334 2.4231

2.5835 2.5669 2.5524 2.5395 2.5280

2.6730 2.6550 2.6391 2.6251 2.6126

2.7482 2.7289 2.7119 2.6969 2.6834

2.8131 2.7925 2.7745 2.7586 2.7444

2.8700 2.8484 2.8295 2.8127 2.7978

2.9208 2.8982 2.8784 2.8609 2.8453

3.1150 3.0885 3.0653 3.0447 3.0264

21 22 23 24 25

2.0796 2.0739 2.0687 2.0639 2.0595

2.2775 2.2703 2.2637 2.2577 2.2523

2.4138 2.4055 2.3979 2.3909 2.3846

2.5176 2.5083 2.4999 2.4922 2.4851

2.6013 2.5912 2.5820 2.5736 2.5660

2.6714 2.6606 2.6507 2.6418 2.6336

2.7316 2.7201 2.7097 2.7002 2.6916

2.7844 2.7723 2.7614 2.7514 2.7423

2.8314 2.8188 2.8073 2.7969 2.7874

3.0101 2.9953 2.9820 2.9698 2.9587

26 27 28 29 30

2.0555 2.0518 2.0484 2.0452 2.0423

2.2472 2.2426 2.2383 2.2343 2.2306

2.3788 2.3734 2.3685 2.3638 2.3596

2.4786 2.4727 2.4671 2.4620 2.4573

2.5589 2.5525 2.5465 2.5409 2.5357

2.6260 2.6191 2.6127 2.6068 2.6012

2.6836 2.6763 2.6695 2.6632 2.6574

2.7340 2.7263 2.7191 2.7126 2.7064

2.7787 2.7707 2.7633 2.7564 2.7500

2.9485 2.9391 2.9305 2.9225 2.9150

35 40 45 50 55

2.0301 2.0211 2.0141 2.0086 2.0040

2.2154 2.2041 2.1954 2.1885 2.1829

2.3420 2.3289 2.3189 2.3109 2.3044

2.4377 2.4233 2.4121 2.4033 2.3961

2.5145 2.4989 2.4868 2.4772 2.4694

2.5786 2.5618 2.5489 2.5387 2.5304

2.6334 2.6157 2.6021 2.5913 2.5825

2.6813 2.6627 2.6485 2.6372 2.6280

2.7238 2.7045 2.6896 2.6778 2.6682

2.8845 2.8620 2.8447 2.8310 2.8199

60 70 80 90 100

2.0003 1.9944 1.9901 1.9867 1.9840

2.1782 2.1709 2.1654 2.1612 2.1579

2.2990 2.2906 2.2844 2.2795 2.2757

2.3901 2.3808 2.3739 2.3685 2.3642

2.4630 2.4529 2.4454 2.4395 2.4349

2.5235 2.5128 2.5047 2.4985 2.4936

2.5752 2.5639 2.5554 2.5489 2.5437

2.6203 2.6085 2.5996 2.5928 2.5873

2.6603 2.6479 2.6387 2.6316 2.6259

2.8107 2.7964 2.7857 2.7774 2.7709

110 120 250 500 1000

1.9818 1.9799 1.9695 1.9647 1.9623

2.1551 2.1528 2.1399 2.1339 2.1310

2.2725 2.2699 2.2550 2.2482 2.2448

2.3607 2.3578 2.3414 2.3338 2.3301

2.4311 2.4280 2.4102 2.4021 2.3980

2.4896 2.4862 2.4673 2.4586 2.4543

2.5394 2.5359 2.5159 2.5068 2.5022

2.5829 2.5792 2.5582 2.5487 2.5439

2.6213 2.6174 2.5956 2.5857 2.5808

2.7655 2.7611 2.7359 2.7244 2.7187

1.9600

2.1280

2.2414

2.3263

2.3940

2.4500

2.4977

2.5392

2.5758

2.7131

74


Table 15

Bonferroni Critical Values (continued)



2

3

4

5

6

7

8

9

10

15

2 3 4 5

6.2053 4.1765 3.4954 3.1634

7.6488 4.8567 3.9608 3.5341

8.8602 5.3919 4.3147 3.8100

9.9248 5.8409 4.6041 4.0321

10.886 6.2315 4.8510 4.2193

11.769 6.5797 5.0675 4.3818

12.590 6.8952 5.2611 4.5257

13.360 7.1849 5.4366 4.6553

14.089 7.4533 5.5976 4.7733

17.277 8.5752 6.2541 5.2474

6 7 8 9 10

2.9687 2.8412 2.7515 2.6850 2.6338

3.2875 3.1276 3.0158 2.9333 2.8701

3.5212 3.3353 3.2060 3.1109 3.0382

3.7074 3.4995 3.3554 3.2498 3.1693

3.8630 3.6358 3.4789 3.3642 3.2768

3.9971 3.7527 3.5844 3.4616 3.3682

4.1152 3.8552 3.6766 3.5465 3.4477

4.2209 3.9467 3.7586 3.6219 3.5182

4.3168 4.0293 3.8325 3.6897 3.5814

4.6979 4.3553 4.1224 3.9542 3.8273

11 12 13 14 15

2.5931 2.5600 2.5326 2.5096 2.4899

2.8200 2.7795 2.7459 2.7178 2.6937

2.9809 2.9345 2.8961 2.8640 2.8366

3.1058 3.0545 3.0123 2.9768 2.9467

3.2081 3.1527 3.1070 3.0688 3.0363

3.2949 3.2357 3.1871 3.1464 3.1118

3.3702 3.3078 3.2565 3.2135 3.1771

3.4368 3.3714 3.3177 3.2727 3.2346

3.4966 3.4284 3.3725 3.3257 3.2860

3.7283 3.6489 3.5838 3.5296 3.4837

16 17 18 19 20

2.4729 2.4581 2.4450 2.4334 2.4231

2.6730 2.6550 2.6391 2.6251 2.6126

2.8131 2.7925 2.7745 2.7586 2.7444

2.9208 2.8982 2.8784 2.8609 2.8453

3.0083 2.9840 2.9627 2.9439 2.9271

3.0821 3.0563 3.0336 3.0136 2.9958

3.1458 3.1186 3.0948 3.0738 3.0550

3.2019 3.1735 3.1486 3.1266 3.1070

3.2520 3.2224 3.1966 3.1737 3.1534

3.4443 3.4102 3.3804 3.3540 3.3306

21 22 23 24 25

2.4138 2.4055 2.3979 2.3909 2.3846

2.6013 2.5912 2.5820 2.5736 2.5660

2.7316 2.7201 2.7097 2.7002 2.6916

2.8314 2.8188 2.8073 2.7969 2.7874

2.9121 2.8985 2.8863 2.8751 2.8649

2.9799 2.9655 2.9525 2.9406 2.9298

3.0382 3.0231 3.0095 2.9970 2.9856

3.0895 3.0737 3.0595 3.0465 3.0346

3.1352 3.1188 3.1040 3.0905 3.0782

3.3097 3.2909 3.2739 3.2584 3.2443

26 27 28 29 30

2.3788 2.3734 2.3685 2.3638 2.3596

2.5589 2.5525 2.5465 2.5409 2.5357

2.6836 2.6763 2.6695 2.6632 2.6574

2.7787 2.7707 2.7633 2.7564 2.7500

2.8555 2.8469 2.8389 2.8316 2.8247

2.9199 2.9107 2.9023 2.8945 2.8872

2.9752 2.9656 2.9567 2.9485 2.9409

3.0237 3.0137 3.0045 2.9959 2.9880

3.0669 3.0565 3.0469 3.0380 3.0298

3.2313 3.2194 3.2084 3.1982 3.1888

35 40 45 50 55

2.3420 2.3289 2.3189 2.3109 2.3044

2.5145 2.4989 2.4868 2.4772 2.4694

2.6334 2.6157 2.6021 2.5913 2.5825

2.7238 2.7045 2.6896 2.6778 2.6682

2.7966 2.7759 2.7599 2.7473 2.7370

2.8575 2.8355 2.8187 2.8053 2.7944

2.9097 2.8867 2.8690 2.8550 2.8436

2.9554 2.9314 2.9130 2.8984 2.8866

2.9960 2.9712 2.9521 2.9370 2.9247

3.1502 3.1218 3.1000 3.0828 3.0688

60 70 80 90 100

2.2990 2.2906 2.2844 2.2795 2.2757

2.4630 2.4529 2.4454 2.4395 2.4349

2.5752 2.5639 2.5554 2.5489 2.5437

2.6603 2.6479 2.6387 2.6316 2.6259

2.7286 2.7153 2.7054 2.6978 2.6918

2.7855 2.7715 2.7610 2.7530 2.7466

2.8342 2.8195 2.8086 2.8002 2.7935

2.8768 2.8615 2.8502 2.8414 2.8344

2.9146 2.8987 2.8870 2.8779 2.8707

3.0573 3.0393 3.0259 3.0156 3.0073

110 120 250 500 1000

2.2725 2.2699 2.2550 2.2482 2.2448

2.4311 2.4280 2.4102 2.4021 2.3980

2.5394 2.5359 2.5159 2.5068 2.5022

2.6213 2.6174 2.5956 2.5857 2.5808

2.6868 2.6827 2.6594 2.6488 2.6435

2.7414 2.7370 2.7124 2.7012 2.6957

2.7880 2.7835 2.7577 2.7460 2.7402

2.8287 2.8240 2.7972 2.7850 2.7790

2.8648 2.8599 2.8322 2.8195 2.8133

3.0007 2.9951 2.9637 2.9494 2.9423

2.2414

2.3940

2.4977

2.5758

2.6383

2.6901

2.7344

2.7729

2.8070

2.9352

75


Table 15

Bonferroni Critical Values (continued)



2

3

4

5

6

7

8

9

10

15

2 3 4 5

14.089 7.4533 5.5976 4.7733

17.277 8.5752 6.2541 5.2474

19.962 9.4649 6.7583 5.6042

22.327 10.215 7.1732 5.8934

24.464 10.869 7.5287 6.1384

26.429 11.453 7.8414 6.3518

28.258 11.984 8.1216 6.5414

29.975 12.471 8.3763 6.7126

31.599 12.924 8.6103 6.8688

38.710 14.819 9.5679 7.4990

6 7 8 9 10

4.3168 4.0293 3.8325 3.6897 3.5814

4.6979 4.3553 4.1224 3.9542 3.8273

4.9807 4.5946 4.3335 4.1458 4.0045

5.2076 4.7853 4.5008 4.2968 4.1437

5.3982 4.9445 4.6398 4.4219 4.2586

5.5632 5.0815 4.7590 4.5288 4.3567

5.7090 5.2022 4.8636 4.6224 4.4423

5.8399 5.3101 4.9570 4.7058 4.5184

5.9588 5.4079 5.0413 4.7809 4.5869

6.4338 5.7954 5.3737 5.0757 4.8547

11 12 13 14 15

3.4966 3.4284 3.3725 3.3257 3.2860

3.7283 3.6489 3.5838 3.5296 3.4837

3.8945 3.8065 3.7345 3.6746 3.6239

4.0247 3.9296 3.8520 3.7874 3.7328

4.1319 4.0308 3.9484 3.8798 3.8220

4.2232 4.1169 4.0302 3.9582 3.8975

4.3028 4.1918 4.1013 4.0263 3.9630

4.3735 4.2582 4.1643 4.0865 4.0209

4.4370 4.3178 4.2208 4.1405 4.0728

4.6845 4.5496 4.4401 4.3495 4.2733

16 17 18 19 20

3.2520 3.2224 3.1966 3.1737 3.1534

3.4443 3.4102 3.3804 3.3540 3.3306

3.5805 3.5429 3.5101 3.4812 3.4554

3.6862 3.6458 3.6105 3.5794 3.5518

3.7725 3.7297 3.6924 3.6595 3.6303

3.8456 3.8007 3.7616 3.7271 3.6966

3.9089 3.8623 3.8215 3.7857 3.7539

3.9649 3.9165 3.8744 3.8373 3.8044

4.0150 3.9651 3.9216 3.8834 3.8495

4.2084 4.1525 4.1037 4.0609 4.0230

21 22 23 24 25

3.1352 3.1188 3.1040 3.0905 3.0782

3.3097 3.2909 3.2739 3.2584 3.2443

3.4325 3.4118 3.3931 3.3761 3.3606

3.5272 3.5050 3.4850 3.4668 3.4502

3.6043 3.5808 3.5597 3.5405 3.5230

3.6693 3.6448 3.6226 3.6025 3.5842

3.7255 3.7000 3.6770 3.6561 3.6371

3.7750 3.7487 3.7249 3.7033 3.6836

3.8193 3.7921 3.7676 3.7454 3.7251

3.9892 3.9589 3.9316 3.9068 3.8842

26 27 28 29 30

3.0669 3.0565 3.0469 3.0380 3.0298

3.2313 3.2194 3.2084 3.1982 3.1888

3.3464 3.3334 3.3214 3.3102 3.2999

3.4350 3.4210 3.4082 3.3962 3.3852

3.5069 3.4922 3.4786 3.4660 3.4544

3.5674 3.5520 3.5378 3.5247 3.5125

3.6197 3.6037 3.5889 3.5753 3.5626

3.6656 3.6491 3.6338 3.6198 3.6067

3.7066 3.6896 3.6739 3.6594 3.6460

3.8635 3.8446 3.8271 3.8110 3.7961

35 40 45 50 55

2.9960 2.9712 2.9521 2.9370 2.9247

3.1502 3.1218 3.1000 3.0828 3.0688

3.2577 3.2266 3.2028 3.1840 3.1688

3.3400 3.3069 3.2815 3.2614 3.2451

3.4068 3.3718 3.3451 3.3239 3.3068

3.4628 3.4263 3.3984 3.3763 3.3585

3.5110 3.4732 3.4442 3.4214 3.4029

3.5534 3.5143 3.4845 3.4609 3.4418

3.5911 3.5510 3.5203 3.4960 3.4764

3.7352 3.6906 3.6565 3.6297 3.6080

60 70 80 90 100

2.9146 2.8987 2.8870 2.8779 2.8707

3.0573 3.0393 3.0259 3.0156 3.0073

3.1562 3.1366 3.1220 3.1108 3.1018

3.2317 3.2108 3.1953 3.1833 3.1737

3.2927 3.2707 3.2543 3.2417 3.2317

3.3437 3.3208 3.3037 3.2906 3.2802

3.3876 3.3638 3.3462 3.3326 3.3218

3.4260 3.4015 3.3833 3.3693 3.3582

3.4602 3.4350 3.4163 3.4019 3.3905

3.5901 3.5622 3.5416 3.5257 3.5131

110 120 250 500 1000

2.8648 2.8599 2.8322 2.8195 2.8133

3.0007 2.9951 2.9637 2.9494 2.9423

3.0945 3.0885 3.0543 3.0387 3.0310

3.1660 3.1595 3.1232 3.1066 3.0984

3.2235 3.2168 3.1785 3.1612 3.1526

3.2717 3.2646 3.2248 3.2067 3.1977

3.3130 3.3057 3.2644 3.2457 3.2365

3.3491 3.3416 3.2991 3.2798 3.2703

3.3812 3.3735 3.3299 3.3101 3.3003

3.5028 3.4943 3.4462 3.4245 3.4137

2.8070

2.9352

3.0233

3.0902

3.1440

3.1888

3.2272

3.2608

3.2905

3.4029

76


Table 16

Upper Critical Values of the Studentized Range : q (r, )

= .10 r 2

3

4

5

6

7

8

9

10

1 2 3 4 5

8.929 4.130 3.328 3.015 2.850

13.44 5.733 4.467 3.976 3.717

16.36 6.773 5.199 4.586 4.264

18.49 7.538 5.738 5.035 4.664

20.15 8.139 6.162 5.388 4.979

21.51 8.633 6.511 5.679 5.238

22.64 9.049 6.806 5.926 5.458

23.62 9.409 7.062 6.139 5.648

24.48 9.725 7.287 6.327 5.816

6 7 8 9 10

2.748 2.680 2.630 2.592 2.563

3.559 3.451 3.374 3.316 3.270

4.065 3.931 3.834 3.761 3.704

4.435 4.280 4.169 4.084 4.018

4.726 4.555 4.431 4.337 4.264

4.966 4.780 4.646 4.545 4.465

5.168 4.972 4.829 4.721 4.636

5.344 5.137 4.987 4.873 4.783

5.499 5.283 5.126 5.007 4.913

11 12 13 14 15

2.540 2.521 2.505 2.491 2.479

3.234 3.204 3.179 3.158 3.140

3.658 3.621 3.589 3.563 3.540

3.965 3.922 3.885 3.854 3.828

4.205 4.156 4.116 4.081 4.052

4.401 4.349 4.305 4.267 4.235

4.568 4.511 4.464 4.424 4.390

4.711 4.652 4.602 4.560 4.524

4.838 4.776 4.724 4.680 4.641

16 17 18 19 20

2.469 2.460 2.452 2.445 2.439

3.124 3.110 3.098 3.087 3.078

3.520 3.503 3.488 3.474 3.462

3.804 3.784 3.767 3.751 3.736

4.026 4.004 3.984 3.966 3.950

4.207 4.183 4.161 4.142 4.124

4.360 4.334 4.311 4.290 4.271

4.492 4.464 4.440 4.418 4.398

4.608 4.579 4.554 4.531 4.510

24 30 40 60 120

2.420 2.400 2.381 2.363 2.344 2.326

3.047 3.017 2.988 2.959 2.930 2.902

3.423 3.386 3.349 3.312 3.276 3.240

3.692 3.648 3.605 3.562 3.520 3.478

3.900 3.851 3.803 3.755 3.707 3.661

4.070 4.016 3.963 3.911 3.859 3.808

4.213 4.155 4.099 4.042 3.987 3.931

4.336 4.275 4.215 4.155 4.096 4.037

4.445 4.381 4.317 4.254 4.191 4.129

r 11

12

13

14

15

16

17

18

19

20

1 2 3 4 5

25.24 10.01 7.487 6.495 5.966

25.92 10.26 7.667 6.645 6.101

26.54 10.49 7.832 6.783 6.223

27.10 10.70 7.982 6.909 6.336

27.62 10.89 8.120 7.025 6.440

28.10 11.07 8.249 7.133 6.536

28.54 11.24 8.368 7.233 6.626

28.96 11.39 8.479 7.327 6.710

29.35 11.54 8.584 7.414 6.789

29.71 11.68 8.683 7.497 6.863

6 7 8 9 10

5.637 5.413 5.250 5.127 5.029

5.762 5.530 5.362 5.234 5.134

5.875 5.637 5.464 5.333 5.229

5.979 5.735 5.558 5.423 5.317

6.075 5.826 5.644 5.506 5.397

6.164 5.910 5.724 5.583 5.472

6.247 5.988 5.799 5.655 5.542

6.325 6.061 5.869 5.723 5.607

6.398 6.130 5.935 5.786 5.668

6.467 6.195 5.997 5.846 5.726

11 12 13 14 15

4.951 4.886 4.832 4.786 4.746

5.053 4.986 4.930 4.882 4.841

5.146 5.077 5.019 4.970 4.927

5.231 5.160 5.100 5.050 5.006

5.309 5.236 5.176 5.124 5.079

5.382 5.308 5.245 5.192 5.147

5.450 5.374 5.311 5.256 5.209

5.514 5.436 5.372 5.316 5.269

5.573 5.495 5.429 5.373 5.324

5.630 5.550 5.483 5.426 5.376

16 17 18 19 20

4.712 4.682 4.655 4.631 4.609

4.805 4.774 4.746 4.721 4.699

4.890 4.858 4.829 4.803 4.780

4.968 4.935 4.905 4.879 4.855

5.040 5.005 4.975 4.948 4.924

5.107 5.071 5.040 5.012 4.987

5.169 5.133 5.101 5.073 5.047

5.227 5.190 5.158 5.129 5.103

5.282 5.244 5.211 5.182 5.155

5.333 5.295 5.262 5.232 5.205

24 30 40 60 120

4.541 4.474 4.408 4.342 4.276 4.211

4.628 4.559 4.490 4.421 4.353 4.285

4.708 4.635 4.564 4.493 4.422 4.351

4.780 4.706 4.632 4.558 4.485 4.412

4.847 4.770 4.695 4.619 4.543 4.468

4.909 4.830 4.752 4.675 4.597 4.519

4.966 4.886 4.807 4.727 4.647 4.568

5.021 4.939 4.857 4.775 4.694 4.612

5.071 4.988 4.905 4.821 4.738 4.654

5.119 5.034 4.949 4.864 4.779 4.694

77


Table 16

Upper Critical Values of the Studentized Range : q (r, ) (continued)

= .05 r 2

3

4

5

6

7

8

9

10

1 2 3 4 5

17.97 6.085 4.501 3.927 3.635

26.98 8.331 5.910 5.040 4.602

32.82 9.798 6.825 5.757 5.218

37.08 10.88 7.502 6.287 5.673

40.41 11.74 8.037 6.707 6.033

43.12 12.44 8.478 7.053 6.330

45.40 13.03 8.853 7.347 6.582

47.36 13.54 9.177 7.602 6.802

49.07 13.99 9.462 7.826 6.995

6 7 8 9 10

3.461 3.344 3.261 3.199 3.151

4.339 4.165 4.041 3.949 3.877

4.896 4.681 4.529 4.415 4.327

5.305 5.060 4.886 4.756 4.654

5.628 5.359 5.167 5.024 4.912

5.895 5.606 5.399 5.244 5.124

6.122 5.815 5.597 5.432 5.305

6.319 5.998 5.767 5.595 5.461

6.493 6.158 5.918 5.739 5.599

11 12 13 14 15

3.113 3.082 3.055 3.033 3.014

3.820 3.773 3.735 3.702 3.674

4.256 4.199 4.151 4.111 4.076

4.574 4.508 4.453 4.407 4.367

4.823 4.751 4.690 4.639 4.595

5.028 4.950 4.885 4.829 4.782

5.202 5.119 5.049 4.990 4.940

5.353 5.265 5.192 5.131 5.077

5.487 5.395 5.318 5.254 5.198

16 17 18 19 20

2.998 2.984 2.971 2.960 2.950

3.649 3.628 3.609 3.593 3.578

4.046 4.020 3.997 3.977 3.958

4.333 4.303 4.277 4.253 4.232

4.557 4.524 4.495 4.469 4.445

4.741 4.705 4.673 4.645 4.620

4.897 4.858 4.824 4.794 4.768

5.031 4.991 4.956 4.924 4.896

5.150 5.108 5.071 5.038 5.008

24 30 40 60 120

2.919 2.888 2.858 2.829 2.800 2.772

3.532 3.486 3.442 3.399 3.356 3.314

3.901 3.845 3.791 3.737 3.685 3.633

4.166 4.102 4.039 3.977 3.917 3.858

4.373 4.302 4.232 4.163 4.096 4.030

4.541 4.464 4.389 4.314 4.241 4.170

4.684 4.602 4.521 4.441 4.363 4.286

4.807 4.720 4.635 4.550 4.468 4.387

4.915 4.824 4.735 4.646 4.560 4.474

r 11

12

13

14

15

16

17

18

19

20

1 2 3 4 5

50.59 14.39 9.717 8.027 7.168

51.96 14.75 9.946 8.208 7.324

53.20 15.08 10.15 8.373 7.466

54.33 15.38 10.35 8.525 7.596

55.36 15.65 10.53 8.664 7.717

56.32 15.91 10.69 8.794 7.828

57.22 16.14 10.84 8.914 7.932

58.04 16.37 10.98 9.028 8.030

58.83 16.57 11.11 9.134 8.122

59.56 16.77 11.24 9.233 8.208

6 7 8 9 10

6.649 6.302 6.054 5.867 5.722

6.789 6.431 6.175 5.983 5.833

6.917 6.550 6.287 6.089 5.935

7.034 6.658 6.389 6.186 6.028

7.143 6.759 6.483 6.276 6.114

7.244 6.852 6.571 6.359 6.194

7.338 6.939 6.653 6.437 6.269

7.426 7.020 6.729 6.510 6.339

7.508 7.097 6.802 6.579 6.405

7.587 7.170 6.870 6.644 6.467

11 12 13 14 15

5.605 5.511 5.431 5.364 5.306

5.713 5.615 5.533 5.463 5.404

5.811 5.710 5.625 5.554 5.493

5.901 5.798 5.711 5.637 5.574

5.984 5.878 5.789 5.714 5.649

6.062 5.953 5.862 5.786 5.720

6.134 6.023 5.931 5.852 5.785

6.202 6.089 5.995 5.915 5.846

6.265 6.151 6.055 5.974 5.904

6.326 6.209 6.112 6.029 5.958

16 17 18 19 20

5.256 5.212 5.174 5.140 5.108

5.352 5.307 5.267 5.231 5.199

5.439 5.392 5.352 5.315 5.282

5.520 5.471 5.429 5.391 5.357

5.593 5.544 5.501 5.462 5.427

5.662 5.612 5.568 5.528 5.493

5.727 5.675 5.630 5.589 5.553

5.786 5.734 5.688 5.647 5.610

5.843 5.790 5.743 5.701 5.663

5.897 5.842 5.794 5.752 5.714

24 30 40 60 120

5.012 4.917 4.824 4.732 4.641 4.552

5.099 5.001 4.904 4.808 4.714 4.622

5.179 5.077 4.977 4.878 4.781 4.685

5.251 5.147 5.044 4.942 4.842 4.743

5.319 5.211 5.106 5.001 4.898 4.796

5.381 5.271 5.163 5.056 4.950 4.845

5.439 5.327 5.216 5.107 4.998 4.891

5.494 5.379 5.266 5.154 5.044 4.934

5.545 5.429 5.313 5.199 5.086 4.974

5.594 5.475 5.358 5.241 5.126 5.012

78


Table 16

Upper Critical Values of the Studentized Range : q (r, ) (continued)

= .01 r 2

3

4

5

6

7

8

9

10

1 2 3 4 5

90.03 14.04 8.261 6.512 5.702

135.0 19.02 10.62 8.120 6.976

164.3 22.29 12.17 9.173 7.804

185.6 24.72 13.33 9.958 8.421

202.2 26.63 14.24 10.58 8.913

215.8 28.20 15.00 11.10 9.321

227.2 29.53 15.64 11.55 9.669

237.0 30.68 16.20 11.93 9.972

245.6 31.69 16.69 12.27 10.24

6 7 8 9 10

5.243 4.949 4.746 4.596 4.482

6.331 5.919 5.635 5.428 5.270

7.033 6.543 6.204 5.957 5.769

7.556 7.005 6.625 6.348 6.136

7.973 7.373 6.960 6.658 6.428

8.318 7.679 7.237 6.915 6.669

8.613 7.939 7.474 7.134 6.875

8.869 8.166 7.681 7.325 7.055

9.097 8.368 7.863 7.495 7.213

11 12 13 14 15

4.392 4.320 4.260 4.210 4.168

5.146 5.046 4.964 4.895 4.836

5.621 5.502 5.404 5.322 5.252

5.970 5.836 5.727 5.634 5.556

6.247 6.101 5.981 5.881 5.796

6.476 6.321 6.192 6.085 5.994

6.672 6.507 6.372 6.258 6.162

6.842 6.670 6.528 6.409 6.309

6.992 6.814 6.667 6.543 6.439

16 17 18 19 20

4.131 4.099 4.071 4.046 4.024

4.786 4.742 4.703 4.670 4.639

5.192 5.140 5.094 5.054 5.018

5.489 5.430 5.379 5.334 5.294

5.722 5.659 5.603 5.554 5.510

5.915 5.847 5.788 5.735 5.688

6.079 6.007 5.944 5.889 5.839

6.222 6.147 6.081 6.022 5.970

6.349 6.270 6.201 6.141 6.087

24 30 40 60 120

3.956 3.889 3.825 3.762 3.702 3.643

4.546 4.455 4.367 4.282 4.200 4.120

4.907 4.799 4.696 4.595 4.497 4.403

5.168 5.048 4.931 4.818 4.709 4.603

5.374 5.242 5.114 4.991 4.872 4.757

5.542 5.401 5.265 5.133 5.005 4.882

5.685 5.536 5.392 5.253 5.118 4.987

5.809 5.653 5.502 5.356 5.214 5.078

5.919 5.756 5.599 5.447 5.299 5.157

r 11

12

13

14

15

16

17

18

19

20

1 2 3 4 5

253.2 32.59 17.13 12.57 10.48

260.0 33.40 17.53 12.84 10.70

266.2 34.13 17.89 13.09 10.89

271.8 34.81 18.22 13.32 11.08

277.0 35.43 18.52 13.53 11.24

281.8 36.00 18.81 13.73 11.40

286.3 36.53 19.07 13.91 11.55

290.4 37.03 19.32 14.08 11.68

294.3 37.50 19.55 14.24 11.81

298.0 37.95 19.77 14.40 11.93

6 7 8 9 10

9.301 8.548 8.027 7.647 7.356

9.485 8.711 8.176 7.784 7.485

9.653 8.860 8.312 7.910 7.603

9.808 8.997 8.436 8.025 7.712

9.951 9.124 8.552 8.132 7.812

10.08 9.242 8.659 8.232 7.906

10.21 9.353 8.760 8.325 7.993

10.32 9.456 8.854 8.412 8.076

10.43 9.554 8.943 8.495 8.153

10.54 9.646 9.027 8.573 8.226

11 12 13 14 15

7.128 6.943 6.791 6.664 6.555

7.250 7.060 6.903 6.772 6.660

7.362 7.167 7.006 6.871 6.757

7.465 7.265 7.101 6.962 6.845

7.560 7.356 7.188 7.047 6.927

7.649 7.441 7.269 7.126 7.003

7.732 7.520 7.345 7.199 7.074

7.809 7.594 7.417 7.268 7.142

7.883 7.665 7.485 7.333 7.204

7.952 7.731 7.548 7.395 7.264

16 17 18 19 20

6.462 6.381 6.310 6.247 6.191

6.564 6.480 6.407 6.342 6.285

6.658 6.572 6.497 6.430 6.371

6.744 6.656 6.579 6.510 6.450

6.823 6.734 6.655 6.585 6.523

6.898 6.806 6.725 6.654 6.591

6.967 6.873 6.792 6.719 6.654

7.032 6.937 6.854 6.780 6.714

7.093 6.997 6.912 6.837 6.771

7.152 7.053 6.968 6.891 6.823

24 30 40 60 120

6.017 5.849 5.686 5.528 5.375 5.227

6.106 5.932 5.764 5.601 5.443 5.290

6.186 6.008 5.835 5.667 5.505 5.348

6.261 6.078 5.900 5.728 5.562 5.400

6.330 6.143 5.961 5.785 5.614 5.448

6.394 6.203 6.017 5.837 5.662 5.493

6.453 6.259 6.069 5.886 5.708 5.535

6.510 6.311 6.119 5.931 5.750 5.574

6.563 6.361 6.165 5.974 5.790 5.611

6.612 6.407 6.209 6.015 5.827 5.645

79


Table 17

Critical Values d for Multiple Comparisons based on the Kruskal-Wallis Test k = the number of groups (treatments) n = the common sample size of each group

Nominal Family Type I Error Rate (more accurate significance levels are in brackets) k

n

.10

.05

.03

.01

3 3 3 3 3

2 3 4 5 6

- 14 (.111) 21 (.106) 29 (.106) 38 (.103)

8 (.067) 15 (.064) 24 (.045) 33 (.048) 43 (.049)

8 (.067) 16 (.029) 25 (.031) 35 (.031) 46 (.030)

8 (.067) 17 (.011) 27 (.011) 39 (.009) 51 (.011)

4 4 4

2 3 4

11 (.086) 20 (.108) 31 (.097)

12 (.029) 22 (.043) 34 (.049)

12 (.029) 23 (.023) 36 (.026)

12 (.029) 24 (.012) 38 (.012)

5 5 5

2 3 4

14 (.121) 27 (.084) - -

15 (.048) 28 (.060) 44 (.056)

16 (.016) 30 (.023) 46 (.033)

16 (.016) 32 (.007) 50 (.010)

6 6

2 3

-

-

19 (.030) 35 (.055)

19 (.030) 37 (.024)

20 (.010) 39 (.009)

7 7

2 3

-

-

22 (.056) 42 (.054)

23 (.021) 44 (.026)

24 (.007) 46 (.012)

8 8

2 3

-

-

26 (.041) 49 (.055)

26 (.041) 51 (.029)

28 (.005) 54 (.010)

9 10 11 12 13 14 15

2 2 2 2 2 2 2

-

-

29 (.063) 33 (.050) 37 (.040) 40 (.062) 44 (.052) 48 (.044) 52 (.038)

30 (.031) 34 (.025) 38 (.020) 41 (.033) 45 (.028) 49 (.024) 52 (.038)

31 (.012) 35 (.009) 39 (.008) 43 (.006) 46 (.014) 50 (.012) 54 (.010)

80


Table 18

Critical Values e for Multiple Comparisons based on the Friedman Test

k = the number of treatments , b = the number of blocks

Nominal Family Type I Error Rate (more accurate significance levels are in brackets) k

b

.10

.05

.03

.01

3

3 4 5 6 7 8 9 10 11 12 13 14 15

6 (.028) 6 (.125) 7 (.093) 8 (.072) 8 (.112) 9 (.079) - - - - - - - -

6 (.028) 7 (.042) 8 (.039) 9 (.029) 9 (.051) 10 (.039) 10 (.048) 11 (.037) 11 (.049) 12 (.038) 12 (.049) 13 (.038) 13 (.047)

6 (.028) 7 (.042) 8 (.039) 9 (.029) 10 (.023) 11 (.018) 11 (.026) 12 (.019) 12 (.028) 13 (.022) 13 (.030) 14 (.023) 14 (.028)

6 (.028) 8 (.005) 9 (.008) 10 (.009) 11 (.008) 12 (.007) 12 (.013) 13 (.010) 14 (.008) 14 (.012) 15 (.009) 16 (.007) 16 (.010)

4

2 3 4 5 6 7 8 9 10 11 12 13 14 15

6 (.083) 8 (.049) 9 (.078) 10 (.082) 11 (.078) 11 (.126) 12 (.111) - - - - - - - -

6 (.083) 8 (.049) 10 (.026) 11 (.037) 12 (.037) 13 (.037) 14 (.034) 15 (.032) 15 (.046) 16 (.041) 17 (.038) 18 (.032) 18 (.042) 19 (.037)

6 (.083) 8 (.049) 10 (.026) 11 (.037) 13 (.018) 14 (.020) 15 (.019) 15 (.032) 16 (.029) 17 (.026) 18 (.023) 19 (.021) 19 (.028) 20 (.024)

6 (.083) 9 (.007) 11 (.005) 12 (.013) 14 (.006) 15 (.008) 16 (.009) 17 (.010) 18 (.010) 19 (.009) 20 (.008) 21 (.008) 21 (.011) 22 (.010)

5

2 3 4 5 6 7 8 9 10 11 12 13 14 15

8 (.050) 10 (.067) 11 (.116) 13 (.076) 14 (.088) 15 (.093) 16 (.094) - - - - - - - -

8 (.050) 10 (.067) 12 (.054) 14 (.040) 15 (.049) 16 (.052) 18 (.036) 19 (.037) 20 (.038) 21 (.038) 22 (.038) 23 (.035) 24 (.034) 24 (.045)

8 (.050) 11 (.018) 13 (.020) 14 (.040) 16 (.028) 17 (.033) 19 (.022) 20 (.024) 21 (.025) 22 (.025) 23 (.025) 24 (.024) 25 (.024) 26 (.022)

8 (.050) 12 (.002) 14 (.006) 16 (.006) 17 (.013) 19 (.009) 20 (.012) 22 (.008) 23 (.009) 24 (.010) 25 (.011) 26 (.011) 27 (.011) 28 (.010)

6

2 3 4 5 6 7 8 9 10 11 12 13 14 15

9 (.150) 12 (.084) 14 (.088) 16 (.075) 7 (.098) - - - - - - - - - -

10 (.033) 13 (.030) 15 (.047) 17 (.047) 19 (.040) 20 (.049) 22 (.039) 23 (.043) 24 (.047) 26 (.036) 27 (.039) 28 (.039) 29 (.040) 30 (.040)

10 (.033) 13 (.030) 16 (.018) 18 (.022) 20 (.021) 21 (.032) 23 (.026) 24 (.030) 26 (.023) 27 (.026) 28 (.028) 29 (.028) 30 (.030) 32 (.023)

10 (.033) 14 (.008) 17 (.006) 19 (.010) 21 (.010) 23 (.010) 25 (.008) 26 (.012) 28 (.009) 29 (.012) 31 (.009) 32 (.010) 33 (.011) 34 (.012)

81


Table 19

Upper Critical Values for Kendall's Rank Correlation Coefficient τˆ

Note: In the table below, the critical values give significance levels as close as possible to, but not exceeding the nominal . Nominal n

.10

.05

.025

.01

.005

.001

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1.000 0.800 0.600 0.524 0.429 0.389 0.378 0.345 0.303 0.308 0.275 0.276 0.250 0.250 0.242 0.228 0.221 0.210 0.203 0.202 0.196 0.193 0.188 0.179 0.180 0.172 0.172 0.166 0.165 0.163 0.159 0.156 0.152 0.150 0.149 0.147 0.144 0.141 0.141 0.138 0.137 0.135 0.132 0.132 0.129 0.129 0.127 0.126 0.124 0.123 0.122 0.121 0.119 0.118 0.117 0.116 0.115

1.000 0.800 0.733 0.619 0.571 0.500 0.467 0.418 0.394 0.359 0.363 0.333 0.317 0.309 0.294 0.287 0.274 0.267 0.264 0.257 0.246 0.240 0.237 0.231 0.228 0.222 0.218 0.213 0.210 0.205 0.201 0.197 0.194 0.192 0.189 0.188 0.185 0.180 0.178 0.176 0.173 0.172 0.169 0.167 0.167 0.163 0.162 0.161 0.158 0.157 0.156 0.154 0.152 0.152 0.149 0.148 0.147

1.000 0.867 0.714 0.643 0.556 0.511 0.491 0.455 0.436 0.407 0.390 0.383 0.368 0.346 0.333 0.326 0.314 0.307 0.296 0.290 0.287 0.280 0.271 0.265 0.261 0.255 0.252 0.246 0.242 0.237 0.234 0.232 0.228 0.223 0.220 0.218 0.215 0.213 0.209 0.207 0.204 0.202 0.199 0.197 0.196 0.192 0.191 0.189 0.187 0.185 0.182 0.181 0.179 0.177 0.176 0.174

1.000 0.867 0.810 0.714 0.667 0.600 0.564 0.545 0.513 0.473 0.467 0.433 0.426 0.412 0.392 0.379 0.371 0.359 0.352 0.341 0.333 0.329 0.322 0.312 0.310 0.301 0.295 0.290 0.288 0.280 0.277 0.273 0.267 0.263 0.260 0.256 0.254 0.250 0.247 0.243 0.240 0.239 0.236 0.232 0.230 0.228 0.225 0.223 0.221 0.219 0.216 0.214 0.212 0.210 0.209 0.207

1.000 0.905 0.786 0.722 0.644 0.600 0.576 0.564 0.516 0.505 0.483 0.471 0.451 0.439 0.421 0.410 0.394 0.391 0.377 0.367 0.360 0.356 0.344 0.340 0.333 0.325 0.323 0.314 0.312 0.304 0.302 0.297 0.292 0.287 0.285 0.280 0.275 0.274 0.268 0.267 0.264 0.260 0.257 0.253 0.251 0.249 0.246 0.244 0.241 0.239 0.236 0.234 0.232 0.230 0.228

1.000 0.857 0.833 0.778 0.709 0.667 0.641 0.604 0.581 0.567 0.544 0.529 0.509 0.495 0.486 0.472 0.455 0.449 0.440 0.428 0.419 0.413 0.404 0.393 0.389 0.379 0.375 0.369 0.361 0.359 0.351 0.346 0.341 0.338 0.334 0.329 0.324 0.321 0.317 0.314 0.310 0.307 0.303 0.300 0.297 0.294 0.290 0.287 0.285 0.282 0.279 0.276 0.274 0.272

82


Table 20

Upper Critical Values of Spearman's Rank Correlation Coefficient Rs

Note: In the table below, the critical values give significance levels as close as possible to, but not exceeding the nominal .

Nominal n

.10

.05

.025

.01

.005

.001

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1.000 0.800 0.657 0.571 0.524 0.483 0.455 0.427 0.406 0.385 0.367 0.354 0.341 0.328 0.317 0.309 0.299 0.292 0.284 0.278 0.271 0.265 0.259 0.255 0.250 0.245 0.240 0.236 0.232 0.229 0.225 0.222 0.219 0.215 0.212 0.210 0.207 0.204 0.202 0.199 0.197 0.194 0.192 0.190 0.188 0.186 0.184 0.182 0.180 0.179 0.177 0.175 0.174 0.172 0.171 0.169 0.168

1.000 0.900 0.829 0.714 0.643 0.600 0.564 0.536 0.503 0.484 0.464 0.446 0.429 0.414 0.401 0.391 0.380 0.370 0.361 0.353 0.344 0.337 0.331 0.324 0.318 0.312 0.306 0.301 0.296 0.291 0.287 0.283 0.279 0.275 0.271 0.267 0.264 0.261 0.257 0.254 0.251 0.248 0.246 0.243 0.240 0.238 0.235 0.233 0.231 0.228 0.226 0.224 0.222 0.220 0.218 0.216 0.214

1.000 0.886 0.786 0.738 0.700 0.648 0.618 0.587 0.560 0.538 0.521 0.503 0.488 0.472 0.460 0.447 0.436 0.425 0.416 0.407 0.398 0.390 0.383 0.375 0.368 0.362 0.356 0.350 0.345 0.340 0.335 0.330 0.325 0.321 0.317 0.313 0.309 0.305 0.301 0.298 0.294 0.291 0.288 0.285 0.282 0.279 0.276 0.274 0.271 0.268 0.266 0.264 0.261 0.259 0.257 0.255

1.000 0.943 0.893 0.833 0.783 0.745 0.709 0.678 0.648 0.626 0.604 0.582 0.566 0.550 0.535 0.522 0.509 0.497 0.486 0.476 0.466 0.457 0.449 0.441 0.433 0.425 0.419 0.412 0.405 0.400 0.394 0.388 0.383 0.378 0.373 0.368 0.364 0.359 0.355 0.351 0.347 0.343 0.340 0.336 0.333 0.329 0.326 0.323 0.320 0.317 0.314 0.311 0.308 0.306 0.303 0.301

1.000 0.929 0.881 0.833 0.794 0.755 0.727 0.703 0.679 0.654 0.635 0.618 0.600 0.584 0.570 0.556 0.544 0.532 0.521 0.511 0.501 0.492 0.483 0.475 0.467 0.459 0.452 0.446 0.439 0.433 0.427 0.421 0.415 0.410 0.405 0.400 0.396 0.391 0.386 0.382 0.378 0.374 0.370 0.366 0.363 0.359 0.356 0.352 0.349 0.346 0.343 0.340 0.337 0.334 0.331

1.000 0.952 0.917 0.879 0.845 0.818 0.791 0.771 0.750 0.729 0.711 0.692 0.675 0.662 0.647 0.633 0.621 0.609 0.597 0.586 0.576 0.567 0.558 0.549 0.540 0.532 0.525 0.517 0.510 0.503 0.497 0.491 0.485 0.479 0.473 0.468 0.462 0.457 0.452 0.448 0.443 0.439 0.434 0.430 0.426 0.422 0.418 0.414 0.411 0.407 0.404 0.400 0.397 0.394

83


Table 21

Sample Size for the T-test

Sample size n required to achieve power K(d) for d

=

µ − µ 0

(standard deviations from µ0). σ These are approximate n for the two-sided test.

One-sided

= .005

(Two-sided

= .01)

One-sided

Power K(d) = 1 -

= .01

(Two-sided

= .02)

Power K(d) = 1 -

d

.99

.95

.90

.80

d

.99

.95

.90

.80

d

0.25

389

290

243

191

0.25

350

256

212

164

0.25

0.30 0.35 0.40 0.45 0.50

272 201 155 123 100

202 150 115 92 75

170 125 97 77 63

134 99 77 62 51

0.30 0.35 0.40 0.45 0.50

244 180 139 110 90

179 132 101 81 66

148 109 85 68 55

115 85 66 53 43

0.30 0.35 0.40 0.45 0.50

0.55 0.60 0.65 0.70 0.75

83 71 61 53 47

63 53 46 40 36

53 45 39 34 30

42 36 31 28 25

0.55 0.60 0.65 0.70 0.75

75 63 55 47 42

55 47 41 35 31

46 39 34 30 27

36 31 27 24 21

0.55 0.60 0.65 0.70 0.75

0.80 0.85 0.90 0.95 1.00

41 37 34 31 28

32 29 26 24 22

27 24 22 20 19

22 20 18 17 16

0.80 0.85 0.90 0.95 1.00

37 33 29 27 25

28 25 23 21 19

24 21 19 18 16

19 17 16 14 13

0.80 0.85 0.90 0.95 1.00

1.10 1.20 1.30 1.40 1.50

24 21 18 16 15

19 16 15 13 12

16 14 13 12 11

14 12 11 10 9

1.10 1.20 1.30 1.40 1.50

21 18 16 14 13

16 14 13 11 10

14 12 11 10 9

12 10 9 9 8

1.10 1.20 1.30 1.40 1.50

1.60 1.70 1.80 1.90 2.00

13 12 12 11 10

11 10 10 9 8

10 9 9 8 8

8 8 8 7 7

1.60 1.70 1.80 1.90 2.00

12 11 10 10 9

10 9 8 8 7

9 8 7 7 7

7 7 7 6 6

1.60 1.70 1.80 1.90 2.00

84


Table 21

Sample Size for the T-test (continued)

Sample size n required to achieve power K(d) for d =

µ − µ 0

(standard deviations from µ0). σ These are approximate n for the two-sided test.

One-sided

= .025

(Two-sided

= .05)

One-sided

Power K(d) = 1 -

= .05

(Two-sided

= .1)

Power K(d) = 1 -

d

.99

.95

.90

.80

d

.99

.95

.90

.80

d

0.25

296

210

171

128

0.25

255

176

139

101

0.25

0.30 0.35 0.40 0.45 0.50

207 152 117 93 76

147 109 84 67 54

119 88 68 54 44

90 67 51 41 34

0.30 0.35 0.40 0.45 0.50

178 131 101 80 65

122 90 70 55 45

97 72 55 44 36

71 52 40 33 27

0.30 0.35 0.40 0.45 0.50

0.55 0.60 0.65 0.70 0.75

63 53 46 40 35

45 38 33 29 26

37 32 27 24 21

28 24 21 19 16

0.55 0.60 0.65 0.70 0.75

54 46 39 34 30

38 32 28 24 21

30 26 22 19 17

22 19 17 15 13

0.55 0.60 0.65 0.70 0.75

0.80 0.85 0.90 0.95 1.00

31 28 25 23 21

22 21 19 17 16

19 17 16 14 13

15 13 12 11 10

0.80 0.85 0.90 0.95 1.00

27 24 21 19 18

19 17 15 14 13

15 14 13 11 11

12 11 10 9 8

0.80 0.85 0.90 0.95 1.00

1.10 1.20 1.30 1.40 1.50

18 15 14 12 11

13 12 10 9 8

11 10 9 8 7

9 8 7 7 6

1.10 1.20 1.30 1.40 1.50

15 13 11 10 9

11 10 8 8 7

9 8 7 7 6

7 6 6 5 5

1.10 1.20 1.30 1.40 1.50

1.60 1.70 1.80 1.90 2.00

10 9 8 8 7

8 7 7 6 6

7 6 6 6 5

6 5 5 5 5

1.60 1.70 1.80 1.90 2.00

8 8 7 7 6

6 6 6 5 5

6 5 5 5 4

5 4 4 4 4

1.60 1.70 1.80 1.90 2.00

85


Table 22

Sample Size for the Pooled T-test

Sample size n X = n Y required to achieve power K(d) for d = These are approximate nX and n Y

One-sided

= .005

(Two-sided

= .01)

µ X

−

µ Y

(standard deviations apart). σ for the two-sided test.

One-sided

Power K(d) = 1 -

= .01

(Two-sided

= .02)

Power K(d) = 1 -

d

.99

.95

.90

.80

d

.99

.95

.90

.80

d

0.35 0.40 0.45 0.50

395 303 240 195

293 225 178 145

245 188 149 122

193 148 118 96

0.35 0.40 0.45 0.50

356 273 216 176

260 200 158 129

215 165 131 106

166 128 101 82

0.35 0.40 0.45 0.50

0.55 0.60 0.65 0.70 0.75

161 136 116 100 88

120 101 87 75 66

101 85 73 63 55

79 67 57 50 44

0.55 0.60 0.65 0.70 0.75

146 123 104 90 79

106 90 77 66 58

88 74 64 55 48

68 58 49 43 38

0.55 0.60 0.65 0.70 0.75

0.80 0.85 0.90 0.95 1.00

77 69 62 55 50

58 51 46 42 38

49 43 39 35 32

39 35 31 28 26

0.80 0.85 0.90 0.95 1.00

70 62 55 50 45

51 46 41 37 33

43 38 34 31 28

33 30 27 24 22

0.80 0.85 0.90 0.95 1.00

1.10 1.20 1.30 1.40 1.50

42 36 31 27 24

32 27 23 20 18

27 23 20 17 15

22 18 16 14 13

1.10 1.20 1.30 1.40 1.50

38 32 28 24 21

28 24 21 18 16

23 20 17 15 14

19 16 14 12 11

1.10 1.20 1.30 1.40 1.50

1.60 1.70 1.80 1.90 2.00

21 19 17 16 14

16 15 13 12 11

14 13 11 11 10

11 10 10 9 8

1.60 1.70 1.80 1.90 2.00

19 17 15 14 13

14 13 12 11 10

12 11 10 9 9

10 9 8 8 7

1.60 1.70 1.80 1.90 2.00

86


Table 22

Sample Size for the Pooled T-test (continued)

Sample size nX = n Y required to achieve power K(d) for d = These are approximate nX and n Y

One-sided

= .025

(Two-sided

= .05)

µ X

−

µ Y

(standard deviations apart). σ for the two-sided test.

One-sided

Power K(d) = 1 -

= .05

(Two-sided

= .10)

Power K(d) = 1 -

d

.99

.95

.90

.80

d

.99

.95

.90

.80

d

0.35 0.40 0.45 0.50

301 231 183 148

214 164 130 106

173 133 105 86

130 100 79 64

0.35 0.40 0.45 0.50

259 199 157 128

178 137 108 88

141 108 86 70

102 78 62 51

0.35 0.40 0.45 0.50

0.55 0.60 0.65 0.70 0.75

123 104 88 76 67

87 74 63 55 48

71 60 51 44 39

53 45 39 34 29

0.55 0.60 0.65 0.70 0.75

112 89 76 66 57

73 61 52 45 40

58 49 42 36 32

42 36 30 26 23

0.55 0.60 0.65 0.70 0.75

0.80 0.85 0.90 0.95 1.00

59 52 47 42 38

42 37 34 30 27

34 31 27 25 23

26 23 21 19 17

0.80 0.85 0.90 0.95 1.00

50 45 40 36 33

35 31 28 25 23

28 25 22 20 18

21 18 16 15 14

0.80 0.85 0.90 0.95 1.00

1.10 1.20 1.30 1.40 1.50

32 27 23 20 18

23 20 17 15 13

19 16 14 12 11

14 12 11 10 9

1.10 1.20 1.30 1.40 1.50

27 23 20 17 15

19 16 14 12 11

15 13 11 10 9

12 10 9 8 7

1.10 1.20 1.30 1.40 1.50

1.60 1.70 1.80 1.90 2.00

16 14 13 12 11

12 11 10 9 8

10 9 8 7 7

8 7 6 6 6

1.60 1.70 1.80 1.90 2.00

14 12 11 10 9

10 9 8 7 7

8 7 7 6 6

6 6 5 5 4

1.60 1.70 1.80 1.90 2.00

87


Table 23

Sample Size for One-Way ANOVA, Fixed Effects, Normal Model

Sample size n (for each group) required to achieve power K(d) for d

=

max ( µ i )

− min( µ i )

(standard deviations apart).

σ

( k = the number of gr oups )

K(d) = 1 – (d) = .70

K(d) = 1 – (d) = .80

d

=

max ( µ i )

− min ( µ i ) σ

k

1.00

1.25

1.50

1.75

2.00

2.50

3.00

1.00

1.25

1.50

1.75

2.00

2.50

3.00

.01

3 4 5 6 7 8 9 10

25 28 30 32 34 35 37 38

17 19 20 21 22 23 24 25

12 13 14 15 16 17 17 18

10 10 11 12 12 13 13 14

8 8 9 9 10 10 10 11

6 6 6 7 7 7 7 7

5 5 5 5 5 5 6 6

30 33 35 38 39 41 43 44

20 22 23 25 26 27 28 29

14 16 17 18 18 19 20 21

11 12 13 13 14 15 15 16

9 10 10 11 11 12 12 12

7 7 7 8 8 8 8 8

5 5 6 6 6 6 6 6

.05

3 4 5 6 7 8 9 10

17 19 21 22 24 25 26 27

11 13 14 15 16 16 17 18

8 9 10 11 11 12 12 13

7 7 8 8 9 9 9 10

5 6 6 7 7 7 8 8

4 4 5 5 5 5 5 6

3 4 4 4 4 4 4 4

21 23 25 27 29 30 31 33

14 15 17 18 19 20 21 21

10 11 12 13 14 14 15 15

8 9 9 10 10 11 11 12

6 7 7 8 8 9 9 9

5 5 5 6 6 6 6 6

4 4 4 4 5 5 5 5

.10

3 4 5 6 7 8 9 10

13 15 17 18 19 20 21 22

9 10 11 12 13 13 14 14

7 7 8 9 9 10 10 10

5 6 6 7 7 7 8 8

4 5 5 5 6 6 6 6

3 4 4 4 4 4 4 5

3 3 3 3 3 3 4 4

17 19 21 22 24 25 26 27

11 13 14 15 16 16 17 18

8 9 10 11 11 12 12 13

6 7 8 8 9 9 9 10

5 6 6 7 7 7 7 8

4 4 4 5 5 5 5 5

3 3 4 4 4 4 4 4

.20

3 4 5 6 7 8 9 10

9 11 12 13 14 15 15 16

6 7 8 9 9 10 10 11

5 5 6 6 7 7 7 8

4 4 5 5 5 6 6 6

3 4 4 4 4 5 5 5

3 3 3 3 3 3 3 4

2 2 3 3 3 3 3 3

12 14 16 17 18 19 20 21

8 9 10 11 12 12 13 14

6 7 8 8 9 9 9 10

5 5 6 6 7 7 7 8

4 4 5 5 5 6 6 6

3 3 4 4 4 4 4 4

3 3 3 3 3 3 3 3

88


Table 23

Sample Size for One-Way ANOVA, Fixed Effects, Normal Model (continued)

Sample size n (for each group) required to achieve power K(d) for d =

max ( µ i )

− min( µ i )

(standard deviations apart).

σ

( k = the number of groups )

K(d) = 1 - (d) = .90

K(d) = 1 - (d) = .95

d

=

max ( µ i )

− min ( µ i ) σ

k

1.00

1.25

1.50

1.75

2.00

2.50

3.00

1.00

1.25

1.50

1.75

2.00

2.50

3.00

.01

3 4 5 6 7 8 9 10

37 40 43 46 48 50 52 54

24 27 28 30 31 33 34 35

18 19 20 21 22 23 24 25

13 15 15 16 17 17 18 19

11 12 12 13 13 14 14 15

8 8 9 9 9 9 10 10

6 6 7 7 7 7 7 7

43 47 51 53 56 58 60 62

29 31 33 35 36 38 39 40

20 22 23 25 26 27 28 29

16 17 18 19 19 20 21 21

12 13 14 15 15 16 16 17

9 9 10 10 10 11 11 11

7 7 7 8 8 8 8 8

.05

3 4 5 6 7 8 9 10

27 30 32 34 36 38 40 41

18 20 21 23 24 25 26 27

13 14 15 16 17 18 18 19

10 11 12 12 13 13 14 14

8 9 9 10 10 11 11 11

6 6 6 7 7 7 8 8

5 5 5 5 5 6 6 6

32 36 39 41 43 45 47 48

21 23 25 27 28 29 30 31

15 17 18 19 20 21 22 22

12 13 14 14 15 16 16 17

9 10 11 11 12 12 13 13

7 7 7 8 8 8 9 9

5 5 6 6 6 6 6 7

.10

3 4 5 6 7 8 9 10

22 25 27 29 31 32 33 35

15 16 18 19 20 21 22 23

11 12 13 14 14 15 16 16

8 9 10 10 11 11 12 12

7 7 8 8 9 9 9 10

5 5 5 6 6 6 6 7

4 4 4 4 5 5 5 5

27 30 33 35 37 39 40 42

18 20 22 23 24 25 26 27

13 14 15 16 17 18 19 19

10 11 12 12 13 14 14 15

8 9 9 10 10 11 11 11

6 6 6 7 7 7 8 8

4 5 5 5 5 5 6 6

.20

3 4 5 6 7 8 9 10

17 20 21 22 24 26 27 28

11 13 14 15 16 17 17 18

8 9 10 11 11 12 13 13

6 7 8 8 9 9 9 10

5 6 6 7 7 7 8 8

4 4 4 5 5 5 5 5

3 3 4 4 4 4 4 4

22 25 27 29 30 32 33 34

14 16 18 19 20 21 22 22

10 12 13 13 14 15 15 16

8 9 10 10 11 11 12 12

6 7 8 8 8 9 9 9

5 5 5 6 6 6 6 6

4 4 4 4 4 5 5 5

89


Table 24

Table of Pseudo-random Permutations of Size 9

Column Row

1-5

6 - 10

1 - 15

16 - 20

21 - 25

26 - 30

1

6 4 8 7 5 3 2 9 1

7 1 9 6 8 3 5 2 4

9 2 4 1 7 5 8 3 6

2 3 7 1 9 8 4 5 6

5 7 6 8 3 9 4 1 2

3 4 1 6 5 2 9 7 8

9 4 1 5 2 6 8 7 3

8 6 5 4 2 3 9 7 1

9 4 7 3 5 1 8 2 6

9 2 7 3 1 5 8 4 6

2 1 6 5 4 7 9 8 3

3 4 7 1 6 9 5 8 2

1 4 8 3 5 9 6 7 2

1 6 7 9 8 3 5 4 2

8 9 3 7 1 6 5 2 4

1 8 5 3 4 2 6 9 7

4 8 1 5 6 3 7 9 2

1 6 4 5 8 7 2 3 9

8 3 2 5 4 6 1 9 7

9 7 5 8 6 4 2 1 3

6 5 8 1 4 7 2 3 9

7 6 3 9 5 8 4 2 1

5 7 2 4 9 1 8 3 6

9 4 6 5 7 8 3 1 2

2 4 3 7 9 8 5 1 6

6 5 9 3 8 7 1 4 2

8 2 3 4 1 7 9 6 5

7 5 4 1 8 6 9 2 3

3 2 4 8 9 1 5 6 7

7 3 9 5 4 2 6 8 1

2

6 7 1 2 5 4 9 3 8

6 1 9 4 2 5 8 3 7

9 3 7 1 4 5 6 2 8

6 5 4 1 3 7 2 8 9

3 6 7 2 5 8 1 4 9

5 6 3 9 1 2 4 7 8

9 3 2 4 5 6 7 8 1

4 1 3 8 2 7 5 9 6

9 5 4 6 3 7 1 2 8

3 9 7 2 5 6 1 8 4

5 4 6 7 1 8 2 3 9

3 6 9 4 7 2 1 5 8

8 7 5 9 6 1 4 3 2

7 6 1 8 3 4 5 2 9

1 3 8 7 4 6 5 9 2

8 7 1 3 5 4 9 2 6

9 4 6 2 3 1 7 8 5

4 2 3 7 9 8 1 6 5

2 6 5 9 3 1 7 4 8

7 8 6 9 5 3 2 1 4

6 7 2 9 5 3 4 1 8

3 6 9 5 1 8 4 7 2

1 4 8 3 7 2 5 6 9

9 8 2 7 6 3 4 1 5

1 4 5 8 9 2 7 3 6

2 4 6 3 5 7 9 8 1

2 7 1 6 3 4 9 5 8

1 3 2 7 8 9 6 4 5

3 2 5 9 7 4 8 1 6

1 2 5 3 4 7 9 6 8

3

9 3 4 2 8 6 1 5 7

9 2 7 1 3 4 8 6 5

7 6 8 5 3 1 9 2 4

3 8 4 2 6 5 7 9 1

4 3 5 1 6 2 9 8 7

5 4 9 3 7 2 1 6 8

1 6 4 5 9 3 8 7 2

5 2 3 8 4 9 1 6 7

1 2 6 8 5 7 3 4 9

2 6 8 1 4 5 9 7 3

6 3 4 8 2 7 9 1 5

6 7 2 1 4 3 9 8 5

9 2 5 1 7 3 8 6 4

7 9 5 2 6 1 8 3 4

2 6 5 8 1 4 3 7 9

3 5 7 2 4 9 6 8 1

3 4 7 1 5 9 2 6 8

8 9 6 3 1 4 2 7 5

3 2 6 5 8 7 1 9 4

2 8 1 9 6 5 4 3 7

6 4 8 1 7 2 3 9 5

2 9 3 1 5 7 4 6 8

3 7 4 9 8 1 5 6 2

6 8 1 9 3 2 7 4 5

9 6 8 4 5 3 2 1 7

2 6 4 9 5 8 7 3 1

3 5 6 7 4 1 9 8 2

3 2 7 5 8 6 1 9 4

9 7 6 3 4 5 1 8 2

6 1 5 3 9 4 7 8 2

4

9 4 3 5 2 8 1 7 6

8 6 2 5 1 4 7 9 3

3 5 1 2 8 4 6 7 9

7 6 3 8 4 9 5 1 2

9 8 2 3 6 7 4 5 1

8 1 3 2 5 9 4 7 6

6 5 8 3 1 9 7 2 4

3 5 4 8 1 6 9 2 7

2 7 5 9 1 8 4 6 3

5 7 2 3 1 4 9 8 6

2 8 4 6 7 9 1 3 5

1 7 9 2 3 4 6 5 8

3 5 1 4 9 6 7 2 8

5 1 4 8 9 2 7 6 3

5 4 1 9 6 7 8 2 3

2 6 7 5 9 3 4 8 1

3 9 8 2 6 4 7 5 1

3 8 4 7 2 9 6 1 5

1 9 5 6 4 8 7 2 3

4 8 1 2 5 3 6 9 7

8 6 3 7 5 1 2 9 4

6 5 9 3 7 2 8 4 1

1 4 7 3 9 6 5 8 2

3 8 9 4 2 5 7 1 6

5 1 4 6 7 2 8 9 3

7 5 4 1 3 6 9 2 8

7 9 6 5 4 2 8 3 1

3 5 7 8 4 9 6 1 2

2 3 8 4 7 9 1 5 6

7 2 4 8 9 6 1 3 5

5

7 8 2 3 1 4 6 5 9

8 6 3 4 2 7 1 9 5

4 3 8 2 5 1 7 9 6

6 8 3 4 9 7 5 2 1

4 9 5 8 6 2 3 7 1

1 8 7 5 6 4 2 3 9

8 6 3 5 7 4 1 2 9

3 8 2 4 7 6 1 9 5

8 7 1 2 3 4 6 9 5

6 9 2 3 4 1 8 7 5

5 3 7 6 8 4 2 1 9

5 8 7 6 3 1 4 9 2

2 5 1 4 6 7 9 3 8

7 8 5 2 1 9 4 3 6

7 6 5 1 3 4 9 8 2

6 9 8 4 3 1 7 2 5

6 4 3 8 5 2 7 9 1

5 6 7 3 4 1 2 9 8

9 3 7 6 8 5 2 1 4

5 8 9 1 3 7 4 6 2

4 8 1 3 7 2 9 5 6

3 4 8 7 2 6 1 9 5

4 3 6 7 2 8 9 5 1

7 4 2 9 6 5 1 3 8

2 8 5 6 7 3 4 1 9

9 3 6 4 1 2 8 7 5

3 5 1 2 7 4 9 8 6

7 4 6 9 5 8 1 2 3

6 8 7 4 3 1 5 9 2

5 2 7 6 4 1 8 9 3

6

7 4 8 3 5 6 1 9 2

7 1 6 8 2 9 3 5 4

9 4 6 3 1 7 8 5 2

6 3 9 1 8 7 5 2 4

1 5 8 6 4 7 9 2 3

6 3 2 9 8 4 1 5 7

2 8 3 7 6 9 5 4 1

6 5 7 8 4 2 3 9 1

2 1 9 7 3 8 4 5 6

8 5 2 7 4 1 3 6 9

4 3 6 9 5 1 8 7 2

3 8 6 7 2 5 4 1 9

1 4 5 3 7 8 9 2 6

3 8 1 4 5 2 6 9 7

8 1 2 5 7 3 6 4 9

2 6 3 1 5 9 7 8 4

1 3 9 5 7 8 2 4 6

7 4 8 5 3 9 1 6 2

8 2 1 7 6 9 5 4 3

1 3 8 4 5 6 2 7 9

8 7 5 4 3 9 2 6 1

7 9 3 1 2 5 8 4 6

5 6 2 7 8 1 4 3 9

4 3 6 8 5 1 7 9 2

3 5 4 6 7 8 1 2 9

6 8 5 7 3 2 1 9 4

7 9 1 5 2 4 8 6 3

1 4 7 9 2 6 3 5 8

3 2 6 4 5 8 9 7 1

9 2 7 8 3 1 4 6 5

90


Table 25

Table of Pseudo-random Digits

Column Line

1-5

6-10

11-15

16-20

21-25

26-30

31-35

36-40

41-45

46-50

1 2 3 4 5

08025 43025 58121 72630 99141

89354 77001 23577 20838 19701

66119 02272 36517 00723 25161

12852 89797 64643 12788 01414

61345 66588 17939 18794 17377

06509 39793 70833 68202 93195

02701 29896 93099 03931 09454

79607 44604 94744 72078 02719

80234 85341 91332 30915 24297

19263 92320 48159 65350 97428

6 7 8 9 10

61248 86046 92019 30784 80938

09975 50522 07352 55302 38097

46169 74470 93315 32329 53567

17215 30125 69852 40934 13720

34288 95272 07395 52882 36450

01118 65600 92103 85483 89775

14681 85604 08348 68921 08951

99216 28241 76089 27908 27155

85172 31735 55143 68489 27759

59198 15291 15496 19321 23319

11 12 13 14 15

18881 19610 77620 72537 51886

60759 77947 59682 17153 61216

71055 72653 58917 23011 82211

62716 47772 72290 02602 44246

97314 57484 72917 70210 55572

60851 48855 74614 45615 52746

26428 57222 41046 28454 44254

84725 13081 65690 32116 44267

66137 21690 45455 31278 10302

74139 28187 19840 97910 55859

16 17 18 19 20

05650 42153 71336 14063 45594

84958 38890 65235 55790 45992

74972 78980 56208 96174 25431

19326 10323 82491 79234 06642

36808 38457 69220 88520 94686

15344 85522 33615 08859 65565

05039 94858 07848 72512 23940

06076 26063 11847 88798 64499

23302 02393 72427 31954 34013

55065 83151 82828 44967 81164

21 22 23 24 25

54338 88713 23615 86160 97640

13992 83287 73976 06090 98709

94521 29349 21076 91947 22933

39220 99464 23476 43277 89872

20355 95499 09987 39543 16271

11344 92499 23591 32532 22821

32881 33074 58266 54269 17688

01038 27496 94120 14948 20347

42038 16658 40624 16269 35989

23483 14281 94174 56600 05584

26 27 28 29 30

06946 90210 84529 11644 35483

45370 59549 66423 41528 55588

50945 31365 78040 90086 23051

48067 80765 26211 02595 85364

44892 58726 32685 85984 05871

67609 71099 23461 13908 90463

01998 64022 56782 18227 51042

16973 96701 96864 63685 28996

29628 00752 35006 57142 07757

50044 24962 49181 77014 62484

31 32 33 34 35

49976 62819 62714 16347 89107

17369 48093 26321 54320 23856

59115 86554 44330 40744 92270

16680 46644 50378 47838 03999

10735 27786 10788 05193 87388

45136 28694 55520 94386 29279

66690 07566 69743 30284 44757

36683 42762 22297 47898 56034

20723 25516 17304 52647 69813

79252 99507 83174 42887 92039

36 37 38 39 40

70833 51177 70376 50686 95135

42469 86088 02710 25769 83236

63483 31614 13186 56334 90950

51138 23778 05649 09464 27914

95099 86117 18534 76655 20376

79174 89430 12061 92663 92565

33444 45737 29313 26480 48109

25899 88533 90739 43425 78847

38456 44982 01809 62041 62391

64150 40705 17546 49220 84305

41 42 43 44 45

86418 56531 68642 61921 10852

45001 43586 82406 51815 33613

47466 32988 43975 65068 38873

28022 98735 87638 70294 78795

84482 66483 68109 22748 85872

68299 47607 15218 62150 54927

34866 25798 11123 63238 04235

41402 49431 92261 67770 63069

28045 97092 75650 53421 18209

68438 04877 78929 08217 13667

46 47 48 49 50

44645 50604 58564 91656 13840

95861 64568 50497 11778 46431

12068 44880 01322 27049 63082

99406 94258 15771 47254 81615

96019 14329 94224 86342 05334

32551 16618 52629 44802 81525

73231 54920 71269 28823 41660

70296 37843 79096 35996 92580

63774 21385 73844 69714 81693

66304 77860 45855 92375 85560

91


92


Acknowledgements

Table 11:

From "A simple test of symmetry about an unknown median", by Paul Cabilio and Joe Masaro, Canadian Journal of Statistics, 24 (1996).

Table 12:

Adapted from "Use of the Correlation Coefficient with Normal Probability Plots" The American Statistician, 39 (1985) with permission of the authors, Stephen W. Looney and Thomas R. Gulledge.

Table 13:

Adapted from "Exact probability levels for the Kruskall-Wallis Test" by Ronald L. Iman, Dana Quade and Douglas A. Alexander, in Selected Tables in Mathematical Statistics, Volume 3, 1975, edited by H. L. Harter and D. B. Owen for the IMS, with permission of Ronald L. Iman and Dana Quade.

Table 14:

Adapted from "Technical Report #SW 16/72" of the Mathematical Centre, Amsterdam, with permission of the author Dana Quade.

Table 16:

Adapted from "Tables of range and studentized range", by H. Leon Harter, Ann. Math. Statist., 31 (1960) with permission of the author H. Leon Harter.

Tables 17-18:

Adapted from "Rank sum multiple comparisons in one- and two-way classifications", Biometrika, 54 (1967) with permission of the authors B. J. MacDonald and W. A. Thompson Jr.

Table 23:

Adapted from "Tables of sample sizes in the analysis of variance" by T. L. Bratcher, M. A. Moran and W. J. Zimmer, Journal of Quality Technology , 2 (1970), with permission of Tom Bratcher and W. J. Zimmer.

Tables 1-11, 15, 19, 20, 24, 25 were generated with the aid of honour students Steven Astels and Kathy Swinamer. Tables 21 and 22 were generated using an approximation found in "Introduction to Statistical Analysis" by Wilfred J. Dixon and Frank J. Massey. Jr.. 5th edition, McGraw-Hill, 1983. The values obtained were checked and corrected using the software package Statable by C.R. Mehta and N.R. Patel. Special thanks to Kathy Swinamer and Julia MacCluskey for typing of the manuscript.

93

Table Statistics

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