Statistics for Management and Economics

Statistics for Management and Economics, Eighth Edition Formulas

Numerical Descriptive techniques Population mean N

∑x

µ =

i

i =1

N Sample mean n

x=

∑x

i

i =1

n

Range Largest observation - Smallest observation Population variance N

σ2 =

∑( x − µ )

2

i

i =1

N Sample variance n

s2 =

∑(x i =1

i

− x)2

n −1 Population standard deviation

σ

=

σ2

Sample standard deviation s=

s2

Population covariance

N

σ xy =

∑( x

i

− µ x )( y i − µ y )

i =1

N

Sample covariance n

s xy =

∑( x

i

− x )( y i − y )

i =1

n −1

Population coefficient of correlation

ρ=

σ xy σ xσ y

Sample coefficient of correlation

r=

s xy sxsy

Coefficient of determination R2 = r2 Slope coefficient

b1 =

s xy s x2

y-intercept

b0 = y − b1 x Probability Conditional probability P(A|B) = P(A and B)/P(B) Complement rule P(

A C ) = 1 – P(A)

Multiplication rule P(A and B) = P(A|B)P(B)

Addition rule P(A or B) = P(A) + P(B) - P(A and B) Bayes’ Law Formula

P ( A i | B) =

P( A i ) P ( B | A i ) P( A 1 ) P ( B | A 1 ) + P( A 2 ) P( B | A 2 ) + . . . + P ( A k ) P( B | A k )

Random Variables and Discrete Probability Distributions Expected value (mean)

µ=

E(X) =

∑xP ( x )

all

x

Variance V(x) =

σ2 =

∑( x − µ )

2

all

P( x )

x

Standard deviation σ = σ2

Covariance COV(X, Y) = σxy =

∑( x −µ

Coefficient of Correlation

ρ=

COV ( X , Y )

σxσ y

Laws of expected value 1. E(c) = c 2. E(X + c) = E(X) + c 3. E(cX) = cE(X) Laws of variance 1.V(c) = 0 2. V(X + c) = V(X) 3. V(cX) = c 2 V(X)

x

)( y −µy )P( x , y )

Laws of expected value and variance of the sum of two variables 1. E(X + Y) = E(X) + E(Y) 2. V(X + Y) = V(X) + V(Y) + 2COV(X, Y) Laws of expected value and variance for the sum of more than two variables 1.

2.

k

k

i =1

i =1

k

k

i =1

i =1

E (∑ X i ) = ∑ E ( X i ) V (∑ X i ) = ∑V ( X i ) if the variables are independent

Mean and variance of a portfolio of two stocks E(Rp) = w1E(R1) + w2E(R2) V(Rp) = w12 V(R1) + w22 V(R2) + 2 w1 w 2 COV(R1, R2) = w12 σ12 + w22 σ22 + 2 w1 w 2 ρ σ1 σ2 Mean and variance of a portfolio of k stocks k

E(Rp) =

∑w E ( R ) i

i

i =1 k

V(Rp) =

∑w

2 2 i σi

i =1

k

+2

k

∑∑w w i

j COV

i =1 j =i +1

Binomial probability P(X = x) =

n! p x ( 1 − p ) n −x x! ( n − x )!

µ = np σ2 = np ( 1 − p ) σ = np ( 1 − p )

Poisson probability P(X = x) =

e −µµx x!

( Ri , R j )

Continuous Probability Distributions Standard normal random variable

X −µ

Z =

σ

Exponential distribution µ = σ =1 / λ

P( X > x ) = e −λx P( X < x ) =1 −e −λx

P ( x 1 < X < x 2 ) = P( X < x 2 ) − P( X < x 1 ) = e − λ x1 − e − λ x 2 F distribution

F1−A,ν1 ,ν2 =

1 FA,ν2 ,ν1

Sampling Distributions Expected value of the sample mean E( X ) =µx

=µ

Variance of the sample mean V ( X ) =σ 2x =

σ2 n

Standard error of the sample mean σx =

σ n

Standardizing the sample mean Z =

X −µ σ/

n

Expected value of the sample proportion ˆ ) = µ pˆ = p E (P

Variance of the sample proportion

p (1 − p ) V ( Pˆ ) = σ2pˆ = n Standard error of the sample proportion

p(1 − p) n

σ pˆ =

Standardizing the sample proportion Z =

ˆ −p P p (1 − p ) n

Expected value of the difference between two means E ( X 1 − X 2 ) = µ x1 −x2 = µ1 − µ 2

Variance of the difference between two means

V ( X 1 − X 2 ) = σ x21 − x2 =

σ 12 σ 22 + n1 n2

Standard error of the difference between two means

σ x1 − x2 =

σ12 σ 22 + n1 n 2

Standardizing the difference between two sample means

Z=

( X 1 − X 2 ) − ( µ1 − µ 2 )

σ 12 σ 22 + n1 n2

Introduction to Estimation Confidence interval estimator of µ x ± zα / 2

σ n

Sample size to estimate µ

z σ n =  α / 2  W

  

2

Introduction to Hypothesis Testing Test statistic for µ

z=

x −µ

σ/

n

Inference about One Population Test statistic for µ

t=

x −µ s/

n

Confidence interval estimator of µ s

x ± tα / 2

n

Test statistic for σ2

χ2 =

( n −1 )s 2

σ2

Confidence interval Estimator of σ2 LCL =

UCL =

( n −1 )s 2

χα2 / 2 ( n −1 )s 2

χ12−α / 2

Test statistic for p z=

ˆp − p p( 1 − p ) / n

Confidence interval estimator of p ˆp ± zα / 2

ˆp( 1 − ˆp ) / n

Sample size to estimate p

2

z ˆp( 1 − ˆp )   n = α/ 2   W  

Confidence interval estimator of the total of a large finite population  s  N x ± t α / 2  n 

Confidence interval estimator of the total number of successes in a large finite population  N  ˆp ± zα / 2  

ˆp( 1 − ˆp   n  

Confidence interval estimator of µ when the population is small s

x ± tα / 2

n

N −n N −1

Confidence interval estimator of the total in a small population  s N  x ± tα / 2  n 

N −n N −1

   

Confidence interval estimator of p when the population is small

 p ± zα / 2

ˆp( 1 − ˆp ) n

N −n N −1

Confidence interval estimator of the total number of successes in a small population  N  ˆp ± z α / 2  

ˆp( 1 − ˆp ) n

N −n N −1

   

Inference About Two Populations Equal-variances t-test of µ1 − µ2

t=

( x1 − x 2 ) − ( µ1 − µ 2 )  1 1   s 2p  +  n1 n 2 

ν = n1 + n 2 − 2

Equal-variances interval estimator of µ1 − µ2

 1 1   ν = n1 + n 2 − 2 ( x1 − x 2 ) ± t α / 2 s 2p  +  n1 n 2  Unequal-variances t-test of µ1 − µ2

t=

( x1 − x 2 ) − ( µ 1 − µ 2 )  s12 s 22   +   n1 n 2   

( s12 / n1 + s22 / n2 )2 ν = 2 ( s1 / n1 )2 ( s22 / n2 )2 + n1 − 1 n2 − 1

Unequal-variances interval estimator of µ1 − µ2

( x1 − x 2 ) ± t α / 2

s12 s 22 + n1 n 2

ν =

( s12 / n1 + s22 / n2 )2 ( s12 / n1 )2 ( s22 / n2 )2 + n1 − 1 n2 − 1

t-Test of µD

t=

xD − µ D

ν = nD −1

s D / nD

t-Estimator of µD

x D ± tα / 2

sD

ν = nD −1

nD

F-test of σ12 / σ22

s12 F= 2 s2

ν 1 = n1 − 1 and ν 2 = n 2 − 1

F-Estimator of σ12 / σ22

 s2 

1 1 LCL =  2   s  Fα / 2 ,ν ,ν  2 1 2  s12 UCL =  2 s  2

  Fα / 2 ,ν ,ν 2 1  

z-Test and estimator of p1 − p 2

( ˆp1 − ˆp 2 )

z=

 1 1  ˆp( 1 − ˆp ) +  n  1 n2 

Case 1:

z=

Case 2:

( ˆp1 − ˆp2 ) − ( p1 − p2 ) ˆp1( 1 − ˆp1 ) ˆp2 ( 1 − ˆp2 ) + n1 n2

z-estimator of p1 − p 2

ˆp1( 1 − ˆp1 ) ˆp2 ( 1 − ˆp2 ) + n1 n2

( ˆp1 − ˆp2 ) ± zα / 2

Analysis of Variance One-way analysis of variance k

∑n ( x

SST =

j

− x )2

j

j =1

k

nj

∑ ∑( x

SSE =

j =1

i =1

MST =

SST k −1

MSE =

SSE n −k

F=

− x j )2

ij

MST MSE

Two-way analysis of variance (randomized block design of experiment) k

SS(Total) =

b

∑ ∑( x j =1

− x )2

i =1

k

SST =

ij

∑b( x [ T ]

j

− x )2

i =1 b

SSB =

∑k ( x [ B ]

i

i =1

− x )2

k

SSE =

b

∑ ∑( x

ij

j =1

− x [ T ] j − x [ B ]i + x )2

i =1

MST =

SST k −1

MSB =

SSB b −1

MSE =

SSE n − k − b +1

F=

MST MSE

F=

MSB MSE

Two-factor experiment a

SS(Total) =

b

r

∑∑∑( x

ijk

− x )2

i =1 j =1 k =1

a

SS(A) = rb

∑( x [ A ]

i

− x )2

j

− x )2

i =1 b

SS(B) = ra

∑( x [ B ] j =1 a

SS(AB) = r

b

∑∑( x [ AB ]

ij

− x [ A ]i − x [ B ] j + x )2

i =1 j =1

a

SSE =

b

r

∑∑∑( x

ijk

− x [ AB ]ij )2

i =1 j =1 k =1

F=

MS ( A) MSE

F=

MS ( B) MSE

F=

MS ( AB ) MSE

Least Significant Difference Comparison Method

 1 1   +  ni  n j  

LSD = t α / 2 MSE 

Tukey’s multiple comparison method

ω = qα ( k ,ν )

MSE ng

Chi-Squared Tests Test statistic for all procedures k

∑

χ2 =

i =1

(f i − e i ) 2 ei

Simple Linear Regression Sample slope

b1 =

s xy s x2

Sample y-intercept

b0 = y − b1 x Sum of squares for error n

SSE =

∑( y

i

− ˆy i ) 2

i =1

Standard error of estimate sε =

SSE n −2

Test statistic for the slope

t=

b1 − β1 s b1

Standard error of b1

s b1 =

sε ( n −1 )s x2

Coefficient of determination

R2 =

2 s xy

s x2 s 2y

=1 −

SSE

∑( y

i

− y )2

Prediction interval

ˆy ± t α / 2 ,n −2 s ε 1 +

2 1 ( xg − x ) + n ( n − 1 )s x2

Confidence interval estimator of the expected value of y

ˆy ± t α / 2 ,n −2 s ε

2 1 ( xg − x ) + n ( n − 1 )s x2

Sample coefficient of correlation

r=

s xy sxsy

Test statistic for testing ρ = 0

t =r

n −2 1−r 2

Multiple Regression Standard Error of Estimate sε =

SSE n − k −1

Test statistic for βi

t=

bi − β i s bi

Coefficient of Determination 2 s xy

2

R =

=1 −

s x2 s 2y

SSE

∑( y

− y )2

i

Adjusted Coefficient of Determination 2 Adjusted R = 1 −

SSE /( n − k −1 )

∑( y

i

− y )2 /( n −1 )

Mean Square for Error MSE = SSE/k Mean Square for Regression MSR = SSR/(n-k-1) F-statistic F = MSR/MSE

Durbin-Watson statistic n

d=

∑( e

i

− ei −1 ) 2

i =2

n

∑e

2 i

i =1

Nonparametric Statistical techniques Wilcoxon rank sum test statistic

T = T1 E(T) =

σT = z=

n1 ( n1 + n 2 + 1 ) 2 n1 n 2 ( n1 + n 2 + 1 ) 12

T − E( T )

Sign test statistic

σT

x = number of positive differences x − .5 n

z=

.5 n

Wilcoxon signed rank sum test statistic

T =T + E(T) =

σT = z=

n( n +1 ) 4 n( n +1 )( 2n +1 ) 24

T − E( T )

σT

Kruskal-Wallis Test  12 H = n( n +1 )  

k

T j2   − 3( n +1 ) nj  

∑ j =1

Friedman Test  12 Fr =  b( k )( k +1 )  

 T j2  −3b( k +1 )  j =1  k

∑

Spearman rank correlation coefficient

rS =

s ab s a sb

Spearman Test statistic for n > 30 z = rS

n −1

Time Series Analysis and Forecasting Exponential smoothing

S t = wy t + (1 − w) S t −1

Statistical Process Control

Centerline and control limits for x chart using S Centerline = x S

Lower control limit = x − 3

Upper control limit = x + 3

n S n

Centerline and control limits for the p chart Centerline = p Lower control limit = p − 3

p( 1 − p ) n

Upper control limit = p + 3

p( 1 − p ) n

Decision Analysis Expected Value of perfect Information EVPI = EPPI - EMV* Expected Value of Sample Information EVSI = EMV' - EMV*