Testing of Hypothesis Hypothesis: A statistical hypothesis is an assumption that we make about a population parameter, which may or may not be true concerning one or more variables. According to Pr...
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Lesson note #
Statistical Inference Testing of Hypothesis Type I Error: Rejection of the null hypothesis when it is true is called a type I error.
Type II Error: Acceptance of the null hypothesis when it is false is called a type II error. Decision of the test for the Null The Null Hypothesis is Hypothesis True False Accept Correct decision Incorrect decision
Type II Error Reject
Incorrect decision .Type I Error
Correct decision
Test Concerning Mean One and Two tailed Tests: A test procedure is called a one tailed test procedure if the alternative hypothesis is one sided. The test will be two tailed if the alternative hypothesis is two sided.
Example: Let a specified value of population mean is 45. Construct the null and alternative hypothesis for the the following following questions; a) Do the the sam sampl plee data data pro provi vide de suf suffi fici cien entt evid eviden ence ce to to indi indica cate te that that the the popu popula lati tion on mean is greater than 45. H 0 : µ = 45 H A : µ > 45
b) b)
Do the the sam sampl plee data data provi provide de suf suffi fici cien entt evide evidenc ncee to ind indic icat atee that that the the popu popula lati tion on mean is less than 45. H 0 : µ = 45 H A : µ < 45
c)
leads to one tailed test (or right tailed test)
leads to one tailed test (or left tailed test)
Do the the sam sampl plee data data pro provi vide de suf suffi fici cien entt evid eviden ence ce to to indi indica cate te that that the the popu popula lati tion on mean is not equal to 45. H 0 : µ = 45 H A : µ ≠ 45
leads to two tailed test (or both tailed test)
Level of Significance and Power of a Test:
•
•
The probability of making type I error is called the level of significance of the test denoted by α . The probability of making a type two error is deno ted by β and (1 - β ) is called the power of the test.
Decision of the test for the Null Hypothesis Accept
The Null Hypothesis is True Correct decision P(Correct decision)=1- α .
False Incorrect decision Type two error P(type II error) = β Correct decision P(correct decision)=1- β =Power of test.
Incorrect decision P(type I error)= α . = level of significance of the test
Reject
Rejection Rule and Conclusion: Points to Note i) Rejection of H indicates that an extremely unlikely sample has been drawn which implies that H is very likely to be false. ii) Failing to reject H does not prove that H is true. It implies that H may be true. iii) In testing hypotheses, the assumption is always made that the sample used in the test process is a random sample. iv) It is assumed that the sampling distribution of the test statistic is known. H , H and α . are determined before the test is carried out. v) 0
0
0
0
0
0
A
Formal Testing Procedure: A hypothesis testing procedure involves the following six steps: Step 1:Set up the null and alternative hypothesis ( H & H ). The alternative hypothesis decides whether the test is one tailed or two tailed. 0
A
Step 2:Specify a level of significance (α . ). Step 3:Select an appropriate test – statistic (z or t-test) and compute the value of the test statistic using sample data assuming null hypothesis to be true. Step 4:Dtermine the critical values and the critical region of the test (using z or t table). Step 5:State a rule to reject the null hypothesis. Step 6:Decide if the null hypothesis is to be rejected and write the conclusion of the test.
Testing the Mean of a Population: 1.
Null Hypothesis : H 0 : µ µ 0 ( µ Alternative Hypothesis: H : µ ≠ µ 0 OR H : µ > µ 0 OR H : µ < µ 0 Significance level: α . = 0.01 or 0.05 Test Statistic: =
A
A
A
2. 3.
z
x
−
=
0
is the hypothesized value of µ ) (Two tailed test) (One tailed test) (One tailed test) or 0.10 etc.
µ 0
σ
(when σ is known)
n z
OR
x =
−
µ 0
s
(when σ is unknown and n
≥
30)
(when σ is unknown and n
<
30
n t
OR
x =
−
µ 0
s n
with d. f. = n-1) The test statistic (z or t ) is decided according to the following table. Normal Population Non-Normal Population Sample Size σ known σ unknown σ known σ unknown (n) n>30 Z - test Z - test Z - test None large sample n ≤small Z - test t - test None None sample Thus t-test is used only if i) The population is normal σ is unknown ( but s is known or can be computed) ii) iii) n < 30
4. Critical Region: The critical region shown in the curve of normal distribution (z table) or t-distribution. The value find out by z or t table, and the area right and left to that value according to the alternative hypothesis in the case of one tailed test and the area to either left or right according to the alternative hypothesis in one tailed test to be shaded. These values taken by the table.
If the value of Z or t calculated lie in the shaded region or considered as critical region then null hypothesis will reject otherwise accept.
.
α
Test
Z
Two tail
α
2
Z
One tail
ε
0
0.01
0.02
0.05
0.10
2.576
2.33
1.96
1.645
2.33
2.05
1.645
1.28
Critical values of t for (n-1) degree of freedom are obtained from the t-table.
5. Rejection Rule: If Z
cal
If t
cal
Z cal Z tab
>
Z tab
> t tab
and and
t cal t tab
; reject H and accept H OR ; reject H and accept H 0
0
A
A
are calculated values of the test statistic. are critical or tabulated values of the test statistic.
6. Conclusion: Reject or do not reject the null hypothesis on the basis of the above rejection rule. The test will be significant if H is rejected otherwise the test will be insignificant. 0
Test Concerning Variance In testing hypothesis concerning Variance the basic six steps of hypothesis will remain same only the difference of distribution and critical region defining procedure. In testing concerning variance, we use χ -distribution (Chi square distribution). 2 (n 1) s 2 2
χ
−
=
σ
2
For finding the value from table; the degree of freedom will be υ
=
n
−
1
Note: it is incomplete. Write about critical region and rejection rule from page 320, 3 rd paragraph, Walpole
