1. Hypothesis Testing – Two Population Variances
Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc.
All rights reserved.
HAWKES LEARNING SYSTEMS
math courseware specialists
Section 11.5
Hypothesis Testing – Two Population Variances

2. The two populations must be normally distributed.
HAWKES LEARNING SYSTEMS
math courseware specialists
Hypothesis Testing (Two or More Populations)
11.5 Hypothesis Testing – Two Population Variances
Criteria for hypothesis testing with variances:
The two populations must be independent, not matched or paired in any way.
The two populations must be normally distributed.

3. The F-distribution is skewed to the right.
HAWKES LEARNING SYSTEMS
math courseware specialists
Hypothesis Testing (Two or More Populations)
11.5 Hypothesis Testing – Two Population Variances
Critical Value, F:
The F-distribution is skewed to the right.
The values of F are always greater than 0.
The shape of the F-distribution is completely determined by its two parameters, the degrees of freedom of the numerator and the degrees of freedom of the denominator of the ratio.
In this section, in order to be consistent with the F-distribution tables, we will round calculated values to four decimal places.

4. HAWKES LEARNING SYSTEMS
math courseware specialists
Hypothesis Testing (Two or More Populations)
11.5 Hypothesis Testing – Two Population Variances
Test Statistic for Population Variances:
F =
To determine if the test statistic calculated from the sample is statistically significant we will need to look at the critical value.
The critical values for population variances are found from the f-distribution table.
The f-distribution table assumes that the area is to the right of f-critical.

5. Reject if F ≤ F1 –  HAWKES LEARNING SYSTEMS
math courseware specialists
Hypothesis Testing (Two or More Populations)
11.5 Hypothesis Testing – Two Population Variances
Rejection Regions for Left-Tailed Tests, Ha contains <:
The f-distribution table assumes that the area is to the right of f-critical. Obtain the critical value F1 – 
Reject if F ≤ F1 – 

6. Reject if F ≥ F HAWKES LEARNING SYSTEMS math courseware specialists
Hypothesis Testing (Two or More Populations)
11.5 Hypothesis Testing – Two Population Variances
Rejection Regions for Right-Tailed Tests, Ha contains >:
The f-distribution table assumes that the area is to the right of f-critical. Obtain the critical value F
Reject if F ≥ F

7. Reject if F ≤ or F ≥ HAWKES LEARNING SYSTEMS
math courseware specialists
Hypothesis Testing (Two or More Populations)
11.5 Hypothesis Testing – Two Population Variances
Rejection Regions for Two-Tailed Tests, Ha contains ≠:
Reject if F ≤ or F ≥

8. State the null and alternative hypotheses.
HAWKES LEARNING SYSTEMS
math courseware specialists
Hypothesis Testing (Two or More Populations)
11.5 Hypothesis Testing – Two Population Variances
Steps for Hypothesis Testing:
State the null and alternative hypotheses.
Set up the hypothesis test by choosing the test statistic and determining the values of the test statistic that would lead to rejecting the null hypothesis.
Gather data and calculate the necessary sample statistics.
Draw a conclusion.

9. HAWKES LEARNING SYSTEMS
math courseware specialists
Hypothesis Testing (Two or More Populations)
11.5 Hypothesis Testing – Two Population Variances
Null and Alternative Hypotheses:

10. First state the hypotheses: H0: Ha:
HAWKES LEARNING SYSTEMS
math courseware specialists
Hypothesis Testing (Two or More Populations)
11.5 Hypothesis Testing – Two Population Variances
Draw a conclusion:
A quality control inspector believes that the machines on Assembly Line A are not adjusted properly and that the variance in the size of the candy produced is greater than the variance in the size of candy produced by Assembly Line B. A sample of 20 pieces of candy is taken from each assembly line and measured. The size of the candy from Assembly Line A has a sample variance of 1.45, while the size of the candy from Assembly Line B has a sample variance of Using a 0.01 level of significance, perform a hypothesis test to test the quality control inspector’s claim.
Solution:
First state the hypotheses:
H0:
Ha:
Next, set up the hypothesis test and state the level of significance:
 = 0.01, d.f. = 19
F =
Reject if F ≥
3.0274

12. Gather the data and calculate the necessary sample statistics:
HAWKES LEARNING SYSTEMS
math courseware specialists
Hypothesis Testing (Two or More Populations)
11.5 Hypothesis Testing – Two Population Variances
Solution (continued):
Gather the data and calculate the necessary sample statistics:
= 1.45, = 0.47
Finally, draw a conclusion:
Since F-statistic (3.0851) is greater than F (3.0274), we will reject the null hypothesis. The evidence does sufficiently support the inspector’s claim.
F =
3.0851

13. First state the hypotheses: H0: Ha:
HAWKES LEARNING SYSTEMS
math courseware specialists
Hypothesis Testing (Two or More Populations)
11.5 Hypothesis Testing – Two Population Variances
Draw a conclusion:
A professor claims that the variances in the test scores on two different versions of an exam are not equal. To test the claim, a sample of 20 scores of version A is chosen and a sample variance of is found, while a sample of 20 scores of version B has a sample variance of Conduct a hypothesis test to test the professor’s claim that the variances of the test scores are not equal. Use a 0.05 level of significance.
Solution:
First state the hypotheses:
H0:
Ha:
Next, set up the hypothesis test and state the level of significance:
 = 0.05, d.f. = 19
= F0.025 =
and
= F0.975 =
0.3958
Reject if F ≤ or F ≥

14. Gather the data and calculate the necessary sample statistics:
HAWKES LEARNING SYSTEMS
math courseware specialists
Hypothesis Testing (Two or More Populations)
11.5 Hypothesis Testing – Two Population Variances
Solution (continued):
Gather the data and calculate the necessary sample statistics:
= 3.815, = 3.391
Finally, draw a conclusion:
Since F-statistic (1.1250) is not in the rejection region(F is between and ), we will fail to reject the null hypothesis. The evidence does not sufficiently support the claim that the two tests have different variances.
F =
1.1250
Ho:
Ha:

15. The table assumes that the area is to the right of f-critical.
Thus, alpha of 0.1 for right tail test corresponds to alpha of 0.9 for left tail tests.
 = DF1 = 19 and DF2 = 40

17. F Statistic = Reject if F < F0.9 or p-value > (1- )
Reject if F < or p-value > 0.9
Computed- (F-Statistic) is and it is less than
p-value is and it is greater than 0.9
Reject the Null Hypothesis. There is enough evidence that σ12 is lower than σ22

18. The table assumes that the area is to the right of f-critical.
DF1 = 17 and DF2 = 15
The table assumes that the area is to the right of f-critical.
But, this is a two-tail test and alpha belongs to both tails.
Thus, alpha of 0.01 for two-tail tests corresponds to:
- alpha of for the right tail  (/2) 
- alpha of for the left tail  (1- /2) 

22. F Statistic =
Reject if F > F or F < F or p-value < (/2) OR p-value > (1- /2)
Reject if F > or F < or p-value < OR p-value > 0.995
Computed- (F-Statistic) is and it is NOT > OR <
p-value is and it is not less than or greater than 0.995
Don’t reject the Null Hypothesis.
There is NOT enough evidence that they are different.

24. F Statistic =
= =
0.006

27. Reject the Null Hypothesis.
There is enough evidence that σ12 is greater than σ22