1. Chapter 11 Hypothesis Testing II
Statistics for
Business and Economics 6th Edition
Chapter 11
Hypothesis Testing II
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

2. Chapter Goals After completing this chapter, you should be able to:
Test hypotheses for the difference between two population means
Two means, matched pairs
Independent populations, population variances known
Independent populations, population variances unknown but equal
Complete a hypothesis test for the difference between two proportions (large samples)
Use the chi-square distribution for tests of the variance of a normal distribution
Use the F table to find critical F values
Complete an F test for the equality of two variances
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

3. Two Sample Tests Two Sample Tests Population Proportions
Population Variances
Proportion 1 vs. Proportion 2
Variance 1 vs.
Variance 2
(Note similarities to Chapter 9)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

4. Test Statistic for Two Population Proportions
The test statistic for
H0: Px – Py = 0
is a z value:
Population proportions
Where
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

5. Decision Rules: Proportions
Population proportions
Lower-tail test:
H0: px – py  0
H1: px – py < 0
Upper-tail test:
H0: px – py ≤ 0
H1: px – py > 0
Two-tail test:
H0: px – py = 0
H1: px – py ≠ 0 a a
a/2
a/2
-za za
-za/2
za/2
Reject H0 if z < -za
Reject H0 if z > za
Reject H0 if z < -za/2
or z > za/2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

6. Example: Two Population Proportions
Is there a significant difference between the proportion of men and the proportion of women who will vote Yes on Proposition A?
In a random sample, 36 of 72 men and 31 of 50 women indicated they would vote Yes
Test at the .05 level of significance
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

7. Example: Two Population Proportions
(continued)
The hypothesis test is:
H0: PM – PW = 0 (the two proportions are equal)
H1: PM – PW ≠ 0 (there is a significant difference between
proportions)
The sample proportions are:
Men: = 36/72 = .50
Women: = 31/50 = .62
The estimate for the common overall proportion is:
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

8. Example: Two Population Proportions
(continued)
Reject H0
Reject H0
The test statistic for PM – PW = 0 is:
.025
.025
-1.96
1.96
-1.31
Decision: Do not reject H0
Conclusion: There is not significant evidence of a difference between men and women in proportions who will vote yes.
Critical Values = ±1.96
For  = .05
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

9. Hypothesis Tests of one Population Variance
Goal: Test hypotheses about the
population variance, σ2
If the population is normally distributed,
follows a chi-square distribution with (n – 1) degrees of freedom
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

10. Confidence Intervals for the Population Variance
(continued)
The test statistic for hypothesis tests about one population variance is
Population
Variance
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

11. Decision Rules: Variance
Population variance
Lower-tail test:
H0: σ2  σ02
H1: σ2 < σ02
Upper-tail test:
H0: σ2 ≤ σ02
H1: σ2 > σ02
Two-tail test:
H0: σ2 = σ02
H1: σ2 ≠ σ02 a a
a/2
a/2
Reject H0 if
Reject H0 if
Reject H0 if or
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

12. Hypothesis Tests for Two Variances
Population
Variances
Goal: Test hypotheses about two
population variances
H0: σx2  σy2
H1: σx2 < σy2
Lower-tail test
F test statistic
H0: σx2 ≤ σy2
H1: σx2 > σy2
Upper-tail test
H0: σx2 = σy2
H1: σx2 ≠ σy2
Two-tail test
The two populations are assumed to be independent and normally distributed
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

13. Hypothesis Tests for Two Variances
(continued)
The random variable
Tests for Two
Population
Variances
F test statistic
Has an F distribution with (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom
Denote an F value with 1 numerator and 2 denominator degrees of freedom by
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

14. Test Statistic Tests for Two
Population
Variances
The critical value for a hypothesis test about two population variances is
F test statistic
where F has (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

15. Decision Rules: Two Variances
Use sx2 to denote the larger variance.
H0: σx2 = σy2
H1: σx2 ≠ σy2
H0: σx2 ≤ σy2
H1: σx2 > σy2
/2  F F
Do not
reject H0
Reject H0
Do not
reject H0
Reject H0
rejection region for a two-tail test is:
where sx2 is the larger of the two sample variances
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

16. Example: F Test
You are a financial analyst for a brokerage firm. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data:
NYSE NASDAQ Number
Mean
Std dev
Is there a difference in the variances between the NYSE & NASDAQ at the  = 0.10 level?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

17. F Test: Example Solution
Form the hypothesis test:
H0: σx2 = σy2 (there is no difference between variances)
H1: σx2 ≠ σy2 (there is a difference between variances)
Find the F critical values for  = .10/2:
Degrees of Freedom:
Numerator
(NYSE has the larger standard deviation):
nx – 1 = 21 – 1 = 20 d.f.
Denominator:
ny – 1 = 25 – 1 = 24 d.f.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

18. F Test: Example Solution
(continued)
The test statistic is:
H0: σx2 = σy2
H1: σx2 ≠ σy2
/2 = .05 F
Do not
reject H0
Reject H0
F = is not in the rejection region, so we do not reject H0
Conclusion: There is not sufficient evidence of a difference in variances at  = .10
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

19. Two-Sample Tests in EXCEL
For variances…
F test for two variances:
Tools | data analysis | F-test: two sample for variances
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

20. Two-Sample Tests in PHStat
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

21. Sample PHStat Output Input Output
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

22. Sample PHStat Output Input Output (continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

23. Chapter Summary Compared two population proportions
Performed z-test for two population proportions
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

24. Chapter Summary
(continued)
Used the chi-square test for a single population variance
Performed F tests for the difference between two population variances
Used the F table to find F critical values
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.