1. Chapter 8
Hypothesis Testing with Two Samples

2. Chapter Outline
8.1 Testing the Difference Between Means (Independent Samples, 1 and 2 Known)
8.2 Testing the Difference Between Means (Independent Samples, 1 and 2 Unknown)
8.3 Testing the Difference Between Means (Dependent Samples)
8.4 Testing the Difference Between Proportions .

3. Testing the Difference Between Means (Independent Samples,
Section 8.2
Testing the Difference Between Means (Independent Samples,
1 and 2 Unknown) .

4. Section 8.2 Objectives
How to perform a t-test for the difference between two means μ1 and μ2 with independent samples with 1 and 2 unknown .

5. Two Sample t-Test for the Difference Between Means
A two-sample t-test is used to test the difference between two population means μ1 and μ2 when
σ2 and σ2 are unknown,
the samples are random,
the samples are independent, and
the populations are normally distributed or both n1 ≥ 30 and n2 ≥ 30. .

6. Two Sample t-Test for the Difference Between Means
The standardized test statistic is
The standard error and the degrees of freedom of the sampling distribution depend on whether the population variances and are equal. .

7. Two Sample t-Test for the Difference Between Means
Variances are equal
Information from the two samples is combined to calculate a pooled estimate of the standard deviation
The standard error for the sampling distribution of is
d.f.= n1 + n2 – 2 .

8. Two Sample t-Test for the Difference Between Means
Variances are not equal
If the population variances are not equal, then the standard error is
d.f = smaller of n1 – 1 or n2 – 1 .

9. Normal (z) or t-Distribution?.

10. Two-Sample t-Test for the Difference Between Means (Small Independent Samples)
In Words In Symbols
Verify that 1 and 2 are unknown, the samples are random and independent, and either the populations are normally distributed or both n1  30 and n2  30 .
State the claim mathematically. Identify the null and alternative hypotheses.
Specify the level of significance.
State H0 and Ha.
Identify . .

11. Two-Sample t-Test for the Difference Between Means (Small Independent Samples)
In Words In Symbols
d.f. = n1+ n2 – 2 or
d.f. = smaller of n1 – 1 or n2 – 1.
Determine the degrees of freedom.
Determine the critical value(s).
Determine the rejection region(s).
Use Table 5 in Appendix B. .

12. Two-Sample t-Test for the Difference Between Means (Small Independent Samples)
In Words In Symbols
Find the standardized test statistic and sketch the sampling distribution.
Make a decision to reject or fail to reject the null hypothesis.
Interpret the decision in the context of the original claim.
If t is in the rejection region, reject H0. Otherwise, fail to reject H0. .

13. Example: Two-Sample t-Test for the Difference Between Means
The results of a state mathematics test for random samples of students taught by two different teachers at the same school are shown below. Can you conclude that there is a difference in the mean mathematics test scores for the students of the two teachers? Use α = Assume the populations are normally distributed and the population variances are not equal.
Teacher 1
Teacher 2
s1 = 39.7
s2 = 24.5
n1 = 8
n2 = 18 .

14. Solution: Two-Sample t-Test for the Difference Between Means
Test Statistic:
H0:
Ha:
 
d.f. =
Rejection Region:
μ1 = μ2
μ1 ≠ μ2
0.10
8 – 1 = 7
Decision:
Fail to Reject H0
At the 10% level of significance, there is not enough evidence to support the claim that the mean mathematics test scores for the students of the two teachers are different. .

15. Example: Two-Sample t-Test for the Difference Between Means
A manufacturer claims that the calling range (in feet) of its 2.4-GHz cordless telephone is greater than that of its leading competitor. You perform a study using 14 randomly selected phones from the manufacturer and 16 randomly selected similar phones from its competitor. The results are shown below. At α = 0.05, can you support the manufacturer’s claim? Assume the populations are normally distributed and the population variances are equal.
Manufacturer (1)
Competition (2)
s1 = 45 ft
s2 = 30 ft
n1 = 14
n2 = 16 .

16. Solution: Two-Sample t-Test for the Difference Between Means
Ha:
 
d.f. =
Rejection Region:
μ1 ≤ μ2
μ1 > μ2
Test Statistic:
0.05
– 2 = 28
Decision: t
1.701
0.05 .

17. Solution: Two-Sample t-Test for the Difference Between Means.

18. Solution: Two-Sample t-Test for the Difference Between Means
Ha:
 
d.f. =
Rejection Region:
μ1 ≤ μ2
μ1 > μ2
Test Statistic:
0.05
– 2 = 28
Decision:
Reject H0
At the 5% level of significance, there is enough evidence to support the manufacturer’s claim that its phone has a greater calling range than its competitors. t
1.701
0.05
1.811 .

19. Section 8.2 Summary
Performed a t-test for the difference between two means μ1 and μ2 with independent samples with 1 and 2 unknown .