1. Chapter 10 Statistical Inference About Means and Proportions With Two Populations
Estimation of the Difference between the Means of Two Populations: Independent Samples
Hypothesis Tests about the Difference between the Means of Two Populations: Independent Samples
Inference about the Difference between the Means of Two Populations: Matched Samples
Inference about the Difference between the Proportions of Two Populations

2. Estimation of the Difference Between the Means of Two Populations: Independent Samples
Point Estimator of the Difference between the Means of Two Populations
Sampling Distribution
Interval Estimate of Large-Sample Case
Interval Estimate of Small-Sample Case

3. Point Estimator of the Difference Between the Means of Two Populations
Let 1 equal the mean of population 1 and 2 equal the mean of population 2.
The difference between the two population means is
1 - 2.
To estimate 1 - 2, we will select a simple random sample of size n1 from population 1 and a simple random sample of size n2 from population 2.
Let equal the mean of sample 1 and equal the mean of sample 2.
The point estimator of the difference between the means of the populations 1 and 2 is

4. Sampling Distribution of
The sampling distribution of has the following properties.
Expected Value:
Standard Deviation:
where 1 = standard deviation of population 1
2 = standard deviation of population 2
n1 = sample size from population
n2 = sample size from population 2

5. Interval Estimate of 1 - 2: Large-Sample Case (n1 > 30 and n2 > 30)
Interval Estimate with 1 and 2 Known
where
1 -  is the confidence coefficient
Interval Estimate with 1 and 2 Unknown

6. Example: Par, Inc. Par, Inc. is a manufacturer of golf equipment. Par
has developed a new golf ball that has been designed to
provide “extra distance.” In a test of driving distance
using a mechanical driving device, a sample of Par golf
balls was compared with a sample of golf balls made by
Rap, Ltd., a competitor. The sample data is below.
Sample # Sample #2
Par, Inc. Rap, Ltd.
Sample Size n1 = 120 balls n2 = 80 balls
Mean = 235 yards = 218 yards
Standard Deviation s1 = 15 yards s2 = 20 yards

7. Example: Par, Inc.
Point Estimate of the Difference Between Two Population Means
1 = mean distance for the population of
Par, Inc. golf balls
2 = mean distance for the population of
Rap, Ltd. golf balls
Point estimate of 1 - 2 = = = 17 yards.

8. Example: Par, Inc.
95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Case, 1 and 2 Unknown
Substituting the sample standard deviations for the population standard deviation:
= or yards to yards.
We are 95% confident that the difference between the mean driving distances of Par, Inc. balls and Rap, Ltd. balls lies in the interval of to yards.

9. Interval Estimate of 1 - 2: Small-Sample Case (n1 < 30 and/or n2 < 30)
Interval Estimate with  2 Known
where
Interval Estimate with  2 Unknown

10. Example: Specific Motors
Specific Motors of Detroit has developed a new
automobile known as the M car. 12 M cars and 8 J cars
(from Japan) were road tested to compare miles-per-
gallon (mpg) performance. The sample data is below.
Sample # Sample #2
M Cars J Cars
Sample Size n1 = 12 cars n2 = 8 cars
Mean = 29.8 mpg = 27.3 mpg
Standard Deviation s1 = 2.56 mpg s2 = 1.81 mpg

11. Example: Specific Motors
Point Estimate of the Difference Between Two Population Means
1 = mean miles-per-gallon for the population of
M cars
2 = mean miles-per-gallon for the population of
J cars
Point estimate of 1 - 2 = = = 2.5 mpg.

12. Example: Specific Motors
95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case
For the small-sample case we will make the following assumption.
1. The miles per gallon rating must be normally
distributed for both the M car and the J car.
2. The variance in the miles per gallon rating must
be the same for both the M car and the J car.
Using the t distribution with n1 + n2 - 2 = 18 degrees
of freedom, the appropriate t value is t.025 =
We will use a weighted average of the two sample
variances as the pooled estimator of  2.

13. Example: Specific Motors
95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case
= or .3 to 4.7 miles per gallon.
We are 95% confident that the difference between the
mean mpg ratings of the two car types is from .3 to 4.7
mpg (with the M car having the higher mpg).

14. Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples
Hypothesis Forms:
H0: 1 - 2 < H0: 1 - 2 > H0: 1 - 2 = 0
Ha: 1 - 2 > Ha: 1 - 2 < Ha: 1 - 2  0
Test Statistic:
Large-Sample Case
Small-Sample Case

15. Example: Par, Inc. Par, Inc. is a manufacturer of golf equipment. Par
has developed a new golf ball that has been designed to
provide “extra distance.” In a test of driving distance
using a mechanical driving device, a sample of Par golf
balls was compared with a sample of golf balls made by
Rap, Ltd., a competitor. The sample data is below.
Sample # Sample #2
Par, Inc. Rap, Ltd.
Sample Size n1 = 120 balls n2 = 80 balls
Mean = 235 yards = 218 yards
Standard Deviation s1 = 15 yards s2 = 20 yards

16. Example: Par, Inc.
Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case
Can we conclude, using a .01 level of significance, that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls?
1 = mean distance for the population of Par, Inc.
golf balls
2 = mean distance for the population of Rap, Ltd.
Hypotheses: H0: 1 - 2 < 0
Ha: 1 - 2 > 0

17. Example: Par, Inc.
Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case
Rejection Rule: Reject H0 if z > 2.33
Conclusion: Reject H0. We are at least 99% confident
that the mean driving distance of Par, Inc. golf balls is
greater than the mean driving distance of Rap, Ltd. golf balls.