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

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

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 .

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

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

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 #1 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

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 = = 235 - 218 = 17 yards.

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: = 17 + 5.14 or 11.86 yards to 22.14 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 11.86 to 22.14 yards.

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

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 #1 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

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 = = 29.8 - 27.3 = 2.5 mpg.

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 = 2.101. We will use a weighted average of the two sample variances as the pooled estimator of  2.

Example: Specific Motors 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case = 2.5 + 2.2 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).

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

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 #1 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

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

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.