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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 Slide 1 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 x1 x2 Interval Estimate of Large-Sample Case Interval Estimate of Small-Sample Case Slide 2 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 x1 equal the mean of sample 1 and x2 equal the mean of sample 2. The point estimator of the difference between the means of the populations 1 and 2 is x1 x2 . Slide 3 Sampling Distribution of x1 x2 The sampling distribution of x1 x2 has the following properties. E( x1 x2 ) 1 2 Expected Value: Standard Deviation: x1 x2 where 12 n1 22 n2 1 = standard deviation of population 1 2 = standard deviation of population 2 n1 = sample size from population n2 = sample size from population 2 Slide 4 Interval Estimate of 1 - 2: Large-Sample Case (n1 > 30 and n2 > 30) Interval Estimate with 1 and 2 Known where x1 x2 z / 2 x1 x2 1 - is the confidence coefficient Interval Estimate with 1 and 2 Unknown where x1 x2 z / 2 sx1 x2 s x1 x2 s12 s22 n1 n2 Slide 5 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 x1 = 235 yards x2 = 218 yards Mean Standard Deviation s1 = 15 yards s2 = 20 yards Slide 6 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 = x1 x2 = 235 - 218 = 17 yards. Slide 7 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: x1 x2 z / 2 12 22 (15)2 (20)2 17 1.96 n1 n2 120 80 = 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. Slide 8 Interval Estimate of 1 - 2: Small-Sample Case (n1 < 30 and/or n2 < 30) Interval Estimate with 2 Known where x1 x2 z / 2 x1 x2 1 1 x1 x2 ( ) n1 n2 Interval Estimate with 2 Unknown 2 where sx1 x2 1 2 1 s ( ) n1 n2 x1 x2 t / 2 s x1 x2 2 2 ( n 1 ) s ( n 1 ) s 1 2 2 s2 1 n1 n2 2 Slide 9 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-pergallon (mpg) performance. The sample data is below. Sample Size Mean Standard Deviation Sample #1 M Cars n1 = 12 cars x1 = 29.8 mpg s1 = 2.56 mpg Sample #2 J Cars n2 = 8 cars x2 = 27.3 mpg s2 = 1.81 mpg Slide 10 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 = x1 x2 = 29.8 - 27.3 = 2.5 mpg. Slide 11 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. Slide 12 Example: Specific Motors 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case 2 2 2 2 ( n 1 ) s ( n 1 ) s 11 ( 2 . 56 ) 7 ( 1 . 81 ) 1 2 2 s2 1 5.28 n1 n2 2 12 8 2 x1 x2 t.025 s 2 ( 1 1 1 1 ) 2.5 2.101 5.28( ) n1 n2 12 8 = 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). Slide 13 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 ( x x2 ) ( 1 2 ) z 1 12 n1 22 n2 • Small-Sample Case t ( x1 x2 ) ( 1 2 ) s 2 (1 n1 1 n2 ) Slide 14 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 x1 = 235 yards x2 = 218 yards Mean Standard Deviation s1 = 15 yards s2 = 20 yards Slide 15 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. golf balls Hypotheses: H0: 1 - 2 < 0 Ha: 1 - 2 > 0 Slide 16 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 z ( x1 x2 ) ( 1 2 ) 2 1 n1 2 2 n2 (235 218) 0 17 6.49 2 2 2.62 (15) (20) 120 80 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. Slide 17