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Chapter 10 Statistical Inference About Means and Proportions With Two Populations

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

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Chapter 10 Statistical Inference About Means and Proportions With Two Populations

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  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 Betweenthe 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 where

  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 #1 Sample #2 Par, Inc.Rap, Ltd. Sample Sizen1 = 120 balls n2 = 80 balls Mean = 235 yards = 218 yards Standard Deviations1 = 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 = = 235 - 218 = 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: = 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.

  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 where

  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 #1 Sample #2 M CarsJ Cars Sample Sizen1 = 12 cars n2 = 8 cars Mean = 29.8 mpg = 27.3 mpg Standard Deviations1 = 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 = = 29.8 - 27.3 = 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 = 2.101. 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 = 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).

  14. Hypothesis Tests About the DifferenceBetween 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

  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 #1 Sample #2 Par, Inc.Rap, Ltd. Sample Sizen1 = 120 balls n2 = 80 balls Mean = 235 yards = 218 yards Standard Deviations1 = 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. golf balls 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.

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