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統計學 Spring 2004

統計學 Spring 2004. 授課教師：統計系余清祥 日期：2004年3月2日 第三週：比較期望值.  1 =  2 ?. ANOVA. Chapter 10 Comparisons Involving Means. Estimation of the Difference Between the Means of Two Populations: Independent Samples Hypothesis Tests about the Difference between the

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統計學 Spring 2004

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  1. 統計學 Spring 2004 授課教師：統計系余清祥 日期：2004年3月2日 第三週：比較期望值

  2. 1 = 2 ? ANOVA Chapter 10 Comparisons Involving Means • 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 • Inferences about the Difference between the Means of Two Populations: Matched Samples • Inferences about the Difference between the Proportions of Two Populations:

  3. 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

  4. 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 .

  5. Sampling Distribution of • Properties of the Sampling Distribution of • Expected Value • Standard Deviation where: 1 = standard deviation of population 1 2 = standard deviation of population 2 n1 = sample size from population 1 n2 = sample size from population 2

  6. 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:

  7. Example: Par, Inc. • Interval Estimate of 1 - 2: Large-Sample Case Par, Inc. is a manufacturer of golf equipment and 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 statistics appear on the next slide.

  8. Example: Par, Inc. • Interval Estimate of 1 - 2: Large-Sample Case • Sample Statistics Sample #1 Sample #2 Par, Inc. Rap, Ltd. Sample Size n1 = 120 balls n2 = 80 balls Mean = 235 yards = 218 yards Standard Dev. s1 = 15 yards s2 = 20 yards

  9. 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.

  10. Simple random sample of n1 Par golf balls x1 = sample mean distance for sample of Par golf ball Simple random sample of n2 Rap golf balls x2 = sample mean distance for sample of Rap golf ball x1 - x2 = Point Estimate of m1 –m2 Point Estimator of the Difference Between the Means of Two Populations Population 1 Par, Inc. Golf Balls m1 = mean driving distance of Par golf balls Population 2 Rap, Ltd. Golf Balls m2 = mean driving distance of Rap golf balls m1 –m2= difference between the mean distances

  11. 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.

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

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

  14. 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 statistics are: Sample #1 Sample #2 M CarsJ 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

  15. 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.

  16. Example: Specific Motors • 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case We will make the following assumptions: • The miles per gallon rating must be normally distributed for both the M car and the J car. • 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.

  17. 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).

  18. Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples • Hypotheses 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 Small-Sample

  19. Example: Par, Inc. • Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case Par, Inc. is a manufacturer of golf equipment and 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 statistics appear on the next slide.

  20. Example: Par, Inc. • Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case • Sample Statistics Sample #1 Sample #2 Par, Inc. Rap, Ltd. Sample Size n1 = 120 balls n2 = 80 balls Mean = 235 yards = 218 yards Standard Dev. s1 = 15 yards s2 = 20 yards

  21. 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 • HypothesesH0: 1 - 2< 0 Ha: 1 - 2 > 0

  22. Example: Par, Inc. • Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case • Rejection RuleReject 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.

  23. Example: Specific Motors • Hypothesis Tests About the Difference Between the Means of Two Populations: Small-Sample Case Can we conclude, using a .05 level of significance, that the miles-per-gallon (mpg) performance of M cars is greater than the miles-per-gallon performance of J cars? 1 = mean mpg for the population of M cars 2 = mean mpg for the population of J cars • HypothesesH0: 1 - 2< 0 Ha: 1 - 2 > 0

  24. Example: Specific Motors • Hypothesis Tests About the Difference Between the Means of Two Populations: Small-Sample Case • Rejection Rule Reject H0 if t > 1.734 (a = .05, d.f. = 18) • Test Statistic where:

  25. Inference About the Difference Between the Means of Two Populations: Matched Samples • With a matched-sample design each sampled item provides a pair of data values. • The matched-sample design can be referred to as blocking. • This design often leads to a smaller sampling error than the independent-sample design because variation between sampled items is eliminated as a source of sampling error.

  26. Example: Express Deliveries • Inference About the Difference Between the Means of Two Populations: Matched Samples A Chicago-based firm has documents that must be quickly distributed to district offices throughout the U.S. The firm must decide between two delivery services, UPX (United Parcel Express) and INTEX (International Express), to transport its documents. In testing the delivery times of the two services, the firm sent two reports to a random sample of ten district offices with one report carried by UPX and the other report carried by INTEX. Do the data that follow indicate a difference in mean delivery times for the two services?

  27. Example: Express Deliveries Delivery Time (Hours) District OfficeUPXINTEXDifference Seattle 32 25 7 Los Angeles 30 24 6 Boston 19 15 4 Cleveland 16 15 1 New York 15 13 2 Houston 18 15 3 Atlanta 14 15 -1 St. Louis 10 8 2 Milwaukee 7 9 -2 Denver 16 11 5

  28. Example: Express Deliveries • Inference About the Difference Between the Means of Two Populations: Matched Samples Let d = the mean of the difference values for the two delivery services for the population of district offices • HypothesesH0: d = 0, Ha: d • Rejection Rule Assuming the population of difference values is approximately normally distributed, the t distribution with n - 1 degrees of freedom applies. With  = .05, t.025 = 2.262 (9 degrees of freedom). Reject H0 if t < -2.262 or if t > 2.262

  29. Example: Express Deliveries • Inference About the Difference Between the Means of Two Populations: Matched Samples • ConclusionReject H0. There is a significant difference between the mean delivery times for the two services.

  30. Inferences About the Difference Between the Proportions of Two Populations • Sampling Distribution of • Interval Estimation of p1 - p2 • Hypothesis Tests about p1 - p2

  31. Sampling Distribution of • Expected Value • Standard Deviation • Distribution Form If the sample sizes are large (n1p1, n1(1 - p1), n2p2, and n2(1 - p2) are all greater than or equal to 5), the sampling distribution of can be approximated by a normal probability distribution.

  32. Interval Estimation of p1 - p2 • Interval Estimate • Point Estimator of

  33. Example: MRA MRA (Market Research Associates) is conducting research to evaluate the effectiveness of a client’s new advertising campaign. Before the new campaign began, a telephone survey of 150 households in the test market area showed 60 households “aware” of the client’s product. The new campaign has been initiated with TV and newspaper advertisements running for three weeks. A survey conducted immediately after the new campaign showed 120 of 250 households “aware” of the client’s product. Does the data support the position that the advertising campaign has provided an increased awareness of the client’s product?

  34. Example: MRA • Point Estimator of the Difference Between the Proportions of Two Populations p1 = proportion of the population of households “aware” of the product after the new campaign p2 = proportion of the population of households “aware” of the product before the new campaign = sample proportion of households “aware” of the product after the new campaign = sample proportion of households “aware” of the product before the new campaign

  35. Example: MRA • Interval Estimate of p1 - p2: Large-Sample Case For = .05, z.025 = 1.96: .08 + 1.96(.0510) .08 + .10 or -.02 to +.18 • Conclusion At a 95% confidence level, the interval estimate of the difference between the proportion of households aware of the client’s product before and after the new advertising campaign is -.02 to +.18.

  36. Hypothesis Tests about p1 - p2 • Hypotheses H0: p1 - p2< 0 Ha: p1 - p2 > 0 • Test statistic • Point Estimator of where p1 = p2 where:

  37. Example: MRA • Hypothesis Tests about p1 - p2 Can we conclude, using a .05 level of significance, that the proportion of households aware of the client’s product increased after the new advertising campaign? p1 = proportion of the population of households “aware” of the product after the new campaign p2 = proportion of the population of households “aware” of the product before the new campaign • HypothesesH0: p1 - p2< 0 Ha: p1 - p2 > 0

  38. Example: MRA • Hypothesis Tests about p1 - p2 • Rejection RuleReject H0 if z > 1.645 • Test Statistic • ConclusionDo not reject H0.

  39. End of Chapter 10

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