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Slides Prepared by JOHN S. LOUCKS St. Edward’s University

Slides Prepared by JOHN S. LOUCKS St. Edward’s University. Chapter 10 Statistical Inferences about Means and Proportions for Two Populations. Estimation of the Difference Between the Means of Two Populations: Independent Samples Hypothesis Tests about the Difference Between the

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Slides Prepared by JOHN S. LOUCKS St. Edward’s University

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  1. Slides Prepared by JOHN S. LOUCKS St. Edward’s University

  2. Chapter 10 Statistical Inferences about Means and Proportions for 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 • 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 Assumed Known where: 1 -  is the confidence coefficient • Interval Estimate with 1 and 2 Estimated by s1 and s2 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 Estimated by s1 and s2 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. Using Excel to Develop an Interval Estimate of m1 – m2: Large-Sample Case • Formula Worksheet Note: Rows 16-121 are not shown.

  13. Using Excel to Develop an Interval Estimate of m1 – m2: Large-Sample Case • Value Worksheet Note: Rows 16-121 are not shown.

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

  15. Interval Estimate of 1 - 2:Small-Sample Case (n1 < 30 and/or n2 < 30) • Interval Estimate with 1 and 2 Estimated by s1 and s2 where:

  16. 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

  17. 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.

  18. 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.

  19. 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 0.3 to 4.7 mpg (with the M car having the higher mpg).

  20. Using Excel to Develop an Interval Estimate of m1 – m2: Small-Sample Case • Formula Worksheet

  21. Using Excel to Develop an Interval Estimate of m1 – m2: Small-Sample Case • Value Worksheet

  22. Hypothesis Tests about the Differencebetween 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

  23. 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.

  24. 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

  25. 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

  26. 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.

  27. Using Excel to Conduct a Hypothesis Test about m1 – m2: Large Sample Case • Excel’s “z-Test: Two Sample for Means” Tool Step 1Select the Tools pull-down menu Step 2Choose the Data Analysis option Step 3 Choose z-Test: Two Sample for Means from the list of Analysis Tools … continued

  28. Using Excel to Conduct a Hypothesis Test about m1 – m2: Large Sample Case • Excel’s “z-Test: Two Sample for Means” Tool Step 4When the z-Test: Two Sample for Means dialog box appears: Enter A1:A121 in the Variable 1 Range box Enter B1:B81 in the Variable 2 Range box Enter 0 in the Hypothesized Mean Difference box Enter 225 in the Variable 1 Variance (known) box Enter 400 in the Variable 2 Variance (known) box … continued

  29. Using Excel to Conduct a Hypothesis Test about m1 – m2: Large Sample Case • Excel’s “z-Test: Two Sample for Means” Tool Step 4 (continued) Select Labels Enter .01 in the Alpha box Select Output Range Enter D4 in the Output Range box (Any upper left-hand corner cell indicating where the output is to begin may be entered) Click OK

  30. Using Excel to Conduct a Hypothesis Test about m1 – m2: Large Sample Case • Value Worksheet Note: Rows 16-121 are not shown.

  31. 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

  32. 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:

  33. Using Excel to Conduct a Hypothesis Test about m1 – m2: Small Sample Case • Excel’s “t-Test: Two Sample Assuming Equal Variances” Tool Step 1Select the Tools pull-down menu Step 2Choose the Data Analysis option Step 3 Choose t-Test: Two Sample Assuming Equal Variances from the list of Analysis Tools … continued

  34. Using Excel to Conduct a Hypothesis Test about m1 – m2: Small Sample Case • Excel’s “t-Test: Two Sample Assuming Equal Variances” Tool Step 4When the t-Test: Two Sample Assuming Equal Variances dialog box appears: Enter A1:A13 in the Variable 1 Range box Enter B1:B9 in the Variable 2 Range box Enter 0 in the Hypothesized Mean Difference box … continued

  35. Using Excel to Conduct a Hypothesis Test about m1 – m2: Small Sample Case • Excel’s “t-Test: Two Sample Assuming Equal Variances” Tool Step 4 (continued) Select Labels Enter .01 in the Alpha box Select Output Range Enter D1 in the Output Range box (Any upper left-hand corner cell indicating where the output is to begin may be entered) Click OK

  36. Using Excel to Conduct a Hypothesis Test about m1 – m2: Small Sample Case • Value Worksheet

  37. 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.

  38. 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?

  39. 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

  40. 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

  41. 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.

  42. Using Excel to Conduct a Hypothesis Test about m1 – m2: Matched Samples • Excel’s “t-Test: Paired Two Sample for Means” Tool Step 1Select the Tools pull-down menu Step 2Choose the Data Analysis option Step 3 Choose t-Test: Paired Two Sample for Means from the list of Analysis Tools … continued

  43. Using Excel to Conduct a Hypothesis Test about m1 – m2: Matched Samples • Excel’s “t-Test: Paired Two Sample for Means” Tool Step 4When the t-Test: Paired Two Sample for Means dialog box appears: Enter B1:B11 in the Variable 1 Range box Enter C1:C11 in the Variable 2 Range box Enter 0 in the Hypothesized Mean Difference box Select Labels Enter .05 in the Alpha box Select Output Range Enter E2 (your choice) in the Output Range box Click OK

  44. Using Excel to Conduct a Hypothesis Test about m1 – m2: Matched Samples • Value Worksheet

  45. Inferences about the Difference between the Proportions of Two Populations • Sampling Distribution of • Interval Estimation of p1 - p2 • Hypothesis Tests about p1 - p2

  46. 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.

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

  48. 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?

  49. 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

  50. 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.

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