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Chapter 10b

Chapter 10b. Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Small-Sample Case Using Excel to Conduct a Hypothesis Test about μ 1 – μ 2: Small Sample Inference About the Difference between the Means of Two Populations: Matched Samples.

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Chapter 10b

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  1. Chapter 10b • Hypothesis Tests About the DifferenceBetween the Means of Two Populations: Independent Samples, Small-Sample Case • Using Excel to Conduct aHypothesis Test about μ1 – μ2: Small Sample • Inference About the Difference between the Means of Two Populations: Matched Samples

  2. Hypothesis Tests About the DifferenceBetween the Means of Two Populations:Independent Samples, Small-Sample Case • Example: Specific Motors • Recall that 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 shown on the next slide.

  3. Hypothesis Tests About the DifferenceBetween the Means of Two Populations:Independent Samples, Small-Sample Case • Example: Specific Motors Sample #1 M Cars Sample #2 J Cars 12 cars 8 cars Sample Size 29.8 mpg 27.3 mpg Sample Mean 2.56 mpg 1.81 mpg Sample Std. Dev. 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?

  4. Hypothesis Tests About the DifferenceBetween the Means of Two Populations:Independent Samples, Small-Sample Case • Using the Test Statistic 1. Determine the hypotheses. H0: 1 - 2< 0 Ha: 1 - 2 > 0 where: 1 = mean mpg for the population of M cars 2 = mean mpg for the population of J cars

  5. HypothesisTests About the DifferenceBetween the Means of Two Populations:Independent Samples, Small-Sample Case • Using the Test Statistic a = .05 2. Specify the level of significance. 3. Select the test statistic. where: 4. State the rejection rule. Reject H0 if t > 1.734 (18 degrees of freedom)

  6. continued Hypothesis Tests About the DifferenceBetween the Means of Two Populations: Independent Samples, Small-Sample Case • Using the Test Statistic 5. Compute the value of the test statistic. Pooled Variance Estimator of s 2

  7. Hypothesis Tests About the DifferenceBetween the Means of Two Populations:Independent Samples, Small-Sample Case • Using the Test Statistic 5. Compute the value of the test statistic. (continued) t Statistic

  8. Hypothesis Tests About the DifferenceBetween the Means of Two Populations:Independent Samples, Small-Sample Case • Using the Test Statistic 6. Determine whether to reject H0. t = 2.384 > t.05 = 1.734, so we reject H0. At the .05 level of significance, the sample evidence indicates that the mean mpg of M cars is greater than the mean mpg of J cars.

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

  10. … continued Using Excel to Conduct aHypothesis Test about m1 – m2: Small Sample • 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 Type 0 in the Hypothesized Mean Difference box

  11. Using Excel to Conduct aHypothesis Test about m1 – m2: Small Sample • Excel’s “t-Test: Two Sample Assuming Equal Variances” Tool Step 4 (continued) Select Labels Type .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

  12. Using Excel to Conduct aHypothesis Test about m1 – m2: Small Sample

  13. Using Excel to Conduct aHypothesis Test about m1 – m2: Small Sample • Value Worksheet

  14. Hypothesis Tests About the DifferenceBetween the Means of Two Populations:Independent Samples, Small-Sample Case • Using the p -Value 4. Compute the value of the test statistic. The Excel worksheet shows t = 2.369 5. Compute the p–value. The Excel worksheet shows p-value = .0146 6. Determine whether to reject H0. Because p–value = .0146 < a = .05, we reject H0.

  15. 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. • 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.

  16. Inference About the Difference between the Means of Two Populations: Matched Samples • Example: Express Deliveries 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.

  17. Inference About the Difference between the Means of Two Populations: Matched Samples • Example: Express Deliveries In testing the delivery times of the two services, the firm sent two reports to a random sample of its district offices with one report carried by UPX and the other report carried by INTEX. Do the data on the next slide indicate a difference in mean delivery times for the two services? Use a .05 level of significance.

  18. Inference About the Difference between the Means of Two Populations: Matched Samples Delivery Time (Hours) District Office UPX INTEX Difference 32 30 19 16 15 18 14 10 7 16 25 24 15 15 13 15 15 8 9 11 7 6 4 1 2 3 -1 2 -2 5 Seattle Los Angeles Boston Cleveland New York Houston Atlanta St. Louis Milwaukee Denver

  19. Inference About the Difference between the Means of Two Populations: Matched Samples • Using the Test Statistic 1. Determine the hypotheses. H0: d = 0 Ha: d Let d = the mean of the difference values for the two delivery services for the population of district offices

  20. Inference About the Difference between the Means of Two Populations: Matched Samples • Using the Test Statistic a = .05 2. Specify the level of significance. 3. Select the test statistic. and where: 4. State the rejection rule. Reject H0 if |t| > 2.262 (9 degrees of freedom)

  21. Inference About the Difference between the Means of Two Populations: Matched Samples • Using the Test Statistic 5. Compute the value of the test statistic.

  22. Inference About the Difference between the Means of Two Populations: Matched Samples • Using the Test Statistic 6. Determine whether to reject H0. t = 2.94 > t.05/2 = 2.262, so we reject H0. At the .05 level of significance, the sample evidence indicates that there is a significant difference between the mean delivery times for the two services.

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

  24. Using Excel to Conduct aHypothesis 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 Type 0 in the Hypothesized Mean Difference box Select Labels Type .05 in the Alpha box Select Output Range Enter E2 (your choice) in the Output Range box Click OK

  25. Using Excel to Conduct aHypothesis Test about m1 – m2: Matched Samples

  26. Using Excel to Conduct aHypothesis Test about m1 – m2: Matched Samples • Value Worksheet

  27. Inference About the Difference between the Means of Two Populations: Matched Samples • Using the p -Value 4. Compute the value of the test statistic. The Excel worksheet shows t = 2.9362 5. Compute the p–value. The Excel worksheet shows p-value = .0166 6. Determine whether to reject H0. Because p–value = .0166 < a = .05, we reject H0.

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