# 1 1 Slide © 2003 South-Western/Thomson Learning™ Slides Prepared by JOHN S. LOUCKS St. Edward’s University.

## Presentation on theme: "1 1 Slide © 2003 South-Western/Thomson Learning™ Slides Prepared by JOHN S. LOUCKS St. Edward’s University."— Presentation transcript:

1 1 Slide © 2003 South-Western/Thomson Learning™ Slides Prepared by JOHN S. LOUCKS St. Edward’s University

2 2 Slide © 2003 South-Western/Thomson Learning™ Chapter 10 Statistical Inferences about Means and Proportions for Two Populations n Estimation of the Difference Between the Means of Two Populations: Independent Samples Two Populations: Independent Samples n Hypothesis Tests about the Difference Between the Means of Two Populations: Independent Samples Means of Two Populations: Independent Samples n Inferences about the Difference Between the Means of Two Populations: Matched Samples of Two Populations: Matched Samples n Inferences about the Difference Between the Proportions of Two Populations Proportions of Two Populations

3 3 Slide © 2003 South-Western/Thomson Learning™ Estimation of the Difference between the Means of Two Populations: Independent Samples n Point Estimator of the Difference between the Means of Two Populations n Sampling Distribution Interval Estimate of      Large-Sample Case Interval Estimate of      Large-Sample Case Interval Estimate of      Small-Sample Case Interval Estimate of      Small-Sample Case

4 4 Slide © 2003 South-Western/Thomson Learning™ 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. 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. The difference between the two population means is  1 -  2. To estimate  1 -  2, we will select a simple random sample of size n 1 from population 1 and a simple random sample of size n 2 from population 2. To estimate  1 -  2, we will select a simple random sample of size n 1 from population 1 and a simple random sample of size n 2 from population 2. n Let equal the mean of sample 1 and equal the mean of sample 2. n The point estimator of the difference between the means of the populations 1 and 2 is.

5 5 Slide © 2003 South-Western/Thomson Learning™ n Properties of the Sampling Distribution of Expected Value Expected Value Standard Deviation Standard Deviation where:  1 = standard deviation of population 1  2 = standard deviation of population 2  2 = standard deviation of population 2 n 1 = sample size from population 1 n 1 = sample size from population 1 n 2 = sample size from population 2 n 2 = sample size from population 2 Sampling Distribution of

6 6 Slide © 2003 South-Western/Thomson Learning™ Interval Estimate with  1 and  2 Assumed Known Interval Estimate with  1 and  2 Assumed Knownwhere: 1 -  is the confidence coefficient Interval Estimate with  1 and  2 Estimated by s 1 and s 2 Interval Estimate with  1 and  2 Estimated by s 1 and s 2where: Interval Estimate of  1 -  2 : Large-Sample Case ( n 1 > 30 and n 2 > 30)

7 7 Slide © 2003 South-Western/Thomson Learning™ Example: Par, Inc. Interval Estimate of  1 -  2 : Large-Sample Case 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 8 Slide © 2003 South-Western/Thomson Learning™ Example: Par, Inc. Interval Estimate of  1 -  2 : Large-Sample Case Interval Estimate of  1 -  2 : Large-Sample Case Sample Statistics Sample Statistics Sample #1 Sample #2 Sample #1 Sample #2 Par, Inc. Rap, Ltd. Par, Inc. Rap, Ltd. Sample Size n 1 = 120 balls n 2 = 80 balls Mean = 235 yards = 218 yards Standard Dev. s 1 = 15 yards s 2 = 20 yards

9 9 Slide © 2003 South-Western/Thomson Learning™ n Point Estimate of the Difference Between Two Population Means  1 = mean distance for the population of Par, Inc. golf balls Par, Inc. golf balls  2 = mean distance for the population of Rap, Ltd. golf balls Rap, Ltd. golf balls Point estimate of  1 -  2 = = 235 - 218 = 17 yards. Example: Par, Inc.

10 Slide © 2003 South-Western/Thomson Learning™ Point Estimator of the Difference between the Means of Two Populations Population 1 Par, Inc. Golf Balls  1 = mean driving distance of Par distance of Par golf balls Population 1 Par, Inc. Golf Balls  1 = mean driving distance of Par distance of Par golf balls Population 2 Rap, Ltd. Golf Balls  2 = mean driving distance of Rap distance of Rap golf balls Population 2 Rap, Ltd. Golf Balls  2 = mean driving distance of Rap distance of Rap golf balls  1 –  2 = difference between the mean distances the mean distances Simple random sample Simple random sample of n 1 Par golf balls of n 1 Par golf balls x 1 = sample mean distance for sample of Par golf ball Simple random sample Simple random sample of n 1 Par golf balls of n 1 Par golf balls x 1 = sample mean distance for sample of Par golf ball Simple random sample Simple random sample of n 2 Rap golf balls of n 2 Rap golf balls x 2 = sample mean distance for sample of Rap golf ball Simple random sample Simple random sample of n 2 Rap golf balls of n 2 Rap golf balls x 2 = sample mean distance for sample of Rap golf ball x 1 - x 2 = Point Estimate of  1 –  2

11 Slide © 2003 South-Western/Thomson Learning™ 95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Case,  1 and  2 Estimated by s 1 and s 2 95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Case,  1 and  2 Estimated by s 1 and s 2 Substituting the sample standard deviations for the population standard deviation: = 17 + 5.14 or 11.86 yards to 22.14 yards. = 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. Example: Par, Inc.

12 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Develop an Interval Estimate of  1 –  2 : Large-Sample Case n Formula Worksheet Note: Rows 16-121 are not shown.

13 Slide © 2003 South-Western/Thomson Learning™ n Value Worksheet Using Excel to Develop an Interval Estimate of  1 –  2 : Large-Sample Case Note: Rows 16-121 are not shown.

14 Slide © 2003 South-Western/Thomson Learning™ Interval Estimate of  1 -  2 : Small-Sample Case ( n 1 < 30 and/or n 2 < 30) Interval Estimate with  2 Assumed Known Interval Estimate with  2 Assumed Knownwhere:

15 Slide © 2003 South-Western/Thomson Learning™ Interval Estimate of  1 -  2 : Small-Sample Case ( n 1 < 30 and/or n 2 < 30) Interval Estimate with  1 and  2 Estimated by s 1 and s 2 Interval Estimate with  1 and  2 Estimated by s 1 and s 2where:

16 Slide © 2003 South-Western/Thomson Learning™ 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 Sample #1 Sample #2 M Cars J Cars M Cars J Cars Sample Size n 1 = 12 cars n 2 = 8 cars Mean = 29.8 mpg = 27.3 mpg Standard Deviation s 1 = 2.56 mpg s 2 = 1.81 mpg

17 Slide © 2003 South-Western/Thomson Learning™ n Point Estimate of the Difference Between Two Population Means  1 = mean miles-per-gallon for the population of M cars M cars  2 = mean miles-per-gallon for the population of J cars J cars Point estimate of  1 -  2 = = 29.8 - 27.3 = 2.5 mpg. Example: Specific Motors

18 Slide © 2003 South-Western/Thomson Learning™ n 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 The miles per gallon rating must be normally distributed for both the M car and the J car. distributed for both the M car and the J car. The variance in the miles per gallon rating must The variance in the miles per gallon rating must be the same for both the M car and the J car. be the same for both the M car and the J car. Using the t distribution with n 1 + n 2 - 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. Example: Specific Motors

19 Slide © 2003 South-Western/Thomson Learning™ n 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. = 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). Example: Specific Motors

20 Slide © 2003 South-Western/Thomson Learning™ n Formula Worksheet Using Excel to Develop an Interval Estimate of  1 –  2 : Small-Sample Case

21 Slide © 2003 South-Western/Thomson Learning™ n Value Worksheet Using Excel to Develop an Interval Estimate of  1 –  2 : Small-Sample Case

22 Slide © 2003 South-Western/Thomson Learning™ Hypothesis Tests about the Difference between the Means of Two Populations: Independent Samples n Hypotheses H 0 :  1 -  2 0 H 0 :  1 -  2 = 0 H a :  1 -  2 > 0 H a :  1 -  2 0 H a :  1 -  2 < 0 H a :  1 -  2  0 n Test Statistic Large-Sample Small-Sample Large-Sample Small-Sample

23 Slide © 2003 South-Western/Thomson Learning™ n 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. Example: Par, Inc.

24 Slide © 2003 South-Western/Thomson Learning™ Example: Par, Inc. n Hypothesis Tests about the Difference between the Means of Two Populations: Large-Sample Case Sample Statistics Sample Statistics Sample #1 Sample #2 Sample #1 Sample #2 Par, Inc. Rap, Ltd. Par, Inc. Rap, Ltd. Sample Size n 1 = 120 balls n 2 = 80 balls Mean = 235 yards = 218 yards Standard Dev. s 1 = 15 yards s 2 = 20 yards

25 Slide © 2003 South-Western/Thomson Learning™ n 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 H 0 :  1 -  2 < 0 Hypotheses H 0 :  1 -  2 < 0 H a :  1 -  2 > 0 Example: Par, Inc.

26 Slide © 2003 South-Western/Thomson Learning™ n Hypothesis Tests about the Difference between the Means of Two Populations: Large-Sample Case Rejection Rule Reject H 0 if z > 2.33 Rejection Rule Reject H 0 if z > 2.33 Conclusion Conclusion Reject H 0. 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. Reject H 0. 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. Example: Par, Inc.

27 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Conduct a Hypothesis Test about  1 –  2 : Large Sample Case n Excel’s “ z -Test: Two Sample for Means” Tool Step 1 Select the Tools pull-down menu Step 2 Choose the Data Analysis option Step 3 Choose z -Test: Two Sample for Means from the list of Analysis Tools … continued

28 Slide © 2003 South-Western/Thomson Learning™ n Excel’s “ z -Test: Two Sample for Means” Tool Step 4 When the z-Test: Two Sample for Means dialog box appears: dialog box appears: Enter A1:A121 in the Variable 1 Range box Enter A1:A121 in the Variable 1 Range box Enter B1:B81 in the Variable 2 Range box Enter B1:B81 in the Variable 2 Range box Enter 0 in the Hypothesized Mean Difference box Enter 0 in the Hypothesized Mean Difference box Enter 225 in the Variable 1 Variance (known) box Enter 225 in the Variable 1 Variance (known) box Enter 400 in the Variable 2 Variance (known) box Enter 400 in the Variable 2 Variance (known) box … continued Using Excel to Conduct a Hypothesis Test about  1 –  2 : Large Sample Case

29 Slide © 2003 South-Western/Thomson Learning™ n Excel’s “ z -Test: Two Sample for Means” Tool Step 4 (continued) Select Labels Select Labels Enter.01 in the Alpha box Enter.01 in the Alpha box Select Output Range Select Output Range Enter D4 in the Output Range box Enter D4 in the Output Range box (Any upper left-hand corner cell indicating (Any upper left-hand corner cell indicating where the output is to begin may be entered) where the output is to begin may be entered) Click OK Click OK Using Excel to Conduct a Hypothesis Test about  1 –  2 : Large Sample Case

30 Slide © 2003 South-Western/Thomson Learning™ n Value Worksheet Using Excel to Conduct a Hypothesis Test about  1 –  2 : Large Sample Case Note: Rows 16-121 are not shown.

31 Slide © 2003 South-Western/Thomson Learning™ n 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 Hypotheses H 0 :  1 -  2 < 0 Hypotheses H 0 :  1 -  2 < 0 H a :  1 -  2 > 0 Example: Specific Motors

32 Slide © 2003 South-Western/Thomson Learning™ Example: Specific Motors n Hypothesis Tests about the Difference between the Means of Two Populations: Small-Sample Case Rejection Rule Rejection Rule Reject H 0 if t > 1.734 Reject H 0 if t > 1.734 (  =.05, d.f. = 18) (  =.05, d.f. = 18) Test Statistic Test Statistic where: where:

33 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Conduct a Hypothesis Test about  1 –  2 : Small Sample Case n Excel’s “ t -Test: Two Sample Assuming Equal Variances” Tool Step 1 Select the Tools pull-down menu Step 2 Choose the Data Analysis option Step 3 Choose t -Test: Two Sample Assuming Equal Variances from the list of Analysis Tools … continued

34 Slide © 2003 South-Western/Thomson Learning™ n Excel’s “ t -Test: Two Sample Assuming Equal Variances” Tool Step 4 When the t -Test: Two Sample Assuming Equal Variances dialog box appears: Enter A1:A13 in the Variable 1 Range box Enter A1:A13 in the Variable 1 Range box Enter B1:B9 in the Variable 2 Range box Enter B1:B9 in the Variable 2 Range box Enter 0 in the Hypothesized Mean Difference box Enter 0 in the Hypothesized Mean Difference box … continued Using Excel to Conduct a Hypothesis Test about  1 –  2 : Small Sample Case

35 Slide © 2003 South-Western/Thomson Learning™ n Excel’s “ t -Test: Two Sample Assuming Equal Variances” Tool Step 4 (continued) Select Labels Select Labels Enter.01 in the Alpha box Enter.01 in the Alpha box Select Output Range Select Output Range Enter D1 in the Output Range box Enter D1 in the Output Range box (Any upper left-hand corner cell indicating (Any upper left-hand corner cell indicating where the output is to begin may be entered) where the output is to begin may be entered) Click OK Click OK Using Excel to Conduct a Hypothesis Test about  1 –  2 : Small Sample Case

36 Slide © 2003 South-Western/Thomson Learning™ n Value Worksheet Using Excel to Conduct a Hypothesis Test about  1 –  2 : Small Sample Case

37 Slide © 2003 South-Western/Thomson Learning™ Inference about the Difference between the Means of Two Populations: Matched Samples n With a matched-sample design each sampled item provides a pair of data values. n The matched-sample design can be referred to as blocking. n 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 Slide © 2003 South-Western/Thomson Learning™ Example: Express Deliveries n 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 Slide © 2003 South-Western/Thomson Learning™ Delivery Time (Hours) Delivery Time (Hours) District OfficeUPX INTEX Difference 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 Example: Express Deliveries

40 Slide © 2003 South-Western/Thomson Learning™ n 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 Let  d = the mean of the difference values for the two delivery services for the population of district offices Hypotheses H 0 :  d = 0, H a :  d  Hypotheses H 0 :  d = 0, H a :  d  Rejection Rule 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). 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 H 0 if t 2.262 Reject H 0 if t 2.262 Example: Express Deliveries

41 Slide © 2003 South-Western/Thomson Learning™ n Inference about the Difference between the Means of Two Populations: Matched Samples Conclusion Reject H 0. Conclusion Reject H 0. There is a significant difference between the mean delivery times for the two services. There is a significant difference between the mean delivery times for the two services. Example: Express Deliveries

42 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Conduct a Hypothesis Test about  1 –  2 : Matched Samples n Excel’s “ t -Test: Paired Two Sample for Means” Tool Step 1 Select the Tools pull-down menu Step 2 Choose the Data Analysis option Step 3 Choose t -Test: Paired Two Sample for Means from the list of Analysis Tools … continued

43 Slide © 2003 South-Western/Thomson Learning™ n Excel’s “ t -Test: Paired Two Sample for Means” Tool Step 4 When the t -Test: Paired Two Sample for Means dialog box appears: dialog box appears: Enter B1:B11 in the Variable 1 Range box Enter B1:B11 in the Variable 1 Range box Enter C1:C11 in the Variable 2 Range box Enter C1:C11 in the Variable 2 Range box Enter 0 in the Hypothesized Mean Difference box Enter 0 in the Hypothesized Mean Difference box Select Labels Select Labels Enter.05 in the Alpha box Enter.05 in the Alpha box Select Output Range Select Output Range Enter E2 (your choice) in the Output Range box Enter E2 (your choice) in the Output Range box Click OK Click OK Using Excel to Conduct a Hypothesis Test about  1 –  2 : Matched Samples

44 Slide © 2003 South-Western/Thomson Learning™ n Value Worksheet Using Excel to Conduct a Hypothesis Test about  1 –  2 : Matched Samples

45 Slide © 2003 South-Western/Thomson Learning™ Inferences about the Difference between the Proportions of Two Populations n Sampling Distribution of n Interval Estimation of p 1 - p 2 n Hypothesis Tests about p 1 - p 2

46 Slide © 2003 South-Western/Thomson Learning™ n Expected Value n Standard Deviation n Distribution Form If the sample sizes are large ( n 1 p 1, n 1 (1 - p 1 ), n 2 p 2, and n 2 (1 - p 2 ) are all greater than or equal to 5), the sampling distribution of can be approximated by a normal probability distribution. Sampling Distribution of

47 Slide © 2003 South-Western/Thomson Learning™ Interval Estimation of p 1 - p 2 n Interval Estimate n Point Estimator of

48 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ Example: MRA n Point Estimator of the Difference between the Proportions of Two Populations p 1 = proportion of the population of households p 1 = proportion of the population of households “aware” of the product after the new campaign “aware” of the product after the new campaign p 2 = proportion of the population of households p 2 = proportion of the population of households “aware” of the product before the new campaign “aware” of the product before the new campaign = sample proportion of households “aware” of the = sample proportion of households “aware” of the product after the new campaign product after the new campaign = sample proportion of households “aware” of the = sample proportion of households “aware” of the product before the new campaign product before the new campaign

50 Slide © 2003 South-Western/Thomson Learning™ Example: MRA n Interval Estimate of p 1 - p 2 : Large-Sample Case For  =.05, z.025 = 1.96: For  =.05, z.025 = 1.96:.08 + 1.96(.0510).08 +.10.08 +.10 or -.02 to +.18 Conclusion 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. 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.

51 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Develop an Interval Estimate of p 1 – p 2 n Formula Worksheet Note: Rows 16-251 are not shown.

52 Slide © 2003 South-Western/Thomson Learning™ n Value Worksheet Using Excel to Develop an Interval Estimate of p 1 – p 2 Note: Rows 16-251 are not shown.

53 Slide © 2003 South-Western/Thomson Learning™ Hypothesis Tests about p 1 - p 2 n Hypotheses H 0 : p 1 - p 2 < 0 H 0 : p 1 - p 2 < 0 H a : p 1 - p 2 > 0 H a : p 1 - p 2 > 0 n Test statistic n Point Estimator of where p 1 = p 2 where:

54 Slide © 2003 South-Western/Thomson Learning™ Example: MRA n Hypothesis Tests about p 1 - p 2 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? p 1 = proportion of the population of households “aware” of the product after the new campaign “aware” of the product after the new campaign p 2 = proportion of the population of households p 2 = proportion of the population of households “aware” of the product before the new campaign “aware” of the product before the new campaign Hypotheses H 0 : p 1 - p 2 < 0 Hypotheses H 0 : p 1 - p 2 < 0 H a : p 1 - p 2 > 0 H a : p 1 - p 2 > 0

55 Slide © 2003 South-Western/Thomson Learning™ Example: MRA n Hypothesis Tests about p 1 - p 2 Rejection Rule Reject H 0 if z > 1.645 Rejection Rule Reject H 0 if z > 1.645 Test Statistic Test Statistic Conclusion Do not reject H 0. Conclusion Do not reject H 0.

56 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Conduct a Hypothesis Test about p 1 – p 2 n Formula Worksheet Note: Rows 17-251 are not shown.

57 Slide © 2003 South-Western/Thomson Learning™ n Value Worksheet Using Excel to Conduct a Hypothesis Test about p 1 – p 2 Note: Rows 17-251 are not shown.

58 Slide © 2003 South-Western/Thomson Learning™ End of Chapter 10

Download ppt "1 1 Slide © 2003 South-Western/Thomson Learning™ Slides Prepared by JOHN S. LOUCKS St. Edward’s University."

Similar presentations