Slides by John Loucks St. Edward’s University

1. Slides by John Loucks St. Edward’s University

2. Chapter 10, Part B Statistical Inference About Means and Proportions With Two Populations • Inferences About the Difference Between Two Population Means: Matched Samples • Inferences About the Difference Between Two Population Proportions

3. Inferences About the Difference BetweenTwo Population Means: 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.

4. Inferences About the Difference BetweenTwo Population Means: 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.

5. Inferences About the Difference BetweenTwo Population Means: 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.

6. Inferences About the Difference BetweenTwo Population Means: 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

7. Inferences About the Difference BetweenTwo Population Means: Matched Samples • p –Value and Critical Value Approaches 1. Develop 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

8. Inferences About the Difference BetweenTwo Population Means: Matched Samples • p –Value and Critical Value Approaches a = .05 2. Specify the level of significance. 3. Compute the value of the test statistic.

9. Inferences About the Difference BetweenTwo Population Means: Matched Samples • p –Value Approach 4. Compute the p –value. For t = 2.94 and d.f. = 9, the p–value is between .02 and .01. (This is a two-tailed test, so we double the upper-tail areas of .01 and .005.) 5. Determine whether to reject H0. Because p–value <a = .05, we reject H0. We are at least 95% confident that there is a difference in mean delivery times for the two services?

10. Inferences About the Difference BetweenTwo Population Means: Matched Samples • Critical Value Approach 4. Determine the critical value and rejection rule. For a = .05 and d.f. = 9, t.025 = 2.262. Reject H0 if t> 2.262 5. Determine whether to reject H0. Because t = 2.94 > 2.262, we reject H0. We are at least 95% confident that there is a difference in mean delivery times for the two services?

11. Excel’s “t-Test: Paired Two Samplefor Means” Tool Step 1Click the Data tab on the Ribbon Step 2In the Analysis group, click Data Analysis Step 3 Choose t-Test: Paired Two Sample for Means from the list of Analysis Tools Step 4 When the t-Test: Paired Two Sample for Means dialog box appears: (see details on next slide)

12. Excel’s “t-Test: Paired Two Samplefor Means” Tool • Excel Dialog Box

13. Excel’s “t-Test: Paired Two Samplefor Means” Tool • Excel Value Worksheet

14. Inferences About the Difference BetweenTwo Population Proportions • Interval Estimation of p1 - p2 • Hypothesis Tests About p1 - p2

15. Sampling Distribution of • Expected Value • Standard Deviation (Standard Error) where: n1 = size of sample taken from population 1 n2 = size of sample taken from population 2

16. If the sample sizes are large, the sampling distribution of can be approximated by a normal probability distribution. Sampling Distribution of The sample sizes are sufficiently large if all of these conditions are met: n1p1> 5 n1(1 - p1) > 5 n2p2> 5 n2(1 - p2) > 5

17. Sampling Distribution of p1 – p2

18. Interval Estimation of p1 - p2 • Interval Estimate

19. Interval Estimation of p1 - p2 • Example: Market Research Associates 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.

20. Interval Estimation of p1 - p2 • Example: Market Research Associates 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?

21. = sample proportion of households “aware” of the product after the new campaign = sample proportion of households “aware” of the product before the new campaign Point Estimator of the Difference BetweenTwo Population Proportions 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

22. Interval Estimation of p1 - p2 For = .05, z.025 = 1.96: .08 + 1.96(.0510) .08 + .10 Hence, the 95% confidence interval for the difference in before and after awareness of the product is -.02 to +.18.

23. A B C D E 1 Sur2 Sur1 Survey 2 (from Popul.1) Survey 1 (from Popul.2) 2 No Yes Sample Size 250 150 3 Yes No No. of "Yes" =COUNTIF(A2:A251,"Yes") =COUNTIF(B2:B151,"Yes") 4 Yes Yes Samp. Propor. =D3/D2 =E3/E2 5 No Yes 6 Yes No Confid. Coeff. 0.95 7 No No Lev. Of Signif. =1-D6 8 No Yes z Value =NORM.S.INV(1-D7/2,TRUE) 9 Yes No 10 No No Std. Error =SQRT(D4*(1-D4)/D2+E4*(1-E4)/E2) 11 Yes Yes Marg. of Error =D8*D10 12 Yes No 13 Yes Yes Pt. Est. of Diff. =D4-E4 14 No Yes Lower Limit =D13-D11 15 Yes Yes Upper Limit =D13+D11 Interval Estimation of p1 - p2 • Excel Formula Worksheet Note: Rows 16-251 are not shown.

24. A B C D E 1 Sur2 Sur1 Survey 2 (from Popul.1) Survey 1 (from Popul.2) 2 No Yes Sample Size 250 150 3 Yes No No. of "Yes" 120 60 4 Yes Yes Samp. Propor. 0.48 0.40 5 No Yes 6 Yes No Confid. Coeff. 0.95 7 No No Lev. Of Signif. 0.05 8 No Yes z Value 1.960 9 Yes No 10 No No Std. Error 0.0510 11 Yes Yes Marg. of Error 0.0999 12 Yes No 13 Yes Yes Pt. Est. of Diff. 0.080 14 No Yes Lower Limit -0.020 15 Yes Yes Upper Limit 0.180 Using Excel to Developan Interval Estimate of p1 – p2 • Excel Value Worksheet Note: Rows 16-251 are not shown.

25. Hypothesis Tests about p1 - p2 • Hypotheses We focus on tests involving no difference between the two population proportions (i.e. p1 = p2) H0: p1 - p2< 0 Ha: p1 - p2 > 0 Left-tailed Right-tailed Two-tailed

26. Pooled Estimate of Standard Error of Hypothesis Tests about p1 - p2 where:

27. Hypothesis Tests about p1 - p2 • Test Statistic

28. Hypothesis Tests about p1 - p2 • Example: Market Research Associates 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?

29. Hypothesis Tests about p1 - p2 • p -Value and Critical Value Approaches 1. Develop the hypotheses. H0: p1 - p2< 0 Ha: p1 - p2 > 0 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

30. Hypothesis Tests about p1 - p2 • p -Value and Critical Value Approaches a = .05 2. Specify the level of significance. 3. Compute the value of the test statistic.

31. Hypothesis Tests about p1 - p2 • p –Value Approach 4. Compute the p –value. For z = 1.56, the p–value = .0594 5. Determine whether to reject H0. Because p–value > a = .05, we cannot reject H0. • We cannot conclude that the proportion of households • aware of the client’s product increased after the new • campaign.

32. Hypothesis Tests about p1 - p2 • Critical Value Approach 4. Determine the critical value and rejection rule. For a = .05, z.05 = 1.645 Reject H0 if z> 1.645 5. Determine whether to reject H0. Because 1.56 < 1.645, we cannot reject H0. • We cannot conclude that the proportion of households • aware of the client’s product increased after the new • campaign.

33. Hypothesis Tests about p1 - p2 • Excel Formula Worksheet A B C D E Survey 2 (from Popul.1) Survey 1 (from Popul.2) 1 Sur2 Sur1 2 No Yes Sample Size =COUNTA(A2:A251) =COUNTA(B2:B151) 3 Yes No Resp. of Interest Yes Yes 4 Yes Yes Count for Resp. =COUNTIF(A2:A251,D3) =COUNTIF(B2:B151,E3) 5 No Yes Sample Propor. =D4/D2 =E4/E2 6 Yes No Note: Rows 17-251 are not shown. 7 No No Hypoth. Value 0 8 No Yes Point Est. of Diff. =D5-E5 9 Yes No 10 No No Pooled Est. of p =(D2*D5+E2*E5)/(D2+E2) 11 Yes Yes Standard Error =SQRT(D10*(1-D10)*(1/D2+1/E2)) 12 Yes No Test Statistic =(D8-D7)/D11 13 Yes Yes 14 No Yes p -Value (lower tail) =NORM.S.DIST(D12,TRUE) 15 Yes Yes p -Value (upper tail) =1-NORM.S.DIST(D12,TRUE) 16 Yes No p -Value (two tail) =2*MIN(D14,D15)

34. A B C D E 1 Sur2 Sur1 Survey 2 (from Popul.1) Survey 1 (from Popul.2) 2 No Yes Sample Size 250 150 Resp. of Interest 3 Yes No Yes Yes 4 Yes Yes Count for Resp. 120 60 5 No Yes Sample Propor. 0.48 0.40 6 Yes No 7 No No Hypoth. Value 0 8 No Yes Point Est. of Diff. 0.08 9 Yes No 10 No No Pooled Est. of p 0.450 11 Yes Yes Standard Error 0.0514 12 Yes No Test Statistic 1.557 13 Yes Yes 14 No Yes p -Value (lower tail) 0.940 15 Yes Yes p -Value (upper tail) 0.060 16 Yes No p -Value (two tail) 0.120 Hypothesis Tests about p1 - p2 • Excel Value Worksheet Note: Rows 17-251 are not shown.

35. End of Chapter 10, Part B