1 1 Slide MA4704 Problem solving 4 Statistical Inference About Means and Proportions With Two Populations n Inferences About the Difference Between Two Population Means:  1 and  2 Known Two Population Means:  1 and  2 Known n Inferences About the Difference Between Two Population Proportions Two Population Proportions n Inferences About the Difference Between Two Population Means: Matched Samples Two Population Means: Matched Samples

2 2 Slide MA4704 n Example: Par, Inc. Interval Estimate of  1 -  2 :  1 and  2 Known Interval Estimate of  1 -  2 :  1 and  2 Known In a test of driving distance using a mechanical 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. Par, Inc. is a manufacturer Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball that has been designed to provide “extra distance.”

3 3 Slide MA4704 n Example: Par, Inc. Interval Estimation of  1 -  2 :  1 and  2 Known Sample Size Sample Mean Sample #1 Par, Inc. Sample #2 Rap, Ltd. 120 balls 80 balls 120 balls 80 balls 275 yards 258 yards Based on data from previous driving distance Based on data from previous driving distance tests, the two population standard deviations are known with  1 = 15 yards and  2 = 20 yards.

4 4 Slide MA4704 Interval Estimation of  1 -  2 :  1 and  2 Known n Example: Par, Inc. Let us develop a 95% confidence interval estimate Let us develop a 95% confidence interval estimate of the difference between the mean driving distances of the two brands of golf ball.

5 5 Slide MA4704 Estimating the Difference Between Two Population Means  1 –  2 = difference between the mean distances the mean distances x 1 - x 2 = Point Estimate of  1 –  2 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 Simple random sample Simple random sample of n 2 Rap golf balls of n 2 Rap golf balls x 2 = sample mean distance for the Rap golf balls for the Rap golf balls Simple random sample Simple random sample of n 2 Rap golf balls of n 2 Rap golf balls x 2 = sample mean distance for the Rap golf balls for the Rap golf balls Simple random sample Simple random sample of n 1 Par golf balls of n 1 Par golf balls x 1 = sample mean distance for the Par golf balls for the Par golf balls Simple random sample Simple random sample of n 1 Par golf balls of n 1 Par golf balls x 1 = sample mean distance for the Par golf balls for the Par golf balls

6 6 Slide MA4704 Point Estimate of  1 -  2 Point estimate of  1   2 = where:  1 = mean distance for the population of Par, Inc. golf balls of Par, Inc. golf balls  2 = mean distance for the population of Rap, Ltd. golf balls of Rap, Ltd. golf balls = 275  258 = 17 yards

7 7 Slide MA4704 Interval Estimation of  1 -  2 :   1 and   2 Known We are 95% confident that the difference between We are 95% confident that the difference between the mean driving distances of Par, Inc. balls and Rap, Ltd. balls is to yards or yards to yards

8 8 Slide MA4704 n Example: Par, Inc. Hypothesis Tests About  1   2 :  1 and  2 Known Can we conclude, using Can we conclude, using  =.01, that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls?

9 9 Slide MA4704 H 0 :  1 -  2 < 0  H a :  1 -  2 > 0 where:  1 = mean distance for the population of Par, Inc. golf balls of Par, Inc. golf balls  2 = mean distance for the population of Rap, Ltd. golf balls of Rap, Ltd. golf balls 1. Develop the hypotheses. Hypothesis Tests About  1   2 :  1 and  2 Known 2. Specify the level of significance.  =.01

10 Slide MA Compute the Alarm Signal : Noise Hypothesis Tests About  1   2 :  1 and  2 Known

11 Slide MA4704 Hypothesis Tests About  1   2 :  1 and  2 Known 5. Determine whether to reject H 0. Because Alarm Signal: Noise= 6.49 > 2.33, we reject H 0. For  =.01, z.01 = Determine the critical value and rejection rule. Reject H 0 if Alarm Signal: Noise > 2.33 The sample evidence indicates the mean driving The sample evidence indicates the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls.

12 Slide MA4704 n Example: Express Deliveries Inferences About the Difference Between Two Population Means: Matched Samples A Chicago-based firm has 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.

13 Slide MA4704 n Example: Express Deliveries Inferences About the Difference Between Two Population Means: Matched Samples In testing the delivery times 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.

14 Slide MA UPXINTEXDifference District Office Seattle Los Angeles Boston Cleveland New York Houston Atlanta St. Louis Milwaukee Denver Delivery Time (Hours) Inferences About the Difference Between Two Population Means: Matched Samples

15 Slide MA4704 H 0 :  d = 0  H a :  d  Let  d = the mean of the difference values for the two delivery services for the population two delivery services for the population of district offices of district offices 1. Develop the hypotheses. Inferences About the Difference Between Two Population Means: Matched Samples

16 Slide MA Specify the level of significance.  =.05 Inferences About the Difference Between Two Population Means: Matched Samples 3. Compute the value of the test statistic.

17 Slide MA Determine the critical value and rejection rule. Inferences About the Difference Between Two Population Means: Matched Samples For  =.05 and df = 9, t.025 = Reject H 0 if t > Determine whether to reject H 0. Because t = 2.94 > 2.262, we reject H 0. We are at least 95% confident that there is a difference in mean delivery times for the two services?

18 Slide MA4704 Market Research Associates is Market Research Associates is conducting research to evaluate the effectiveness of a client’s new adver- tising 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. Interval Estimation of p 1 - p 2 n Example: Market Research Associates The new campaign has been initiated with TV and The new campaign has been initiated with TV and newspaper advertisements running for three weeks.

19 Slide MA4704 A survey conducted immediately A survey conducted immediately after the new campaign showed 120 of 250 households “aware” of the client’s product. Interval Estimation of p 1 - p 2 n Example: Market Research Associates Does the data support the position Does the data support the position that the advertising campaign has provided an increased awareness of the client’s product?

20 Slide MA4704 Point Estimator of the Difference Between Two Population Proportions = 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 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

21 Slide MA (.0510) Interval Estimation of p 1 - p 2 Hence, the 95% confidence interval for the difference Hence, the 95% confidence interval for the difference in before and after awareness of the product is -.02 to For  =.05, z.025 = 1.96:

22 Slide MA4704 Hypothesis Tests about p 1 - p 2 n Hypotheses H 0 : p 1 - p 2 < 0 H a : p 1 - p 2 > 0 Left-tailedRight-tailedTwo-tailed We focus on tests involving no difference between the two population proportions (i.e. p 1 = p 2 )

23 Slide MA4704 Hypothesis Tests about p 1 - p 2 Pooled Estimate of Standard Error of Pooled Estimate of Standard Error ofwhere:

24 Slide MA4704 Hypothesis Tests about p 1 - p 2 Test Statistic Test Statistic

25 Slide MA4704 Can we conclude, using a.05 level 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? Hypothesis Tests about p 1 - p 2 n Example: Market Research Associates

26 Slide MA4704 Hypothesis Tests about p 1 - p 2 1. Develop the hypotheses. H 0 : p 1 - p 2 < 0 H a : p 1 - p 2 > 0 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

27 Slide MA4704 Hypothesis Tests about p 1 - p 2 2. Specify the level of significance.  = Compute the value of the test statistic.

28 Slide MA4704 Hypothesis Tests about p 1 - p 2 5. Determine whether to reject H 0. Because 1.56 < 1.645, we Fail to Reject H 0. For  =.05, z.05 = Determine the critical value and rejection rule. Reject H 0 if z > We cannot conclude that the proportion of households aware of the client’s product increased after the new campaign.