1 1 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. John Loucks St. Edward’s University SLIDES. BY

2 2 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 10: 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 Means: Matched Samples Two Population Means: Matched Samples n Inferences About the Difference Between Two Population Means:  1 and  2 Unknown Two Population Means:  1 and  2 Unknown

3 3 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Inferences About the Difference Between Two Population Means:  1 and  2 Known Interval Estimation of  1 –  2 Interval Estimation of  1 –  2 Hypothesis Tests About  1 –  2 Hypothesis Tests About  1 –  2

4 4 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Estimating the Difference Between Two Population Means Let  1 equal the mean of population 1 and  2 equal Let  1 equal the mean of population 1 and  2 equal the mean of population 2. the mean of population 2. n The difference between the two population means is  1 -  2.  1 -  2. To estimate  1 -  2, we will select a simple random To estimate  1 -  2, we will select a simple random sample of size n 1 from population 1 and a simple sample of size n 1 from population 1 and a simple random sample of size n 2 from population 2. random sample of size n 2 from population 2. n Let equal the mean of sample 1 and equal the mean of sample 2. mean of sample 2. The point estimator of the difference between the The point estimator of the difference between the means of the populations 1 and 2 is. means of the populations 1 and 2 is.

5 5 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Expected Value Sampling Distribution of n Standard Deviation (Standard Error) 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

6 6 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Interval Estimate Interval Estimation of  1 -  2 :  1 and  2 Known where: 1 -  is the confidence coefficient 1 -  is the confidence coefficient

7 7 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interval Estimation 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 of golf equipment and Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball that has been designed to provide “extra distance.” n Example: Par, Inc.

8 8 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

9 9 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

10 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

11 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

12 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

13 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests About  1   2 :  1 and  2 Known Hypotheses Hypotheses Left-tailedRight-tailedTwo-tailed Test Statistic Test Statistic

14 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: Par, Inc. Hypothesis Tests About  1   2 :  1 and  2 Known 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?

15 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. p –Value and Critical Value Approaches p –Value and Critical Value Approaches Hypothesis Tests About  1   2 :  1 and  2 Known 2. Specify the level of significance.  =.01

16 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 3. Compute the value of the test statistic. Hypothesis Tests About  1   2 :  1 and  2 Known p –Value and Critical Value Approaches p –Value and Critical Value Approaches

17 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. p –Value Approach p –Value Approach 4. Compute the p –value. For z = 6.49, the p –value < Hypothesis Tests About  1   2 :  1 and  2 Known 5. Determine whether to reject H 0. Because p –value <  =.01, we reject H 0. At the.01 level of significance, the sample evidence At the.01 level of significance, 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.

18 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Inferences About the Difference Between Two Population Means:  1 and  2 Unknown Interval Estimation of  1 –  2 Interval Estimation of  1 –  2 Hypothesis Tests About  1 –  2 Hypothesis Tests About  1 –  2

19 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interval Estimation of  1 -  2 :  1 and  2 Unknown When  1 and  2 are unknown, we will: replace z  /2 with t  /2. replace z  /2 with t  /2. use the sample standard deviations s 1 and s 2 use the sample standard deviations s 1 and s 2 as estimates of  1 and  2, and

20 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Where the degrees of freedom for t  /2 are: Interval Estimation of  1 -  2 :  1 and  2 Unknown n Interval Estimate

21 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: Specific Motors Difference Between Two Population Means:  1 and  2 Unknown Specific Motors of Detroit has developed a new Specific Motors of Detroit has developed a new Automobile known as the M car. 24 M cars and 28 J cars (from Japan) were road tested to compare miles- per-gallon (mpg) performance. The sample statistics are shown on the next slide.

22 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Difference Between Two Population Means:  1 and  2 Unknown n Example: Specific Motors Sample Size Sample Mean Sample Std. Dev. Sample #1 M Cars Sample #2 J Cars 24 cars 2 8 cars 24 cars 2 8 cars 29.8 mpg 27.3 mpg 2.56 mpg 1.81 mpg

23 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Difference Between Two Population Means:  1 and  2 Unknown Let us develop a 90% confidence interval estimate Let us develop a 90% confidence interval estimate of the difference between the mpg performances of of the difference between the mpg performances of the two models of automobile. the two models of automobile. n Example: Specific Motors

24 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Point estimate of  1   2 = Point Estimate of  1   2 where:  1 = mean miles-per-gallon for the population of M cars population of M cars  2 = mean miles-per-gallon for the population of J cars population of J cars = = 2.5 mpg

25 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interval Estimation of  1   2 :  1 and  2 Unknown The degrees of freedom for t  /2 are: With  /2 =.05 and df = 24, t  /2 = 1.711

26 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interval Estimation of  1   2 :  1 and  2 Unknown We are 90% confident that the difference between We are 90% confident that the difference between the miles-per-gallon performances of M cars and J cars is to mpg or to mpg

27 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests About  1   2 :  1 and  2 Unknown n Hypotheses Left-tailedRight-tailedTwo-tailed n Test Statistic

28 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: Specific Motors Hypothesis Tests About  1   2 :  1 and  2 Unknown 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? 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?

29 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. H 0 :  1 -  2 < 0  H a :  1 -  2 > 0 where:  1 = mean mpg for the population of M cars  2 = mean mpg for the population of J cars 1. Develop the hypotheses. p –Value and Critical Value Approaches p –Value and Critical Value Approaches Hypothesis Tests About  1   2 :  1 and  2 Unknown

30 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 2. Specify the level of significance. 3. Compute the value of the test statistic.  =.05 p –Value and Critical Value Approaches p –Value and Critical Value Approaches Hypothesis Tests About  1   2 :  1 and  2 Unknown

31 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests About  1   2 :  1 and  2 Unknown p –Value Approach p –Value Approach 4. Compute the p –value. The degrees of freedom for t  are: Because t = > t.005 = 1.683, the p –value t.005 = 1.683, the p –value <.005.

32 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 5. Determine whether to reject H 0. We are at least 95% confident that the miles-per- gallon ( mpg ) performance of M cars is greater than the miles-per-gallon performance of J cars?. We are at least 95% confident that the miles-per- gallon ( mpg ) performance of M cars is greater than the miles-per-gallon performance of J cars?. p –Value Approach p –Value Approach Because p –value <  =.05, we reject H 0. Hypothesis Tests About  1   2 :  1 and  2 Unknown

33 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests About  1   2 :  1 and  2 Unknown, and  1 =  2 Estimate of Standard Error

34 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests About  1   2 :  1 and  2 Unknown, and  1 =  2 n Hypotheses Left-tailedRight-tailedTwo-tailed n Test Statistic

35 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. With a matched-sample design each sampled item With a matched-sample design each sampled item provides a pair of data values. provides a pair of data values. This design often leads to a smaller sampling error This design often leads to a smaller sampling error than the independent-sample design because than the independent-sample design because variation between sampled items is eliminated as a variation between sampled items is eliminated as a source of sampling error. source of sampling error. Inferences About the Difference Between Two Population Means: Matched Samples

36 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: Express Deliveries Inferences About the Difference Between Two Population Means: Matched Samples A Chicago-based firm has documents that must 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.

37 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: Express Deliveries Inferences About the Difference Between Two Population Means: Matched Samples In testing the delivery times of the two services, 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.

38 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part 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

39 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 p –Value and Critical Value Approaches p –Value and Critical Value Approaches

40 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 2. Specify the level of significance.  =.05 Inferences About the Difference Between Two Population Means: Matched Samples p –Value and Critical Value Approaches p –Value and Critical Value Approaches 3. Compute the value of the test statistic.

41 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 5. Determine whether to reject H 0. We are at least 95% confident that there is a difference in mean delivery times for the two services? We are at least 95% confident that there is a difference in mean delivery times for the two services? 4. Compute the p –value. For t = 2.94 and df = 9, the p –value is between For t = 2.94 and df = 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.) Because p –value <  =.05, we reject H 0. Inferences About the Difference Between Two Population Means: Matched Samples p –Value Approach p –Value Approach

42 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 4. Determine the critical value and rejection rule. Inferences About the Difference Between Two Population Means: Matched Samples Critical Value Approach Critical Value Approach 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?

43 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Inferences About the Difference Between Two Population Proportions Two Population Proportions Inference About Means and Proportions with Two Populations

44 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Inferences About the Difference Between Two Population Proportions n Interval Estimation of p 1 - p 2 n Hypothesis Tests About p 1 - p 2

45 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Expected Value Sampling Distribution of where: n 1 = size of sample taken from population 1 n 2 = size of sample taken from population 2 n 2 = size of sample taken from population 2 n Standard Deviation (Standard Error)

46 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. If the sample sizes are large, the sampling distribution If the sample sizes are large, the sampling distribution of can be approximated by a normal probability of can be approximated by a normal probability distribution. distribution. If the sample sizes are large, the sampling distribution If the sample sizes are large, the sampling distribution of can be approximated by a normal probability of can be approximated by a normal probability distribution. distribution. The sample sizes are sufficiently large if all of these The sample sizes are sufficiently large if all of these conditions are met: conditions are met: The sample sizes are sufficiently large if all of these The sample sizes are sufficiently large if all of these conditions are met: conditions are met: n1p1 > 5n1p1 > 5n1p1 > 5n1p1 > 5 n 1 (1 - p 1 ) > 5 n2p2 > 5n2p2 > 5n2p2 > 5n2p2 > 5 n 2 (1 - p 2 ) > 5 Sampling Distribution of

47 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sampling Distribution of p 1 – p 2

48 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interval Estimation of p 1 - p 2 n Interval Estimate

49 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Market Research Associates is conducting research 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.

50 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. A survey conducted immediately after the new 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 that the Does the data support the position that the advertising campaign has provided an increased awareness of the client’s product?

51 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

52 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part (.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:

53 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 )

54 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests about p 1 - p 2 Standard Error of when p 1 = p 2 = p Standard Error of when p 1 = p 2 = p Pooled Estimator of p when p 1 = p 2 = p Pooled Estimator of p when p 1 = p 2 = p

55 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests about p 1 - p 2 Test Statistic Test Statistic

56 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Can we conclude, using a.05 level of significance, 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

57 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests about p 1 - p 2 1. Develop the hypotheses. p -Value and Critical Value Approaches p -Value and Critical Value Approaches 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

58 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests about p 1 - p 2 2. Specify the level of significance.  = Compute the value of the test statistic. p -Value and Critical Value Approaches p -Value and Critical Value Approaches

59 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests about p 1 - p 2 5. Determine whether to reject H 0. We cannot conclude that the proportion of households aware of the client’s product increased after the new campaign. 4. Compute the p –value. For z = 1.56, the p –value =.0594 Because p –value >  =.05, we cannot reject H 0. p –Value Approach p –Value Approach

60 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests about p 1 - p 2 Critical Value Approach Critical Value Approach 5. Determine whether to reject H 0. Because 1.56 < 1.645, we cannot 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.