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

2 2 Slide Chapter 10 Inference About Means and Proportions with Two Populations n Inferences About the Difference Between Two Population Means:  1 and  2 Known -- SKIP Two Population Means:  1 and  2 Known -- SKIP 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 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.

4 4 Slide 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

5 5 Slide Sampling Distribution of n Estimate of Standard Error where: S 1 = standard deviation of sample from population 1 S 2 = standard deviation of sample from population 2 S 2 = standard deviation of sample from 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 Hypothesis Tests About  1   2 :  1 and  2 Unknown n Hypotheses Left-tailedRight-tailedTwo-tailed n Test Statistic

7 7 Slide 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?

8 8 Slide 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. Hypothesis Tests About  1   2 :  1 and  2 Unknown

9 9 Slide 2. Specify the level of significance. 3. Compute the value of the test statistic.  =.05 Hypothesis Tests About  1   2 :  1 and  2 Unknown

10 Slide 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.

11 Slide 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

12 Slide Sampling Distribution of Estimate of Standard Error if  1 =  2 Estimate of Standard Error if  1 =  2 where: S p = Pooled estimate of the common standard deviation

13 Slide Hypothesis Tests About  1   2 :  1 and  2 Unknown n Hypotheses Left-tailedRight-tailedTwo-tailed n Test Statistic

14 Slide 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

15 Slide 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.

16 Slide 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.

17 Slide 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

18 Slide 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

19 Slide 2. Specify the level of significance.  =.05 Inferences About the Difference Between Two Population Means: Matched Samples 3. Compute the value of the test statistic.

20 Slide 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

21 Slide 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?

22 Slide n Inferences About the Difference Between Two Population Proportions Two Population Proportions Inference About Means and Proportions with Two Populations

23 Slide Inferences About the Difference Between Two Population Proportions n Interval Estimation of p 1 - p 2 n Hypothesis Tests About p 1 - p 2

24 Slide 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)

25 Slide 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

26 Slide Sampling Distribution of p 1 – p 2

27 Slide Interval Estimation of p 1 - p 2 n Interval Estimate

28 Slide 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.

29 Slide 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 )

30 Slide 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

31 Slide Hypothesis Tests about p 1 - p 2 Test Statistic Test Statistic

32 Slide 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

33 Slide 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

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

35 Slide 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

36 Slide End of Chapter 10