Chapter 10b Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Small-Sample CaseHypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Small-Sample Case Using Excel to Conduct a Hypothesis Test about μ 1 – μ 2: Small SampleUsing Excel to Conduct a Hypothesis Test about μ 1 – μ 2: Small Sample Inference About the Difference between the Means of Two Populations: Matched Samples

Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Small-Sample Case n Example: Specific Motors Recall that 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 shown on the next slide. next slide.

n Example: Specific Motors Sample Size Sample Mean Sample Std. Dev. Sample #1 M Cars Sample #2 J Cars 12 cars 8 cars 12 cars 8 cars 29.8 mpg 27.3 mpg 2.56 mpg 1.81 mpg Can we conclude, using a.05 level of significance, that the miles-per-gallon ( mpg ) performance of M cars is 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? Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Small-Sample Case

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. Determine the hypotheses. Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Small-Sample Case Using the Test Statistic Using the Test Statistic

Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Small-Sample Case 2. Specify the level of significance. 3. Select the test statistic.  = State the rejection rule. Reject H 0 if t > (18 degrees of freedom) where: Using the Test Statistic Using the Test Statistic

Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Small-Sample Case 5. Compute the value of the test statistic. Pooled Variance Estimator of  2 continued Using the Test Statistic Using the Test Statistic

Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Small-Sample Case 5. Compute the value of the test statistic. (continued) t Statistic Using the Test Statistic Using the Test Statistic

6. Determine whether to reject H 0. At the.05 level of significance, the sample At the.05 level of significance, the sample evidence indicates that the mean mpg of M cars is greater than the mean mpg of J cars. t = > t.05 = 1.734, so we reject H 0. Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Small-Sample Case Using the Test Statistic Using the Test Statistic

Step 1 Select the Tools menu Step 2 Choose the Data Analysis option Step 3 Choose t -Test: Two Sample Assuming Equal Variances from the list of Analysis Tools Variances from the list of Analysis Tools … continued n Excel’s “ t -Test: Two Sample Assuming Equal Variances” Tool Using Excel to Conduct a Hypothesis Test about  1 –  2 : Small Sample

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

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

Using Excel to Conduct a Hypothesis Test about  1 –  2 : Small Sample

n Value Worksheet Using Excel to Conduct a Hypothesis Test about  1 –  2 : Small Sample

Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Small-Sample Case Using the p  Value Using the p  Value 4. Compute the value of the test statistic. 5. Compute the p –value. The Excel worksheet shows p -value = Determine whether to reject H 0. Because p –value =.0146 <  =.05, we reject H 0. The Excel worksheet shows t = 2.369

Inference About the Difference between the Means of Two Populations: Matched Samples 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.

n 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. Inference About the Difference between the Means of Two Populations: Matched Samples

n 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. Inference About the Difference between the Means of Two Populations: Matched Samples

UPXINTEXDifference District Office Seattle Los Angeles Boston Cleveland New York Houston Atlanta St. Louis Milwaukee Denver Delivery Time (Hours) Inference About the Difference between the Means of Two Populations: Matched Samples

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. Determine the hypotheses. Using the Test Statistic Using the Test Statistic Inference About the Difference between the Means of Two Populations: Matched Samples

2. Specify the level of significance. 3. Select the test statistic.  = State the rejection rule. Reject H 0 if | t| > (9 degrees of freedom) where: Using the Test Statistic Using the Test Statistic and

Inference About the Difference between the Means of Two Populations: Matched Samples 5. Compute the value of the test statistic. Using the Test Statistic Using the Test Statistic

Inference About the Difference between the Means of Two Populations: Matched Samples 6. Determine whether to reject H 0. At the.05 level of significance, the sample evidence At the.05 level of significance, the sample evidence indicates that there is a significant difference between the mean delivery times for the two services. t = 2.94 > t.05/2 = 2.262, so we reject H 0. Using the Test Statistic Using the Test Statistic

Using Excel to Conduct a Hypothesis Test about  1 –  2 : Matched Samples Step 1 Select the Tools menu Step 2 Choose the Data Analysis option Step 3 Choose t -Test: Paired Two Sample for Means from the list of Analysis Tools from the list of Analysis Tools … continued n Excel’s “ t -Test: Paired Two Sample for Means” Tool

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

Using Excel to Conduct a Hypothesis Test about  1 –  2 : Matched Samples

n Value Worksheet Using Excel to Conduct a Hypothesis Test about  1 –  2 : Matched Samples

Inference About the Difference between the Means of Two Populations: Matched Samples Using the p  Value Using the p  Value 4. Compute the value of the test statistic. 5. Compute the p –value. The Excel worksheet shows p -value = Determine whether to reject H 0. Because p –value =.0166 <  =.05, we reject H 0. The Excel worksheet shows t =