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1 1 Slide Slides Prepared by JOHN S. LOUCKS St. Edward’s University © 2002 South-Western /Thomson Learning

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2 2 Slide Chapter 10 Comparisons Involving Means 1 = 2 ? ANOVA n Estimation of the Difference Between the Means of Two Populations: Independent Samples Two Populations: Independent Samples n Hypothesis Tests about the Difference between the Means of Two Populations: Independent Samples Means of Two Populations: Independent Samples n Inferences about the Difference between the Means of Two Populations: Matched Samples of Two Populations: Matched Samples n Inferences about the Difference between the Proportions of Two Populations: Proportions of Two Populations:

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3 3 Slide Estimation of the Difference Between the Means of Two Populations: Independent Samples n Point Estimator of the Difference between the Means of Two Populations n Sampling Distribution Interval Estimate of Large-Sample Case Interval Estimate of Large-Sample Case Interval Estimate of Small-Sample Case Interval Estimate of Small-Sample Case

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4 4 Slide Point Estimator of the Difference Between the Means of Two Populations Let 1 equal the mean of population 1 and 2 equal the mean of population 2. Let 1 equal the mean of population 1 and 2 equal the mean of population 2. The difference between the two population means is 1 - 2. The difference between the two population means is 1 - 2. To estimate 1 - 2, we will select a simple random sample of size n 1 from population 1 and a simple random sample of size n 2 from population 2. To estimate 1 - 2, we will select a simple random sample of size n 1 from population 1 and a simple random sample of size n 2 from population 2. n Let equal the mean of sample 1 and equal the mean of sample 2. n The point estimator of the difference between the means of the populations 1 and 2 is.

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5 5 Slide n Properties of the Sampling Distribution of Expected Value Expected Value Standard Deviation Standard Deviation 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 Sampling Distribution of

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6 6 Slide Interval Estimate with 1 and 2 Known Interval Estimate with 1 and 2 Knownwhere: 1 - is the confidence coefficient Interval Estimate with 1 and 2 Unknown Interval Estimate with 1 and 2 Unknownwhere: Interval Estimate of 1 - 2 : Large-Sample Case ( n 1 > 30 and n 2 > 30)

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7 7 Slide Example: Par, Inc. Interval Estimate of 1 - 2 : Large-Sample Case Interval Estimate of 1 - 2 : Large-Sample Case Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball that has been designed to provide “extra distance.” 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.

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8 8 Slide Example: Par, Inc. Interval Estimate of 1 - 2 : Large-Sample Case Interval Estimate of 1 - 2 : Large-Sample Case Sample Statistics Sample Statistics Sample #1 Sample #2 Sample #1 Sample #2 Par, Inc. Rap, Ltd. Par, Inc. Rap, Ltd. Sample Size n 1 = 120 balls n 2 = 80 balls Mean = 235 yards = 218 yards Standard Dev. s 1 = 15 yards s 2 = 20 yards

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9 9 Slide n Point Estimate of the Difference Between Two Population Means 1 = mean distance for the population of Par, Inc. golf balls Par, Inc. golf balls 2 = mean distance for the population of Rap, Ltd. golf balls Rap, Ltd. golf balls Point estimate of 1 - 2 = = 235 - 218 = 17 yards. Example: Par, Inc.

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10 Slide Point Estimator of the Difference Between the Means of Two Populations 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 1 – 2 = difference between the mean distances the mean distances Simple random sample Simple random sample of n 1 Par golf balls of n 1 Par golf balls x 1 = sample mean distance for sample of Par golf ball Simple random sample Simple random sample of n 1 Par golf balls of n 1 Par golf balls x 1 = sample mean distance for sample of Par golf ball Simple random sample Simple random sample of n 2 Rap golf balls of n 2 Rap golf balls x 2 = sample mean distance for sample of Rap golf ball Simple random sample Simple random sample of n 2 Rap golf balls of n 2 Rap golf balls x 2 = sample mean distance for sample of Rap golf ball x 1 - x 2 = Point Estimate of 1 – 2

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11 Slide 95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Case, 1 and 2 Unknown 95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Case, 1 and 2 Unknown Substituting the sample standard deviations for the population standard deviation: = 17 + 5.14 or 11.86 yards to 22.14 yards. = 17 + 5.14 or 11.86 yards to 22.14 yards. We are 95% confident that the difference between the mean driving distances of Par, Inc. balls and Rap, Ltd. balls lies in the interval of 11.86 to 22.14 yards. Example: Par, Inc.

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12 Slide Interval Estimate of 1 - 2 : Small-Sample Case ( n 1 < 30 and/or n 2 < 30) Interval Estimate with 2 Known Interval Estimate with 2 Knownwhere:

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13 Slide Interval Estimate of 1 - 2 : Small-Sample Case ( n 1 < 30 and/or n 2 < 30) Interval Estimate with 2 Unknown Interval Estimate with 2 Unknownwhere:

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14 Slide Example: Specific Motors 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: Sample #1 Sample #2 Sample #1 Sample #2 M Cars J Cars M Cars J Cars Sample Size n 1 = 12 cars n 2 = 8 cars Mean = 29.8 mpg = 27.3 mpg Standard Deviation s 1 = 2.56 mpg s 2 = 1.81 mpg

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15 Slide n Point Estimate of the Difference Between Two Population Means 1 = mean miles-per-gallon for the population of M cars M cars 2 = mean miles-per-gallon for the population of J cars J cars Point estimate of 1 - 2 = = 29.8 - 27.3 = 2.5 mpg. Example: Specific Motors

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16 Slide n 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case We will make the following assumptions: The miles per gallon rating must be normally The miles per gallon rating must be normally distributed for both the M car and the J car. distributed for both the M car and the J car. The variance in the miles per gallon rating must The variance in the miles per gallon rating must be the same for both the M car and the J car. be the same for both the M car and the J car. Using the t distribution with n 1 + n 2 - 2 = 18 degrees of freedom, the appropriate t value is t.025 = 2.101. We will use a weighted average of the two sample variances as the pooled estimator of 2. Example: Specific Motors

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17 Slide n 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case = 2.5 + 2.2 or.3 to 4.7 miles per gallon. = 2.5 + 2.2 or.3 to 4.7 miles per gallon. We are 95% confident that the difference between the mean mpg ratings of the two car types is from.3 to 4.7 mpg (with the M car having the higher mpg). Example: Specific Motors

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18 Slide Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples n Hypotheses H 0 : 1 - 2 0 H 0 : 1 - 2 = 0 H a : 1 - 2 > 0 H a : 1 - 2 0 H a : 1 - 2 < 0 H a : 1 - 2 0 n Test Statistic Large-Sample Small-Sample Large-Sample Small-Sample

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19 Slide n Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball that has been designed to provide “extra distance.” 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. Example: Par, Inc.

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20 Slide Example: Par, Inc. n Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case Sample Statistics Sample Statistics Sample #1 Sample #2 Sample #1 Sample #2 Par, Inc. Rap, Ltd. Par, Inc. Rap, Ltd. Sample Size n 1 = 120 balls n 2 = 80 balls Mean = 235 yards = 218 yards Standard Dev. s 1 = 15 yards s 2 = 20 yards

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21 Slide n Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case Can we conclude, using a.01 level of significance, that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls? 1 = mean distance for the population of Par, Inc. golf balls 2 = mean distance for the population of Rap, Ltd. golf balls Hypotheses H 0 : 1 - 2 < 0 Hypotheses H 0 : 1 - 2 < 0 H a : 1 - 2 > 0 Example: Par, Inc.

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22 Slide n Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case Rejection Rule Reject H 0 if z > 2.33 Rejection Rule Reject H 0 if z > 2.33 Conclusion Conclusion Reject H 0. We are at least 99% confident that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls. Reject H 0. We are at least 99% confident that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls. Example: Par, Inc.

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23 Slide n Hypothesis Tests About the Difference Between the Means of Two Populations: Small-Sample Case 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? 1 = mean mpg for the population of M cars 2 = mean mpg for the population of J cars Hypotheses H 0 : 1 - 2 < 0 Hypotheses H 0 : 1 - 2 < 0 H a : 1 - 2 > 0 Example: Specific Motors

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24 Slide Example: Specific Motors n Hypothesis Tests About the Difference Between the Means of Two Populations: Small-Sample Case Rejection Rule Rejection Rule Reject H 0 if t > 1.734 Reject H 0 if t > 1.734 ( =.05, d.f. = 18) ( =.05, d.f. = 18) Test Statistic Test Statistic where: where:

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25 Slide Inference About the Difference Between the Means of Two Populations: Matched Samples n With a matched-sample design each sampled item provides a pair of data values. n The matched-sample design can be referred to as blocking. n 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.

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26 Slide Example: Express Deliveries n Inference About the Difference Between the Means of Two Populations: Matched Samples 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. In testing the delivery times of the two services, the firm sent two reports to a random sample of ten district offices with one report carried by UPX and the other report carried by INTEX. Do the data that follow indicate a difference in mean delivery times for the two services?

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27 Slide Delivery Time (Hours) Delivery Time (Hours) District OfficeUPX INTEX Difference Seattle 32 25 7 Los Angeles 30 24 6 Boston 19 15 4 Cleveland 16 15 1 New York 15 13 2 Houston 18 15 3 Atlanta 14 15 -1 St. Louis 10 8 2 Milwaukee 7 9 -2 Denver 16 11 5 Example: Express Deliveries

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28 Slide n Inference About the Difference Between the Means of Two Populations: Matched Samples Let d = the mean of the difference values for the two delivery services for the population of district offices Let d = the mean of the difference values for the two delivery services for the population of district offices Hypotheses H 0 : d = 0, H a : d Hypotheses H 0 : d = 0, H a : d Rejection Rule Rejection Rule Assuming the population of difference values is approximately normally distributed, the t distribution with n - 1 degrees of freedom applies. With =.05, t.025 = 2.262 (9 degrees of freedom). Assuming the population of difference values is approximately normally distributed, the t distribution with n - 1 degrees of freedom applies. With =.05, t.025 = 2.262 (9 degrees of freedom). Reject H 0 if t 2.262 Reject H 0 if t 2.262 Example: Express Deliveries

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29 Slide n Inference About the Difference Between the Means of Two Populations: Matched Samples Conclusion Reject H 0. Conclusion Reject H 0. There is a significant difference between the mean delivery times for the two services. There is a significant difference between the mean delivery times for the two services. Example: Express Deliveries

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30 Slide Inferences About the Difference Between the Proportions of Two Populations n Sampling Distribution of n Interval Estimation of p 1 - p 2 n Hypothesis Tests about p 1 - p 2

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31 Slide n Expected Value n Standard Deviation n Distribution Form If the sample sizes are large ( n 1 p 1, n 1 (1 - p 1 ), n 2 p 2, and n 2 (1 - p 2 ) are all greater than or equal to 5), the sampling distribution of can be approximated by a normal probability distribution. Sampling Distribution of

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32 Slide Interval Estimation of p 1 - p 2 n Interval Estimate n Point Estimator of

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33 Slide Example: MRA MRA (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. 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?

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34 Slide Example: MRA n Point Estimator of the Difference Between the Proportions of Two Populations 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 = 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

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35 Slide Example: MRA n Interval Estimate of p 1 - p 2 : Large-Sample Case For =.05, z.025 = 1.96: For =.05, z.025 = 1.96:.08 + 1.96(.0510).08 +.10.08 +.10 or -.02 to +.18 Conclusion Conclusion At a 95% confidence level, the interval estimate of the difference between the proportion of households aware of the client’s product before and after the new advertising campaign is -.02 to +.18. At a 95% confidence level, the interval estimate of the difference between the proportion of households aware of the client’s product before and after the new advertising campaign is -.02 to +.18.

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36 Slide Hypothesis Tests about p 1 - p 2 n Hypotheses H 0 : p 1 - p 2 < 0 H 0 : p 1 - p 2 < 0 H a : p 1 - p 2 > 0 H a : p 1 - p 2 > 0 n Test statistic n Point Estimator of where p 1 = p 2 where:

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37 Slide Example: MRA n Hypothesis Tests about p 1 - p 2 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? 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 Hypotheses H 0 : p 1 - p 2 < 0 Hypotheses H 0 : p 1 - p 2 < 0 H a : p 1 - p 2 > 0 H a : p 1 - p 2 > 0

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38 Slide Example: MRA n Hypothesis Tests about p 1 - p 2 Rejection Rule Reject H 0 if z > 1.645 Rejection Rule Reject H 0 if z > 1.645 Test Statistic Test Statistic Conclusion Do not reject H 0. Conclusion Do not reject H 0.

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39 Slide End of Chapter 10

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