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1 1 Slide STATISTICS FOR BUSINESS AND ECONOMICS Seventh Edition AndersonSweeneyWilliams Slides Prepared by John Loucks © 1999 ITP/South-Western College Publishing

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2 2 Slide Chapter 10 Statistical Inference About Means and Proportions With Two Populations n Estimation of the Difference between the Means of Two Populations: Independent Samples n Hypothesis Tests about the Difference between the Means of Two Populations: Independent Samples n Inference about the Difference between the Means of Two Populations: Matched Samples n Inference about the Difference between the 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. n 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 Sampling Distribution of n The sampling distribution of has the following properties. Expected Value: Standard Deviation: where 1 = standard deviation of population 1 2 = standard deviation of population 2 n 1 = sample size from population n 2 = sample size from population 2

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6 6 Slide Interval Estimate of 1 - 2 : Large-Sample Case ( n 1 > 30 and n 2 > 30) 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

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7 7 Slide Example: Par, Inc. Par, Inc. is a manufacturer of golf equipment. Par 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 data is below. 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 Deviation s 1 = 15 yards s 2 = 20 yards

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8 8 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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9 9 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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10 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 Interval Estimate with 2 Unknown Interval Estimate with 2 Unknownwhere

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11 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 data is below. 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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12 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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13 Slide n 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case For the small-sample case we will make the following assumption. 1. 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. 2. 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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14 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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15 Slide Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples n Hypothesis Forms: 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 Case Large-Sample Case Small-Sample Case Small-Sample Case

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16 Slide Par, Inc. is a manufacturer of golf equipment. Par 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 data is below. 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 Deviation s 1 = 15 yards s 2 = 20 yards Example: Par, Inc.

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17 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 H a : 1 - 2 > 0 Example: Par, Inc.

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18 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 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. Example: Par, Inc.

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19 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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20 Slide 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. 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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21 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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22 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 district offices Hypotheses: H 0 : d = 0, H a : d 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). Rejection Rule: Reject H 0 if t 2.262 Example: Express Deliveries

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

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24 Slide Inference About the Difference Between the Proportions of Two Populations n Sampling Distribution of n Interval Estimation of n Hypothesis Tests about

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25 Slide Sampling Distribution of 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.

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26 Slide Interval Estimation of p 1 - p 2 : Large-Sample Case n Interval Estimate n Point Estimator of

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27 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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28 Slide 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 Example: MRA

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29 Slide n Interval Estimate of p 1 - p 2 : Large-Sample Case For =.05, z.025 = 1.96..08 + 1.96(.0510).08 +.10.08 +.10 or -.02 to +.18 or -.02 to +.18 Conclusion: At a 95% level of confidence, 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. Example: MRA

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30 Slide Hypothesis Testing 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 :

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31 Slide 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 H a : p 1 - p 2 > 0 Example: MRA

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

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33 Slide The End of Chapter 10

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