Ch10 - Chapter10 LearningObjectives 1 populationmeanswhen 1...

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Chapter 10 Comparisons Involving Means Learning Objectives 1. Be able to develop interval estimates and conduct hypothesis tests about the difference between two population means when 1 σ and 2 σ are known. 2. Know the properties of the sampling distribution of 1 2 x x - . 3. Be able to use the t distribution to conduct statistical inferences about the difference between two population means when 1 σ and 2 σ are unknown. 4. Learn how to analyze the difference between two population means when the samples are independent and when the samples are matched. 5. Understand the principles of experimental design. 6. Be able to analyze a completely randomized design. 7. Understand how the analysis of variance procedure can be used to determine if the means of more than two populations are equal. 8. Know the assumptions necessary to use the analysis of variance procedure. 9. Understand the use of the F distribution in performing the analysis of variance procedure. 10. Know how to set up an ANOVA table and interpret the entries in the table. 11. Be able to use output from computer software packages to solve analysis of variance problems. Solutions: 10 - 1
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Chapter 10 1. a. x x 1 2 - = 13.6 - 11.6 = 2 b. / 2 .05 z z α = = 1.645 2 2 1 2 1 2 1 2 1.645 x x n n σ σ - ± + 2 2 (2.2) (3) 2 1.645 50 35 ± + 2 ± .98 (1.02 to 2.98) c. / 2 .025 z z α = = 1.96 2 2 (2.2) (3) 2 1.96 50 35 ± + 2 ± 1.17 (.83 to 3.17) 2. a. ( 29 1 2 0 2 2 2 2 1 2 1 2 (25.2 22.8) 0 2.03 (5.2) 6 40 50 x x D z n n σ σ - - - - = = = + + b. p -value = 1.0000 - .9788 = .0212 c. p -value .05, reject H 0 . 3. a. ( 29 1 2 0 2 2 2 2 1 2 1 2 (104 106) 0 1.53 (8.4) (7.6) 80 70 x x D z n n σ σ - - - - = = = - + + b. p -value = 2(.0630) = .1260 c. p -value > .05, do not reject H 0 . 4. a. 1 2 x x - = 2.04 - 1.72 = .32 b. 2 2 1 2 .025 1 2 z n n σ σ + 2 2 (.10) (.08) 1.96 1.96(.0208) .04 40 35 + = = c. .32 ± .04 (.28 to .36) 5. a. 1 2 x x - = 135.67 – 68.64 = 67.03 10 - 2
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Comparisons Involving Means b. 2 2 2 2 1 2 / 2 1 2 (35) (20) 2.576 17.08 40 30 z n n α σ σ + = + = c. 67.03 ± 17.08 (49.95 to 84.11) We estimate that men spend $67.03 more than women on Valentine’s Day with a margin of error of $17.08. 6. 1 μ = Mean loan amount for 2002 2 μ = Mean loan amount for 2001 H 0 : 1 2 0 μ μ - H a : 1 2 0 μ μ - ( 29 1 2 0 2 2 2 2 1 2 1 2 (175 165) 0 2.17 55 50 270 250 x x D z n n σ σ - - - - = = = + + p -value = 1.0000 - .9850 = .0150 p -value .05; reject H 0 . The mean loan amount has increased between 2001 and 2002. 7. a. 1 μ = Population mean 2002 2 μ = Population mean 2003 H 0 : 1 2 0 μ μ - H a : 1 2 0 μ μ - b. With time in minutes, 1 2 x x - = 172 - 166 = 6 minutes c. ( 29 1 2 0 2 2 2 2 1 2 1 2 (172 166) 0 2.61 12 12 60 50 x x D z n n σ σ - - - - = = = + + p -value = 1.0000 - .9955 = .0045 p -value .05; reject H 0 . The population mean duration of games in 2003 is less than the population mean in 2002. d. 2 2 1 2 1 2 .025 1 2 x x z n n σ σ - ± + 10 - 3
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Chapter 10 2 2 12 12 (172 166) 1.96 60 50 - ± + 6 ± 4.5 (1.5 to 10.5) e. Percentage reduction: 6/172 = 3.5%. Management should be encouraged by the fact that steps taken in 2003 reduced the population mean duration of baseball games. However, the statistical analysis shows that the reduction in the mean duration is only 3.5%. The interval estimate shows the reduction in the population mean is 1.5 minutes (.9%) to 10.5 minutes (6.1%). Additional data collected by the end of the 2003 season would provide a more precise estimate. In any case, most likely the issue will continue in future years. It is expected that major league baseball would prefer that additional steps be taken to further reduce the mean duration of games.
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