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# sol4 - Problem 294/12(a The parameter of interest is =true...

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Problem 294/12 (a) The parameter of interest is μ =true average braking distance at 40 mph for the new design . The hypothesis are H 0 : μ = 120 vs H a : μ < 120. (b) E x ) ( = 120 , if H 0 is true, < 120 , if H a is true. Hence R 2 should be used. (c) σ ¯ X = σ n = 10 6 = 1 . 6667, so α = P ( R 2 when H 0 is true) = P ( ¯ X 155 . 20when μ = 120) = P ( ¯ X - 120 1 . 6667 115 . 20 - 120 1 . 6667 ) = N ( - 2 . 88; 0 , 1) = 0 . 002 . To obtain α = 0 . 001, we need to replace 115.20 by a number c such that P ( ¯ X c when μ = 120) = N ( c ; μ = 120 , σ ¯ X = 1 . 6667) = 0 . 001. Hence, we use JMP to compute c = Normal Quantile(0.001,120,1.667) = 114 . 87. (d) P (accepting H 0 when μ = 115) = P ( ¯ X 115 . 20when μ = 115) = P ( ¯ X - 115 1 . 6667 115 . 20 - 115 1 . 6667 ) = 1 - N (0 . 12; 0 , 1) = 0 . 4522. For the first region, α = P ( Z ≤ - 2 . 33) = N ( - 2 . 33; 0 , 1) = 0 . 01. For the second region, α = P ( Z ≤ - 2 . 88) = 0 . 002. Problem 305/24 μ = true average viscosity. H 0 : μ = 3000 vs H a : μ 6 = 3000. t obs = ¯ x - 3000 s/ n = 2887 . 6 - 3000 84 / 5 = - 2 . 99. At level α , the rejection region is | t obs | = 2 . 99 > t α 2 , 4 . For example, using the table below, we can see that we reject at α = 0 . 05 but don’t reject at α = 0 . 01. Hence, the requirement is not satisfied at α = 0 . 05. On the other hand, there’s no evidence that indicates that the requirement is satisfied at α = 0 . 01. (For the homework, it suffices to report the test for your favorite choice of α ). 1

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Problem 311/44 p = proportion of defectives, by robots. H 0 : p = 0 . 035 vs H a : p < 0 . 035. We use z obs = ˆ p - 0 . 035 (0 . 035)(0 . 965) / 500 = z obs = 15 500 - 0 . 035 (0 . 035)(0 . 965) / 500 = - 0 . 005 0 . 0082 = - 0 . 61. At α = 0 . 01, the rejection region is z obs z α = - 2 . 33. Since z obs = - 0 . 61 > - 2 . 33, H 0 is not rejected and robots have not demonstrated their superiority. Using JMP, we can construct the following table and check the appropriate test by doing Analize¿Distribution¿Test probabilities. The results are exhibited below, with a p - value much larger than 0.01. This also shows that there is no evidence at all that the robots are
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sol4 - Problem 294/12(a The parameter of interest is =true...

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