04_02ans - STAT 410 Examples for Spring 2008 1 A car...

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Unformatted text preview: STAT 410 Examples for 04/02/2008 Spring 2008 1. A car manufacturer claims that, when driven at a speed of 50 miles per hour, the miles- per-gallon rating of a certain model follows a normal distribution with mean 30 miles per gallon and standard deviation σ = 4 miles per gallon. A consumer advocate thinks that the manufacturer is overestimating average mileage. H : μ ≥ 30 vs. H 1 : μ < 30. a) Suppose the consumer advocate tests a sample of n = 25 cars. What is the significance level associated with the rejection region “Reject H if x < 28.6” ? α = significance level = P ( Type I Error ) = P( Reject H | H true ). Need P ( X ≤ 28.6 | μ = 30 ) = ? . n Z X =- σ μ P ( X < 28.6 | μ = 30 ) = & & & ¡ ¢ £ £ £ ¤ ¥- < 25 4 30 6 . 28 Z P = P ( Z < - 1.75 ) = Φ (- 1.75 ) = 0.0401 . b) Suppose the consumer advocate tests a sample of n = 25 cars. Find the rejection region with the significance level α = 0.05. n = 25. α = 0.05. Rejection Region: Z = n & ¡ X- < – z α . Z = 25 4 30 X- < – 1.645. 25 4 645 . 1 30 X ⋅- < = 28.684. 2 c) Suppose the consumer advocate tests a sample of n = 25 cars and uses a 5% level of significance. Find the power of the test if the true mean is μ 1 = 29.5. P ( X < 28.684 | μ 1 = 29.5 ) = & & & & ¡ ¢ £ £ £ £ ¤ ¥- < 25 4 5 . 29 684 . 28 Z P = P ( Z < – 1.02 ) = 0.1539 . d) Repeat part (c) for the case when the true value of the mean is μ 1 = 28.5. P ( X < 28.684 | μ 1 = 28.5 ) = & & & & ¡ ¢ £ £ £ £ ¤ ¥- < 25 4 5 . 28 684 . 28 Z P = P ( Z < 0.23 ) = 0.5910 . e) Repeat parts (b) and (c) using a larger sample size of n = 49. n = 49. α = 0.05. Rejection Region: Z = n & ¡ X- < – z α . Z = 49 4 30 X- < – 1.645. 49 4 645 . 1 30 X ⋅- < = 29.06. 3 P ( X < 29.06 | μ 1 = 29.5 ) = & & & ¡ ¢ £ £ £ ¤ ¥- < 49 4 5 . 29 06 . 29 Z P = P ( Z < – 0.77 ) = 0.2206 ....
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This note was uploaded on 02/11/2011 for the course STAT 410 taught by Professor Monrad during the Spring '08 term at University of Illinois, Urbana Champaign.

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04_02ans - STAT 410 Examples for Spring 2008 1 A car...

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