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

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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 0 : 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 0 if x < 28.6” ? α = significance level = P ( Type I Error ) = P ( Reject H 0 | H 0 true ) . Need P ( X 28.6 | μ = 30 ) = ? . n Z X = - σ μ P ( X < 28.6 | μ = 30 ) = ° ° ° ± ² ³ ³ ³ ´ µ - < 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 ° ± 0 X - < – z α . Z = 25 4 30 X - < 1.645 . 25 4 645 . 1 30 X - < = 28.684 .

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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 ) = ° ° ° ° ± ² ³ ³ ³ ³ ´ µ - < 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 ) = ° ° ° ° ± ² ³ ³ ³ ³ ´ µ - < 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 ° ± 0 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 ) = ° ° ° ± ² ³ ³ ³ ´ µ - < 49 4 5 . 29 06 . 29 Z P = P ( Z < – 0.77 ) = 0.2206 . P ( X < 29.06 | μ 1 = 28.5 ) = ° ° ° ± ² ³ ³ ³ ´ µ - < 49 4 5 . 28 06 . 29 Z P = P ( Z < 0.98 ) = 0.8365 .

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4 f) Repeat parts (b) and (c) using a 10% level of significance.
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