10_21_09 - STAT 409 1. Fall 2009 Examples for 10/21/2009 An...

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STAT 409 Examples for 10/21/2009 Fall 2009 1. An advertisement for a particular brand of automobile states that it accelerates from 0 to 60 mph in an average of 5.0 seconds. Makers of a competing automobile feel that the true average number of seconds it takes to reach 60 mph from zero is above 5.0. Suppose the population standard deviation is believed to be 0.43 seconds. We wish to test H 0 : μ 5.0 vs. H 1 : μ > 5.0. a) Find the power of this test if the true average time is 5.15 seconds, a 5% level of significance will be used, and 50 automobiles will be tested (each automobile to be tested a single time). b) Fifty automobiles were tested (each automobile was tested a single time). The sample mean time to reach 60 mph from zero was 5.09 seconds. Find the p-value of this test. 2. Suppose n = 49 observations are taken from a normal distribution where σ = 8.0 for the purpose of testing H 0 : μ = 60 versus H 1 : μ 60 at the α = 0.05 level of significance. What is the power of the appropriate test when μ = 59.2? 3. Let X 1 , X 2 , … , X 16 be a random sample of size n = 16 from a N ( μ , σ 2 ) distribution. We are interested in testing H 0 : σ = 39 vs. H 1 : σ > 39. Recall: If X 1 , X 2 , … , X n are i.i.d. N ( μ , σ 2 ), then ( 29 2 2 σ S 1 n - is χ 2 ( n – 1 ). a) Find the “best” critical ( rejection ) region with the significance level α = 0.05. b) Find the power of the test from part (a) at σ = 66.7. c) What is the probability of Type II Error if σ = 66.7? The Chi-Square Distribution
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10_21_09 - STAT 409 1. Fall 2009 Examples for 10/21/2009 An...

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