10_21_09ans - STAT 409 1. Fall 2009 Examples for 10/21/2009...

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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). σ = 0.43. n = 50. α = 0.05. σ is known. Test Statistic: 50 43 . 0 0 . 5 X X Z σ μ 0 - = - = n . Rejection Region: Right - tailed. Reject H 0 if Z z α = 1.645. 50 43 . 0 645 . 1 0 . 5 X + = 5.1 . Power = P ( Reject H 0 | H 0 is NOT true ) = P ( X > 5.1 | μ = 5.15 ) = - 50 43 . 0 15 . 5 1 . 5 Z P = P ( Z > – 0.82 ) = 0.7939 . 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. Test Statistic: σ is known. 50 43 . 0 0 . 5 09 . 5 X Z σ μ 0 - = - = n = 1.48 . Right – tailed test. p-value = ( area of the right tail ) = P ( Z > 1.48 ) = 0.0694 .
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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? H 0 : μ = 60 vs. H 1 : μ 60 2 – tailed. α = 0.05. Rejection Region : Reject H 0 if Z = σ μ 0 X n - < 2 α z - or Z = σ μ 0 X n - > 2 α z σ μ 2 0 α X n z - < or σ μ 2 0 α X n z + 49 8 96 . 1 60 X - < or 49 8 96 . 1 60 X + 57.76 < X or 62.24 X Power = P ( Reject H 0 | H 0 is false ) = P ( 76 . 57 X < | μ = 59.2 ) + P( 24 . 62 X | μ = 59.2 ) = - + - < 49 8 2 . 59 24 . 62 Z P 49 8 2 . 59 76 . 57 Z P = P ( Z < – 1.26
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This note was uploaded on 10/12/2011 for the course STATISTICS stat 410 taught by Professor Stepanov during the Spring '11 term at University of Illinois, Urbana Champaign.

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10_21_09ans - STAT 409 1. Fall 2009 Examples for 10/21/2009...

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