# 10_14_11_2ans - STAT 409 Fall 2011 Examples for(2 H 0 true...

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STAT 409 Examples for 10/14/2011 (2) Fall 2011 H 0 true H 0 is NOT true Do Not Reject H 0 Type II Error Reject H 0 Type I Error α = significance level = P ( Type I Error ) = P ( Reject H 0 | H 0 is true ) β = P ( Type II Error ) = P ( Do Not Reject H 0 | H 0 is NOT true ) Power = 1 – P ( Type II Error ) = P ( Reject H 0 | H 0 is NOT true ) 1. A car manufacturer claims that, when driven at a speed of 50 miles per hour on a highway, the mileage of a certain model follows a normal distribution with mean μ 0 = 30 miles per gallon and standard deviation σ = 4 miles per gallon. A consumer advocate thinks that the manufacturer is overestimating average mileage. The advocate decides to test the null hypothesis H 0 : μ = 30 against the alternative hypothesis H 1 : μ < 30. 0a) Suppose the actual overall average mileage μ is indeed 30 miles per gallon. What is the probability that the sample mean is 29.4 miles per gallon or less, for a random sample of n = 25 cars? P( X 29.4 ) = - 25 4 30 4 . 29 Z P = P( Z - 0.75 ) = Φ ( - 0.75 ) = 0.2266 .

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0b) A random sample of 25 cars yields x = 29.4 miles per gallon. Based on the answer for part (a), is there a reason to believe that the actual overall average mileage is not 30 miles per gallon? If μ = 30, it is not unusual to see the values of the sample mean x at 29.4 miles per gallon or even lower. It does not imply that μ = 30, but we have no reason to doubt the manufacturer’s claim. 0c)
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## This note was uploaded on 11/15/2011 for the course STAT 409 taught by Professor Stephanov during the Fall '11 term at University of Illinois at Urbana–Champaign.

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10_14_11_2ans - STAT 409 Fall 2011 Examples for(2 H 0 true...

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