Sample midterm3 and soln

# Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat312: Sample Midterm II. Moo K. Chung [email protected] March 25, 2003 1. Consider the sample of fat content of 10 randomly selected hot dogs: 25, 21, 22, 17, 29, 25, 16, 20, 19, 22. Suppose that these are from a normal pop- ulation. We want to test if the fat content of hot dogs is 23. (a) State the null and alternate hypotheses (5 points). (b) What is your test statistic and a rejection rule at 95 % level? (5 points) (c) Is the fat content of hot dogs 23? Explain your result (5 points). (d) Compute the probability of type I error when you do the hypothesis test using the test statistic in (b). Explain your result (5 points). (e) Define the P -value and compute the P -value for using the test statistic in (b) (5 points). >X<-c(25,21,22,17,29,25,16,20,19,22) >sum(X) [1] 216 > var(X) [1] 15.6 > qnorm(c(0.025,0.05)) [1] -1.96 -1.64 > qt(0.025,c(9,10)) [1] -2.26 -2.23 > pt(c(1.0,1.1,1.2,1.3,1.4,1.5),9) [1] 0.83 0.85 0.87 0.89 0.90 0.92 Solution. (a) Let μ be the population mean of the fat content in hot dogs. Then H 0 : μ = 23 and H 1 : μ 6 = 23 . (b) Since the population variance is unknown, the test statistic should be the t statis- tic given by T = ( ¯ X - μ ) / ( S/ 10) . Note that t 0 . 025 , 9 = 2 . 26 . So we reject H 0 if | T | > 2 . 26 . (c) s 2 = 15 . 6 . The t -value is t = (¯ x - 23) / ( s/ 10) = - 1 . 12 . Since the value is not in the rejection re- gion, we do not reject H 0 . So we may assume that the fat content of hot dogs is 23. (d) By defini- tion α = P ( reject H 0 | H 0 is true ) = P ( | T | > t 0 . 025 , 9 = 0 . 05 . You should answer this ques- tion without computing anything. (e) The

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