100BHW2S

100BHW2S - STAT 100B HWII Solution Problem 1: Suppose we...

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Unformatted text preview: STAT 100B HWII Solution Problem 1: Suppose we flip a coin 100 times independently and we get 55 heads. (1) Test H0 : p = 1/2 vs H1 : p > 1/2, and calculate the p-value. A: The observed value of p = 55/100 = .55. The Z -score is (.55 − 1/2)/ 1/2(1 − 1/2)/100 = ˆ .05/.05 = 1. The p-value is Pr(Z > 1) = .159. (2) Suppose the decision rule is to reject H0 if p > .6. What is the probability of type I error? ˆ If the true value of p is .7, then what is the probability of type II error? A: The probability of type I error is Pr(ˆ > .6|H0 ) = Pr(Z = (ˆ − p0 )/ p0 (1 − p0 )/n > p p (.6 − 1/2)/ 1/2(1 − 1/2)/100 = 2) = .0228. The probability of type II error is Pr(ˆ < .6|H1 ) = p Pr(Z = (ˆ − p1 )/ p1 (1 − p1 )/n < (.6 − .7)/ .7(1 − .7)/100) = −2.18) = .0146. p 1 ...
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