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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 pvalue.
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 pvalue 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 > .6H0 ) = 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 < .6H1 ) =
p
Pr(Z = (ˆ − p1 )/ p1 (1 − p1 )/n < (.6 − .7)/ .7(1 − .7)/100) = −2.18) = .0146.
p 1 ...
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 Spring '11
 Wu

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