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Unformatted text preview: STAT 100B: Homework 2 Solutions
Denise Tsai
January 31, 2011
1. Suppose we ﬂip a coin 100 times independently and we get 55 heads.
(a) Test H0 : p = 1/2 vs H1 : p > 1/2, and calculate the pvalue.
Solution:
Z= p − p0
ˆ
p0 (1−p0 )
n = 55
− 0.5
100
0.5(1−0.5)
100 =1 The pvalue:
P (Z > 1) = 1 − P (Z ≤ 1) = 1 = 0.8413 = 0.1587
We do not reject the null hypothesis. Therefore, there is strong evidence to
claim that the coin is fair.
(b) Suppose the decision rule is to reject H0 if p > .6. What is the probability of type
ˆ
I error? If the true value of p is .7, then what is the probability of type II error?
Solution:
P (Type I error) = P (reject H0  H0 is true)
1
= P (ˆ > 0.6  p = )
p
2 0.6 − 0.5 = P Z >
= P (Z > 2) = 0.0228
0.5(1−0.5)
100 P (Type II error) = P (accept H0  H1 is true)
= P (ˆ ≤ 0.6  p = 0.7)
p 0.6 − 0.7 = P Z ≤
= P (Z ≤ −2.182) = 0.0146
0.7(1−0.7)
100 1 ...
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 Winter '11
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