100BHWS4

# 100BHWS4 - STAT 100B Homework IV Solution Problem 1 Suppose...

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Unformatted text preview: STAT 100B Homework IV Solution Problem 1: Suppose we observe X 1 ,...,X n ∼ Bernoulli( p ) independently. Let ˆ p = ∑ n i =1 X i /n be the sample proportion. Suppose n = 100, and we want to test H : p = . 5 versus H 1 : p = . 64. Suppose our decision rule is to reject H if the observed value of ˆ p > . 6. (1) (6 points) Calculate the probability of type I error. A: If p = . 5, then P (ˆ p > . 6) = P ( Z = (ˆ p- . 5) / p . 5 × (1- . 5) / 100 > ( . 6- . 5) / p . 5 × (1- . 5) / 100 = 2) = . 0228. (2) (4 points) Calculate the probability of type II error. A: If p = . 64, then P (ˆ p < . 6) = P ( Z = (ˆ p- . 64) / p . 64 × (1- . 64) / 100 < ( . 6- . 64) / p . 64 × (1- . 64) / 100) = . 2023. Problem 2: Suppose we observe X 1 ,...,X n ∼ N( μ 1 ,σ 2 ), and Y 1 ,...,Y m ∼ N( μ 2 ,σ 2 ), and X 1 ,...,X n , Y 1 ,...,Y m are all independent of each other. Suppose we want to test whether μ 1 = μ 2 . (1) (5 points) Let ¯ X = ( X 1 + ... + X n ) /n , and let ¯ Y = ( Y 1 + ... + Y m ) /m . If we known σ 2 , then what is an appropriate test statistic, and what is the distribution of this test statistic under the...
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100BHWS4 - STAT 100B Homework IV Solution Problem 1 Suppose...

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