2010100AHW7

2010100AHW7 - STAT 100A HWVII Problem 1: Suppose we flip a...

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Unformatted text preview: STAT 100A HWVII Problem 1: Suppose we flip a fair coin 1000 times independently. Let X be the number of heads. (1) What is the probability that 480 ≤ X ≤ 520? (2) What is the probability that X > 530? Problem 2: Suppose among the population of voters, 1/3 of the people support a candidate. If we sample 1000 people from the population, and let X be the number of supporters of this candidate among these 1000 people. Let p = X/n be the sample proportion. ˆ (1) What is the probability that p > .35? ˆ (2) What is the probability that p < .3? ˆ Problem 3: Suppose we flip a fair coin n times independently. Let X be the number of heads. √ √ Let k = n/2 + z n/2, or z = (k − n/2)/( n/2). Let g (z ) = P (X = k ). √ 2 (1) Using the Stirling formula n! ∼ 2πnnn e−n , show that g (0) ∼ √1 π √n . a ∼ b means that 2 a/b → 1 as n → ∞. 2 (2) Show that g (z )/g (0) → e−z /2 as n → ∞. √ √ (3) For two integers a < b, let a = (a − n/2)/( n/2), and b = (b − n/2)/( n/2). Show that 2 b P (a ≤ X ≤ b) → a f (z )dz , where f (z ) ≈ √1 π e−z /2 . 2 √ (4) Let Z = (X − n/2)/( n/2). Show that P (a ≤ X ≤ b) = P (a ≤ Z ≤ b ). Argue that in the limit Z ∼ N(0, 1). 1 ...
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