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q3solutions - Statistics 20 Quiz 3 Solutions 1 Suppose X1...

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Statistics 20: Quiz 3 Solutions 1. Suppose X 1 , . . . , X 64 are independent copies of X N (0 , 2 2 ). Find SD ( ¯ X ). With n = 64, SD ( ¯ X ) = SD ( X i ) n = 2 64 = 1 4 . 2. You design a strategy for playing Black Jack so that in each game you expect to win 5 dollars with a standard deviation of 10 dollars. You quit your day job and vow to play professionally; under reasonable assumptions, you can easily play 10 , 000 independent games of Black Jack at casinos in the next 6 months. What is the probability that you win more than 51 , 000 dollars doing so? With n = 10 , 000, let X 1 , . . . , X n denote the winnings (which may be positive, negative, or zero) on the i th game of Black Jack. We are interested in the total winnings X = n i =1 X i . By the CLT, X has an approximately Normal distribution. We need to determine its expected value and standard deviation: E [ X ] = nE [ X i ] = (10 , 000)5 = 50 , 000; SD ( X ) = nSD ( X i ) = 10 , 000(10) = 1000. Therefore, X N (50000 , 1000 2 ). Now we can compute the probability of winning more than 51 , 000 dollars. If we let Z N (0 , 1), then: P ( X > 51 , 000) = P X - E [ X ] SD ( X ) > 51 , 000 - E [ X ] SD (
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