# hw8sol - FALL 2011 ORIE 3500/5500 PROBLEM SET 8...

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FALL 2011 ORIE 3500/5500 PROBLEM SET 8 SOLUTIONS (1) (a) By Chebyshev’s inequality, P [ | X n - p | > 0 . 2] Var( X n ) 0 . 2 2 = p (1 - p ) 0 . 04 n . We choose n large enough that this value is less than 0.1 for any p . Since p (1 - p ) 1 / 4 for 0 p 1 (plot the function g ( p ) = p - p 2 ), we require 0 . 16 n 0 . 1 - 1 , i.e. n 62 . 5. Therefore, we should interview at least 63 people. (d) We seek n such that P [ X n > 0 . 5] 0 . 9. Since one way for X n > 0 . 5 to be true is that X n lies within 0.1 of its mean p = 0 . 6, we have P [ X n > 0 . 5] P [ | X n - 0 . 6 | ≤ 0 . 1] 1 - 0 . 6(0 . 4) 0 . 01 n . Hence, it is enough to choose n to make this at least 0.9, i.e. 0 . 24 / (0 . 01 n ) 0 . 1. This is satisﬁed for n 240. Therefore, when p = 0 . 6, the probability that we predict correctly will be at least 0.9 provided we interview at least 240 people. Alternatively, observing that n X n = n i =1 X i Bin( n, 0 . 6), we could compute this probability exactly for diﬀerent values of n , and stop as soon as we exceed 0.9.

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hw8sol - FALL 2011 ORIE 3500/5500 PROBLEM SET 8...

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