MIT6_042JS10_lec38_sol

MIT6_042JS10_lec38_sol - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science May 10 Prof. Albert R. Meyer revised May 10, 2010, 677 minutes Solutions to In-Class Problems Week 14, Mon. Problem 1. A recent Gallup poll found that 35% of the adult population of the United States believes that the theory of evolution is “well-supported by the evidence.” Gallup polled 1928 Americans selected uniformly and independently at random. Of these, 675 asserted belief in evolution, leading to Gallup’s estimate that the fraction of Americans who believe in evolution is 675 / 1928 0 . 350 . Gallup claims a margin of error of 3 percentage points, that is, he claims to be confident that his estimate is within 0.03 of the actual percentage. (a) What is the largest variance an indicator variable can have? Solution. 1 4 By Lemma 21.4.2 , Var [ H ] = pq . Noting that d p (1 p ) /dp = 2 p 1 is zero when p = 1 / 2 , it follows that the maximum value of p (1 p ) must be at p = 1 / 2 , so the maximum value of Var [ H ] is (1 / 2)(1 (1 / 2)) = 1 / 4 . (b) Use the Pairwise Independent Sampling Theorem to determine a confidence level with which Gallup can make his claim. Solution. By the Pairwise Independent Sampling, the probability that a sample of size n = 1928 is further than x = 0 . 03 of the actual fraction is at most σ 2 1 1 1 0 . 144 x · n 4(0 . 03) 2 · 1928 so we can be confident of Gallup’s estimate at the 85.6% level. (c) Gallup actually claims greater than 99% confidence in his estimate. How might he have ar- rived at this conclusion? (Just explain what quantity he could calculate; you do not need to carry out a calculation.) Solution. Gallup’s sample has a binomial distribution B 1928 ,p for an unknown p he estimates to be about 0 . 35 . So he wants an upper bound on B 1928 ,p Pr > 0 . 03 p 1928 Creative Commons 2010, Prof. Albert R. Meyer .
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2 Solutions to In-Class Problems Week 14, Mon. By part ( a ), the variance of B n,p is largest when p = 1 / 2 , which suggests that the probability that a sample average differs from the actual mean will be largest when p = 1 / 2 . This is in fact the case. So Gallup will calculate Pr > 0 . 03 B 1928 , 1 / 2 1928 2 > 0 . 03(1928) B 1928 , 1 / 2 1 = Pr 1928 2 = Pr 906 B 1928 , 1 / 2 1021 1021 1928 i =906 i = 2 1928 0 . 9912 . Mathematica will actually calculate this sum exactly. There are also simple ways to use Stirling’s formula to get a good estimate of this value. (d) Accepting the accuracy of all of Gallup’s polling data and calculations, can you conclude that there is a high probability that the number of adult Americans who believe in evolution is 35 ± 3 percent? Solution. No. As explained in Notes and lecture, the assertion that fraction p is in the range 0 . 35 ± 0 . 03 is an assertion of fact that is either true or false. The number p is a constant . We don’t know its value, and we don’t know if the asserted fact is true or false, but there is nothing probabilistic about the fact’s truth or falsehood.
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