Recitation 20 Problem 4. A Gallup poll in November 2004 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 1016 people and claims a margin of error of 3 percentage points Let's check Gallups claim. Suppose that there are m adult Americans, of whom pm be- lieve evolution is well-supported and(1-p)m do not. Gallup polls n Americans selected uniformly and independently at random. Of these, gn believe that evolution is well- supported and(1-q)n do not. Gallup then estimates that the fraction of america ans who believe evolution is well-supported is q Note that the only randomization in this experiment is in who gallup chooses to poll. So the sample space is all sequences of n adult Americans. The response of the i-th person polled is"yes"with probability p and"no"with probability 1 -p since the person is selected uniformly at random. Furthermore, the n responses are mutually independent (a) Give an upper bound on the probability that the polls estimate will be 0.03 or more too low. Just write the expression; don' t evaluate yet Solution. We can regard each response as a coin flip that is heads with probability p In these terms, qn is the total number of heads flipped. So we have: Pr(qm≤(p-0.03)n) 2nH(P-003) 51-(=0022=0-(0=0p-py (1-(P-0.03)n (b) Give an upper bound on the probability that the polls estimate will be 0.03 or more too high. Again, just write the expression Solution. Reasoning as before and using the answer to the preceding problem gives Pr(qm>(p+0.03)n) ≤-P+003 nH(p+0.03) (1-(P+0.03)n 1-1=(+0)√2r(p+0.03)(1-(p+0.03)n (c) The sum of these two answers is the probability that Gallup's poll will be off by 3 percentage points or more, one way or the other. Unfortunately, these expres- sions both depend on pthe unknown fraction of evolution-believers that Gallup is trying to estimate! However, the sum of these two expressions is maximized when p=0.5. So evaluate the sum with p=0.5 and n= 1016 to upper bound the probability that Gallup's error is 0.03 or more. Pollsters usually try to ensure that there is a 95% chance that the actual percentage p lies within the polls error range, which is q+0.03 in this Is Gallups evolution poll Solution. The probability that the error is 0.03 or more is about 0.07, which mea that p will lie within the error range of a polled fraction with probability 0.93. So ourRecitation 20 5 Problem 4. A Gallup poll in November 2004 found that 35% of the adult population of the United States believes that the theory of evolution is “well­supported by the evidence”. Gallup polled 1016 people and claims a margin of error of 3 percentage points. Let’s check Gallup’s claim. Suppose that there are m adult Americans, of whom pm be￾lieve evolution is well­supported and (1 − p)m do not. Gallup polls n Americans selected uniformly and independently at random. Of these, qn believe that evolution is well￾supported and (1 − q)n do not. Gallup then estimates that the fraction of Americans who believe evolution is well­supported is q. Note that the only randomization in this experiment is in who Gallup chooses to poll. So the sample space is all sequences of n adult Americans. The response of the i­th person polled is “yes” with probability p and “no” with probability 1 − p since the person is selected uniformly at random. Furthermore, the n responses are mutually independent. (a) Give an upper bound on the probability that the poll’s estimate will be 0.03 or more too low. Just write the expression; don’t evaluate yet! Solution. We can regard each response as a coin flip that is heads with probability p. In these terms, qn is the total number of heads flipped. So we have: Pr (qn ≤ (p − 0.03)n) 2nH(p−0.03) 1 − (p − 0.03) (p−0.03)n(1 − p) (1−(p−0.03))n ≤ · p 1 − (p − 0.03)/p · �2π(p − 0.03)(1 − (p − 0.03))n (b) Give an upper bound on the probability that the poll’s estimate will be 0.03 or more too high. Again, just write the expression. Solution. Reasoning as before and using the answer to the preceding problem gives: Pr (qn > (p + 0.03)n) 2nH(p+0.03) 1−(p+0.03) ≤ 1 − p + 0.03 · �2π(p + 0.03)(1 − (p + 0.03))n · p(p+0.03)n(1 − p) (1−(p+0.03))n 1−p (c) The sum of these two answers is the probability that Gallup’s poll will be off by 3 percentage points or more, one way or the other. Unfortunately, these expres￾sions both depend on p— the unknown fraction of evolution­believers that Gallup is trying to estimate! However, the sum of these two expressions is maximized when p = 0.5. So evaluate the sum with p = 0.5 and n = 1016 to upper bound the probability that Gallup’s error is 0.03 or more. Pollsters usually try to ensure that there is a 95% chance that the actual percentage p lies within the poll’s error range, which is q ± 0.03 in this case. Is Gallup’s evolution poll properly designed? Solution. The probability that the error is 0.03 or more is about 0.07, which means that p will lie within the error range of a polled fraction with probability 0.93. So our