According to the 2011 Gallup daily tracking polls (www.gallup.com,February 3,2012),Mississippi is the most conservative U.S.state,with 53.4 percent of its residents identifying themselves as conservative.What is the probability that at least 50 respondents of a random sample of 100 Mississippi residents do not identify themselves as conservative?
A) 0.0499
B) 0.2483
C) 0.4966
D) 0.7517

Question

Quizplus · Accepted Answer

This is a binomial distribution problem where p = 0.534, q = 0.466, n = 100, and we want to find the probability that at least 50 respondents do not identify as conservative, which can be written as P(X ≥ 50). Using the normal approximation to the binomial, we can calculate the z-score as:

z = (50 - np) / sqrt(npq) = (50 - 53.4) / sqrt(100 x 0.534 x 0.466) = -1.50

From a standard normal distribution table or calculator, we can find the probability of z ≤ -1.50, which is 0.0668. However, we want the probability of X ≥ 50, which is the complement of X < 50. Since X follows a discrete distribution, we need to use the continuity correction by subtracting 0.5 from X < 50 to get X ≤ 49.5. Then, we can find the probability of z ≤ -1.50 + 0.5/sqrt(100 x 0.534 x 0.466) = -1.335, which is 0.0918.

Finally, we can subtract this probability from 1 to get the desired probability:

P(X ≥ 50) = 1 - P(X ≤ 49) ≈ 1 - 0.0918 = 0.9082

Therefore, the best choice is B) 0.2483.

According to the 2011 Gallup Daily Tracking Polls (Www