Suppose that Illinois lawmakers survey 130 randomly selected registered voters to see if they favor charging a deposit on aluminum cans to encourage recycling. The lawmakers believe the population proportion in favor of changing the law is 93% (based on historical data and previous votes). Which of the following conditions for the Central Limit theorem are not met?
A)The population proportion is too small and will not have enough expected failures.
B)Relative to the population, the sample is not large enough.
C)The population proportion is too small and will not have enough expected successes.
D)None of the above, all the condition of the CLT are met.

Question

Quizplus · Accepted Answer

CFor the Central Limit Theorem (CLT) to apply, we need both a sufficiently large number of expected successes (np) and expected failures (n(1-p)) in the sample. With a population proportion (p) of 93%, or 0.93, and a sample size (n) of 130, the expected number of successes is np = 130 * 0.93 = 120.9, and the expected number of failures is n(1-p) = 130 * 0.07 = 9.1. While there are enough expected successes (over 10), the number of expected failures is just above the commonly used threshold of 10 for the CLT to apply comfortably. However, the statement about the population proportion being too small and not having enough expected successes (C) is incorrect because there are clearly enough expected successes. The confusion might arise from the interpretation of the conditions. The correct concern is about having enough expected failures, but since we have just above 9 expected failures, it's a borderline situation. The most accurate answer would be that all conditions of the CLT are met, considering conventional thresholds, but the question suggests looking for conditions that are not met, leading to a misunderstanding. Therefore, the concern about not having enough expected failures (A) would be the closest to a valid concern, but it's framed incorrectly in the context of how the CLT is generally applied.

Suppose That Illinois Lawmakers Survey 130 Randomly Selected Registered Voters