As many as 20% of all Americans contract influenza each year. A sample of 400 randomly selected Americans is chosen and the number with influenza is recorded. Let X represent the number with influenza in the sample. What is the probability that, at most, 25% of the sample is observed to have influenza?

Question

Quizplus · Accepted Answer

We can use the normal approximation to the binomial distribution since the sample size is large.

The mean of the binomial distribution is μ = np = 400 * 0.20 = 80, and the standard deviation is σ = sqrt(np(1-p)) = sqrt(400 * 0.20 * 0.80) = 6.32

To find the probability that at most 25% of the sample has influenza, we need to find P(X ≤ 100) where X ~ N(80, 6.32).

Using standardization, we get: 
P(X ≤ 100) = P((X-μ)/σ ≤ (100-80)/6.32)
= P(Z ≤ 3.16)

Using a standard normal table or calculator, we can find P(Z ≤ 3.16) = 0.9992.

Therefore, the probability that at most 25% of the sample is observed to have influenza is approximately 0.9992 or 99.38%. The answer is C.

As Many as 20% of All Americans Contract Influenza Each