The business college computing center wants to determine the proportion of business students who have laptop computers. If the proportion exceeds 35%, then the lab will scale back a proposed enlargement of its facilities. Suppose 200 business students were randomly sampled and 65 have laptops.

Which of the following conditions is satisfied for this problem?

A) The sample size n satisfies both np0 ≥ 10 and nq0 ≥ 10.
B) The population has an approximately normal distribution.
C) The sample size n satisfies n ≥ 30.
D) The sample proportion is close to 0.5.

answer A

To determine which of the conditions is satisfied for this problem, let's analyze each option:

A) The condition np0 ≥ 10 and nq0 ≥ 10 is related to the sampling distribution being approximately normal. Here, np0 = n * p0 and nq0 = n * (1 - p0), where n is the sample size and p0 is the assumed proportion. In this case, p0 is 35% (or 0.35).

Calculating np0: 200 * 0.35 = 70
Calculating nq0: 200 * (1 - 0.35) = 130

Since both np0 and nq0 are greater than or equal to 10 (70 and 130 respectively), option A is satisfied.

B) The assumption of the population having an approximately normal distribution is not explicitly stated in the problem. We do not have information on whether the proportion of business students with laptops follows a normal distribution, so this condition cannot be determined.

C) The condition of sample size n ≥ 30 is only a general guideline for the Central Limit Theorem to hold. It ensures that the distribution of the sample proportion is close to normal. In this case, the sample size is n = 200, which satisfies n ≥ 30.

D) The sample proportion being close to 0.5 indicates that the distribution would be approximately symmetrical. However, in this problem, the sample proportion is 65/200 = 0.325, which is not close to 0.5.

Based on this analysis, the condition that is satisfied for this problem is option A) The sample size n satisfies both np0 ≥ 10 and nq0 ≥ 10.