Question

1. using the central limit theorem, determine whether the sampling distribution is approximately normal or not. state your reasoning.

a. the distribution of the population is unknown. the sample size is 5.
b. the distribution of the population is right - skewed. the sample size is 35.
c. the distribution of the population is approximately normal. the sample size is 8.

Explanation:

• a. The Central Limit Theorem (CLT) states that if the population distribution is unknown, a sample size \(n\geq30\) is typically required for the sampling distribution of the sample mean to be approximately normal. Here, \(n = 5<30\), so the sampling distribution is not approximately normal.
• b. When the population distribution is skewed (right - skewed in this case), by the Central Limit Theorem, for \(n\geq30\), the sampling distribution of the sample mean is approximately normal. Since \(n = 35\geq30\), the sampling distribution is approximately normal.
• c. If the population distribution is approximately normal, then for any sample size \(n\) (including \(n = 8\)), the sampling distribution of the sample mean is approximately normal.

Answer:

• a. Not approximately normal. Reason: \(n=5<30\) and population distribution is unknown.
• b. Approximately normal. Reason: \(n = 35\geq30\) (CLT for non - normal populations with large \(n\)).
• c. Approximately normal. Reason: Population is approximately normal (CLT for normal populations).