Question

a simple random sample of size n = 40 is obtained from a population with a mean of 20 and a standard deviation of 5. is the sampling distribution approximately normal? why?
yes, the sampling distribution is approximately normal because the sample size is greater than 30.
yes, the sampling distribution is approximately normal because the population is normally distributed.
no, the sampling distribution is not approximately normal because the population is not normally distributed.
no, the sampling distribution is not approximately normal because the population mean is less than 30.

Explanation:

Step1: Recall the Central Limit Theorem

The Central Limit Theorem states that for a sample of size \(n\), if \(n\geq30\), the sampling distribution of the sample mean \(\bar{X}\) is approximately normal, regardless of the shape of the population distribution.

Step2: Analyze the given sample size

Here, the sample size \(n = 40\) and \(40>30\).

Answer:

Yes, the sampling distribution is approximately normal because the sample size is greater than 30.