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.

Answer 1

The sampling distribution is approximately normal because the sample size of 40 exceeds the minimum requirement of 30 according to the Central Limit Theorem, regardless of the population distribution.

Step-by-step explanation:

Yes, the sampling distribution is approximately normal because the sample size is greater than 30. This is a result of the Central Limit Theorem, which states that the distribution of sample means will be approximately normal if the sample size is sufficiently large, which typically means a sample size of at least 30. Additionally, the mean of the sampling distribution will equal the population mean, and the standard deviation of this sampling distribution (known as the standard error of the mean) will be the population standard deviation divided by the square root of the sample size.

In the question presented, a simple random sample of size n=40 is obtained, which meets this criterion. The population mean and standard deviation are given, but the population distribution does not need to be normal if the sample size is large, which in this case, it is.