Question

Consider a population with a quantitative variable for which we would like to estimate ne population mean. We do so by taking random samples and calculating the sample nean. Use this scenario for questions 14 and 15. 14. If the population is normally distributed, what does that imply for the sample mean? a. The sample mean will be non-normal for small samples sizes, but become more normal as the sample size increases. b. The sample mean is guaranteed to be normal for any sample size. c. The standard deviation of the sample mean will increase as the sample " se increases. . The center of the sample mean will increase as the sample size increases. 15. If the population is NOT normally distributed, what does that imply for the sample mean? a. The sample mean will be non-normal for small samples sizes, but become more normal as the sample size increases. b. The sample mean is guaranteed to be normal for any sample size. C. The standard deviation of the sample mean will increase as the sample size increases. d. The center of the sample mean will increase as the sample size increases.

Answer 1

14. If the population is normally distributed, this imply that:

b. The sample mean is guaranteed to be normal for any sample size.

c. The standard deviation of the sample mean will increases as the sample size increases.

Since the population is normally distributed ,the distribution of the sample mean is normally distributed for any sample size. The standard deviation of the sample is directly proportional to the sample size. Hence increasing sample size increases the standard deviation of the sample mean.

15. If the population is not normally distributed, this implies that:

a. The sample mean will be non normal for small sample sizes but become more normal as the sample size increases.

Though the sample size is not normally distributed, according to the central limit theorem, the sampling distribution of the sample mean is approximately normally distributed for sufficiently large sample size say n>30.

c. The standard deviation of the sample mean will increase as the sample size increases.