Question

You are attempting to determine the average grade (out of 100%) of students at MIT. You have taken 100 independent random samples of size 50 from that population and calculated the sample mean (x̄ ) and a sample variance (s^2 ) of each sample.

1. What distribution would you expect the 100 sample means ( x̄) you have calculated to follow? Why?
2. What distribution would you expect the 100 sample variances (s^2 ) to follow?

Answer 1

a. As, the central limit theorem states that, when the sample size increases, the sampling distribution of the sample means will follow approximately Normal distribution regardless of their population distribution.

Since, 100 is enoughly large as a sample size, we would expect the 100 sample means to follow a Normal distribution with mean equal to the population mean and standard deviation equal to the population standard error.

b. Though , there is no CLT for the variances, we can consider a special case. As the sample means are following Normal distribution, so we would expect the sample variances to follow a Chi-sqared distribution.