3 Further Exercises

Use R code to answer each of the following exercises.

Exercise 1

A university offers 8 different statistics courses to its honours level students. The numbers of students in each course are as follows:

• There are three course which have 85 students enrolled
• There are two course which have 112 students enrolled
• There is one course with 108 students enrolled
• There is one course with 95 students enrolled
• There is one course with only 46 students enrolled

Suppose you were to randomly sample three of these statistics courses with replacement. What would the sampling distribution of the sample mean number of students enrolled, \(\bar X\), be? What would the sampling distribution of the sample variance in the number of students enrolled, \(S^2\), be?

Can you use your sampling distributions to find \(E[\bar X]\), \(\sigma_{\bar X}^2\) and \(E[S^2]\)?

Exercise 2

Now suppose you want to randomly sample three of the above statistics courses without replacement. Find the sampling distribution of the sample mean number of students, \(\bar X\) and the sampling distribution of the sample variance in the number of students enrolled, \(S^2\).

Use these sampling distributions to find \(E[\bar X]\), \(\sigma_{\bar X}^2\) and \(E[S^2]\).

How do these values compare to those found in Exercise 1?