Let X1 X2 Xn be a sample of n iid observations population 1

Let X1, X2, ..., Xn be a sample of n iid observations population 1 and Y1, X2, ..., Ym be a sample of m iid observations from population 2. Let 1 and 1 be the mean and SD of population 1 and 2 and 2 be the mean and SD of population 2. Let Xbar denoted the mean of the first sample and Ybar denote the mean of the second sample.

a) What is the expected value of (Xbar – Ybar)?

b) Suppose the X’s and Y’s are independent. What is the variance of (Xbar – Ybar)?

c) Simplify the expression for the variance of (Xbar – Ybar) when 1 = 2 and we denote the common value as ?

d) For the situation in part (c), how does the variance simplify when the sample sizes are equal and we denote the common sample size as n?

e) Suppose you collect the data (two sets of observations) with n = m = 20. The values of sample mean and sample SD for the first set are 10 and 2; and the values of sample mean and sample SD for the second set are 12 and 3. Suppose that it is reasonable to assume that 1 = 2 and equals . Compute the estimate of using the pooled-sample SD method that you learned in the introductory stats course.

f) Suppose it is reasonable to assume that the data are normally distributed. Obtain a 95% confidence interval for 1 – 2, the difference in population means.

Solution