The purpose of this exercise is to compare the variability of x1 and x2 with the variability of 1x1 - x22.

a. Suppose the first sample is selected from a population with mean m1 = 150 and variance s2 1 = 900. Within what range should the sample mean vary about 95% of the time in repeated samples of 100 measurements from this distribution? That is, construct an interval extending 2 standard deviations of x1 on each side of m1.

b. Suppose the second sample is selected independently of the first from a second population with mean m2 = 150 and variance s2 2 = 1,600. Within what range should the sample mean vary about 95% of the time in repeated samples of 100 measurements from this distribution? That is, construct an interval extending 2 standard deviations of x2 on each side of m2.

c. Now consider the difference between the two sample means 1x1 - x22. What are the mean and standard deviation of the sampling distribution of 1x1 - x22?

d. Within what range should the difference in sample means vary about 95% of the time in repeated independent samples of 100 measurements each from the two populations?

e. What, in general, can be said about the variability of the difference between independent sample means relative to the variability of the individual sample means?