4. The article from which the data of Exercise 1 was extracted also gave the accompanying strength observations for cylinders:

6.1 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.3 7.8 8.1 7.4 8.5 8.9 9.8 9.7 14.1 12.6 11.2 Prior to obtaining data, denote the beam strengths by X1, . . . , Xm and the cylinder strengths by Y1, . . . , Yn. Suppose that the Xis constitute a random sample from a distribution with mean 1 and standard deviation 1 and that the Yis form a random sample (independent of the Xis) from another distribution with mean 2 and standard deviation 2.

a. Use rules of expected value to show that X  Y is an unbiased estimator of 1  2. Calculate the estimate for the given data.

b. Use rules of variance from Chapter 5 to obtain an expression for the variance and standard deviation (standard error) of the estimator in part (a), and then compute the estimated standard error.

c. Calculate a point estimate of the ratio 1/2 of the two standard deviations.

d. Suppose a single beam and a single cylinder are randomly selected. Calculate a point estimate of the variance of the difference X  Y between beam strength and cylinder strength.