The article from which the data in Exercise I 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 X,,..., X, and the cylinder strengths by Y, ..., Y. Suppose that the X's constitute a random sample from a distribution with mean , and standard deviation , and that the Y's form a random sample (independent of the X's) from another distribution with mean , and standard deviation 2.

a. Use rules of expected value to show that X - Y is an unbi- ased estimator of . Calculate the estimate for the given data.

b. Use rules of variance from Chapter 5 to obtain an expres- sion 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 /, 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 dif- ference X - Y between beam strength and cylinder strength.