4. The data set of Exercise 1 also includes these thirdgrade verbal IQ observations for males:

117 103 121 112 120 132 113 117 132 149 125 131 136 107 108 113 136 114 and females:

114 102 113 131 124 117 120 90 114 109 102 114 127 127 103 Prior to obtaining data, denote the male values by X1, . . . , Xm and the female values by Y1, . . . , Yn. Suppose that the Xi s constitute a random sample from a distribution with mean m1 and standard deviation s1 and that the Yi s form a random sample (independent of the Xi s) from another distribution with mean m2 and standard deviation s2.

a. Use rules of expected value to show that is an unbiased estimator of m1m2. Calculate the estimate for the given data.

b. Use rules of variance from Chapter 6 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.

X  Y

c. Calculate a point estimate of the ratio s1/s2 of the two standard deviations.

d. Suppose one male third-grader and one female third-grader are randomly selected. Calculate a point estimate of the variance of the difference X  Y between male and female IQ.