41. A particular quality characteristic of items produced using a certain process is known to be normally distributed with mean m and standard deviation 1. Let X denote the value of the characteristic for a randomly selected item. An unbiased estimator for the parameter u  P(X c), where c is a critical threshold, is desired. The estimator will be based on a random sample X1, . . . , Xn.

a. Obtain a suf cient statistic for m.

b. Consider the estimator . Obtain an improved unbiased estimator based on the suf -

cient statistic (it is actually the minimum variance unbiased estimator). [Hint:You may use the following facts: (1) The joint distribution of X1 and is bivariate normal with means m and m, respectively, variances 1 and 1/n, respectively, and correlation r (which you should verify). (2) If Y1 and Y2 have a bivariate normal distribution, then the conditional distribution of Y1 given that Y2 

y2 is normal with mean m1  (rs1/s2)(y2  m2)

and variance s .] 21 11  r2 2