59. Let X1, . . . , Xn be a random sample from a normal distribution with both m and s unknown. An unbiased estimator of u  P(X

c) based on the jointly suf cient statistics is desired. Let and . Then it can be shown that the minimum variance unbiased estimator for u is where T has a t distribution with n  2 df. The article Big and Bad: How the S.U.V. Ran over Automobile Safety (The New Yorker, Jan. 24, 2004) reported that when an engineer with Consumer Union

(the product testing and rating organization that publishes Consumers Reports) performed three different trials in which a Chevrolet Blazer was accelerated to 60 mph and then suddenly braked, the stopping distances (ft) were 146.2, 151.6, and u ˆ

 μ

0 kw

1 Pa T 

kw1n  2 21  k2w2 b 1  kw  1 1 kw  1

∂

w  1c  x2/s k  1n/ 1n  12 sˆ

2 1z1  z2 2/2 g1zi  z 22  1z1  z2 22 z sˆ 2 g1Xi  Yi 22/ 14n2.

153.4, respectively. Assuming that braking distance is normally distributed, obtain the minimum variance unbiased estimate for the probability that distance is at most 150 ft, and compare to the maximum likelihood estimate of this probability.