82. When the population distribution is normal and n is large, the sample standard deviation S has approximately a normal distribution with E(S)
and V(S)
2/(2n). We already know that in this case, for any n, X is normal with E(X)
and V(X)
2/n.

a. Assuming that the underlying distribution is normal, what is an approximately unbiased estimator of the 99th percentile
2.33?

b. When the Xis are normal, it can be shown that X and S are independent rv’s (one measures location whereas the other measures spread). Use this to compute V(ˆ

)

and ˆ

for the estimator ˆ

of part (a). What is the estimated standard error ˆ ˆ

?