When the population distribution is normal and n is large, the sample standard deviation S has approximately a normal distribution with and . We already know that in this case, for any n, is normal with and .

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

b. When the Xi

’s are normal, it can be shown that and S are independent rv’s (one measures location whereas the X

u 5 m 1 2.33s V(X) 5 s2

/n X E(X) 5 m V(S) < s2 E(S) < s /(2n)

Ha H0: s : s2 , .04 2 5 .04 a 5 .01

(n 2 1)s2

/s0 2 $ xa, n21 2 Ha: s2 . s0 2

x n 2 1 2 5 (n 2 1)S2

/s0 2

s 5 s0 H0: s2 5 s0 2

x n 2 1 2 5 (n 2 1)S2

/s2 other measures spread). Use this to compute and for the estimator of part (a). What is the estimated standard error ?

c. Write a test statistic for testing that has approximately a standard normal distribution when H0 is true. If soil pH is normally distributed in a certain region and 64 soil samples yield , does this provide strong evidence for concluding that at most 99%

of all possible samples would have a pH of less than 6.75? Test using .