When the population distribution is normal and n is large, the sample standard deviation S has approximately a normal distribution with E(S) and

When the population distribution is normal and n is large, the sample standard deviation S has approximately a normal distribution with E(S) ≈ σ and V(S) ≈ σ2/(2n). We already know that in this case, for any n,  is normal with E() = μ and V() = σ2/n.
a. Assuming that the underlying distribution is normal, what is an approximately unbiased estimator of the 99th percentile θ = μ + 2.33σ?
b. When the Xi's are normal, it can be shown that  and S are independent rv's (one measures location whereas the other measures spread). Use this to compute V(θ) and σθ for the estimator θ of part (a). What is the estimated standard error θ?
c. Write a test statistic for testing H0: θ = θ0 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  = 6.33, s = .16, 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 α = .01.