When the population distribution is normal and n is large, the sample standard deviation...

When the population distribution is normal and n is large, the sample standard deviation S has approximately a normal distribution with 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 99^th 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 ?

c. Write a test statistic for testing that has H₀: θ = θ₀ 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.