Consider a random sample of size n from a normal distribution, Xi ~ N(m,s^2). m=mean, s^2=variance.

a) If it is known that s^2 = 9, find a 90% confidence interval for m based on the estimate x(bar) = 19.3 with n = 16.

b) Based on the information in (a), find a one-sided lower 90% confidence limit for m. Also, find a one-sided upper 90% confidence limit for m.

c) For a confidence interval of the form [ x(bar) - z1(-alpha/2) standard deviation/square root of n, x(bar) + z1(-alpha/2) standard deviation/square root of n ], derive a formula for the sample size required to obtain an interval of specified length lamda. If s^2 = 9, then what sample size is needed to achieve a 90% confidence interval of length 2?

d) Suppose now that s^2 is unknown. Find a 90% confidence interval for m if x(bar) = 19.3 and s^2 = 10.24 with n = 16

e) Based on the data in (d), find a 99% confidence interval for s^2.