Consider independent random samples from two normal distributions, Xi ~ N(mu1,s1^2) for i = 1, .., n1 and Yj ~ N(mu2, s2^2) for j = 1, ..., n2. Let n1 = n2 = 9, x(bar) = 16, y(bar) = 10, s1^2 = 36, and s2^2 = 45.
a) Assuming equal variances, test Ho: mu1 = mu2 against Ha: mu1 not = to mu2 at the alpha = .10 level of significance.
b) Perform an approximate alpha = .10 level test of Ho: mu1 = mu2 against Ha: mu1 not = to mu2 using the equation T = [Y(bar) - X(bar) - (mu2 - mu1)] / sqrt[S1^2/n1 + S2^2/n2] ~ t(v)
c) Perform a test of these hypotheses at the alpha = .10 significance level using the equation T = [D(bar) - (mu2 - mu1)] / [SD/sqrt(n)] ~ t(n-1), assuming the data were obtained from paired samples with sD^2 = 81.
d) Test Ho: s2^2/s1^2 < or = to 1 versus Ha: s2^2/s1^2 > 1 at the alpha = .05 level.