Random samples of size n were selected from binomial populations with population parameters p given here. Find the mean and the standard deviation of the sampling distribution of the sample proportion ^p = X \bar in each case: a) n = 100, p = 0.3; b) n = 400, p = 0.1; c) n = 250, p = 0.6. . a) Let T have a t-distribution with 10 degrees of freedom. Find P(T \legslant 1.812). b) If F has an F-distribution with 5 and 10 degrees of freedom. find a and b, such that P(a \legslant F \legslant b) = 0.9. c) Let Z be a standard normal random variable, independent of Y which is chi-square random variable with 4 degrees of freedom. Find the value of a so that P(|Z| \legslant a \sqrt{Y} )= 0.9. d) If X ~ X25, determine the constants c and d such that P(c < X