Suppose that random samples of 50 independent observations are drawn from two normal populations with identical means and standard deviations of 10 and 15. Find the probability that the difference between the sample means of the two sets of observations exceeds 0.5 in value.
Hint: Let X, Y be the underlying random variables corresponding to the two normal populations. Then X - Y is a normally distributed random variable. You should be able to calculate the mean and variance of X -Y as a linear combination of two normally distributed random variables. Then use these parameters to estimate the probability that the sample mean of observations of X - Y exceeds 0.5.