1. A population random variable is normally distributed with a mean of 50 and a variance of 16. If a random sample of size n = 25 is drawn from this population, what are the following probabilities? a) that the sample mean will exceed 51; b) that the sample mean will be between 48.5 and 51.5. 2. Repeat Exercise 1, but now with n = 36. What is the effect on the probabilities of increasing the sample size? 3. Random samples of size n were selected from populations with the means and variances given here. Find the mean and standard deviation of the sampling distribution of the sample mean in each case: a) n = 36, \muμ = 10, \sigmaσ2= 9; b) n = 100, \muμ= 5, \sigmaσ2= 4; c) n = 8, \muμ= 120, \sigmaσ2= 1. 4. Refer to Exercise 3. a) If the sampled populations are normal, what is the sampling distribution of X \bar for parts a), b), and c)? b) According to the Central Limit Theorem, if the sampled populations are not normal, what can be said about the sampling distribution of X \bar for parts a), b), and c)?