8.23 The random variable X, representing the number of cherries in a cherry puff, has the following probability distribution:

x 4 5 6 7 P(X = x) 0.2 0.4 0.3 0.1

(a) Find the mean μ and the variance σ2 of X.

(b) Find the mean μX¯ and the variance σ2 X¯ of the mean

¯X for random samples of 36 cherry puffs.

(c) Find the probability that the average number of cherries in 36 cherry puffs will be less than 5.5.