It has been determined empiracally that for a certain egg producer, the number of cracked or broken eggs X, in a randomly slected box containing six eggs, follows approximately the distribution given in the table below:

X: 0 1 2 3 4 5 6

P(X): 0.9 0.07 0.02 0.005 0.003 0.001 0.001

Find the expectation and variance of X.

If a consignment of eggs from this producer contains 1000 boxes, use the central limit theorem to estimate the probability that the total number of cracked or broken eggs

i) exceeds 175

ii) is at least 150 but does not exceed 190.

Indicate any assumptions you have made, and indicate briefly how these might be violated.