Assume that the fill, X, of a water filling machine is 12.2 with a standard deviation of 0.1 with the measurements being in liquid ounces. Assume X is a normal random variable. Suppose we independently take 50 such measurements, say X1, X2, .., X50. Let M be the sample mean of the 50 measurements. Let S be the sum of the 50 measurements.

a) What is the probability that X < 12?

b) What is the probability that at least one of the 50 measurements is less than 12?

c) What is the expected value of S-(50 * 12.2) ?

d) What is the standard deviation of S- (50 * 12.2) ?

e) What is the probability that S- (50 * 12.2) >1?

f) What is the probability that S- (50 12.2) < -0.5?

g) What is the expected value of M?

h) What is the standard deviation of M-12.2?

i) What is the probability that M-12.2>.01?

j) What is the standard deviation of 50*(M-12.2)?

k) What is the variance of 50*(M-12.2)?

l) If the probability of X > k is equal to .05 then what is k.?