A normal random variable `x` has an unknown mean `mu` , and a standard deviation `sigma=2` . If the probability that `x` exceeds 7.5 is 0.8023, find the mean `mu` .
(1) Recall that we convert the value of a random variable to a `z` score by the formula `z=(x-mu)/sigma` .
(2) If the probability that `x` exceeds 7.5 is 0.8023, then the probability that `x` is less than 7.5 is 0.1977; we can get the corresponding `z` score for .1977 from a table or utility to be `z~~-.8499` .
(3) Substitute the known quantities and solve for `mu` :
`-.8499 = (7.5-mu)/2` or `mu~~9.1997`
Thus we can say that the mean is approximately 9.2