Let X be normally distributed with a given mean and standard deviation. Sometimes you want to find two values a and b such that P(a < X < b) is equal to some specific probability such as 0.90 or 0.95. There are many answers to this problem, depending on how much probability you put in each of the two tails. For this question, assume the mean and standard deviation are µ = 100 and σ = 10, and that you want to find a and b such that P(a < X < b) = 0.90.
a. Find a and b so that there is probability 0.05 in each tail.
b. Find a and b so that there is probability 0.025 in the left tail and 0.075 in the right tail.
c. The usual answer to the general problem is the answer from part a, that is, where you put equal probability in the two tails. It turns out that this is the answer that minimizes the length of the interval from a to b. That is, if you solve the following problem: minimize (b - a), subject to P(a < X < b) = 0.90, you will get the same answer as in part a. Verify this by using Excel’s Solver add-in.