Sigma in a normal distribution

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Question from Justin, a student:

Hello,

Here is the question:
Suppose the random variable Y can be described by a normal curve with Mu=40. For what value of the standard deviation is P(20 less than or equal to Y less than or equal to 60) = 0.50

I'm fairly certain the problem is actually solvable because we have the standard deviation. I just don't understand how to solve for it with the seemingly limited information in the problem. Solving for Z comes to mind, but there doesn't seem to be enough information. Thanks for your help!

-Justin

Hi Justin,

Changing the problem to a problem using the standard normal random variable Z is the right way to go. The conversion is which in your case is .

Your probability statement P(20 ≤ Y ≤ 60) = 0.50 can be represented graphically by

normal distribution

Notice that the shaded area is symmetric about the mean 40 so P(40 ≤ Y ≤ 60) = 0.25.

Now convert to the standard normal distribution and the diagram becomes

standard normal distribution

(I only brought across half of the shaded region because most of the standard normal tables I have seen make it easy to deal with areas between Z = 0 and Z = some positive number.) The point s on my diagram is the converted value for Y = 60, that is .

Use the normal table to determine the value of s, substitute into the equation above and solve for σ.

Harley

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