16.3 - Using Normal Probabilities to Find X16.3 - Using Normal Probabilities to Find X
On the last page, we learned how to use the standard normal curve N(0, 1) to find probabilities concerning a normal random variable X with mean \(\mu\) and standard deviation \(\sigma\). What happens if it's not the probability that we want to find, but rather the value of X? That's what we'll investigate on this page. That is, we'll consider what I like to call "inside-out" problems, in which we use known probabilities to find the value of the normal random variable X. Let's start with an example.
Suppose X, the grade on a midterm exam, is normally distributed with mean 70 and standard deviation 10. The instructor wants to give 15% of the class an A. What cutoff should the instructor use to determine who gets an A?
My approach to solving this problem is, of course, going to involve drawing a picture:
The instructor now wants to give 10% of the class an A−. What cutoff should the instructor use to determine who gets an A−?
We'll use the same method as we did previously:
In summary, in order to use a normal probability to find the value of a normal random variable X:
Find the z value associated with the normal probability.
Use the transformation \(x = \mu + z \sigma\) to find the value of x.