16.3 - Using Normal Probabilities to Find X

16.3 - 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.

Example 16-3

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?

Solution

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−?

Solution

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:

1. Find the z value associated with the normal probability.

2. Use the transformation $$x = \mu + z \sigma$$ to find the value of x.

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