Finding Probability Using a Normal Distribution

Question

Suppose the mean income of firms in the industry for a year is 55 million dollars with a standard deviation of 3 million dollars. If incomes for the industry are distributed normally, what is the probability that a randomly selected firm will earn less than 60 million dollars? Round your answer to four decimal places.

Answer 1

To solve this, you can either do manual calculations and use a z-score table or use a computer of some sort (like a TI-84 or Excel). I'll explain how to do this manually.

The problem wants you to find the probability that the firm selected earns less than 60 million dollars. Recall that probability is the same thing as area under the curve. So you're trying to find the area under the Normal curve that is to the left of 60 million dollars.

First, you need to convert your values into z-scores. The equation for that is as follows:

z-score= (x - xbar) / standard deviation

z-score= (60-55) / 3

z-score= 1.67

If doing this problem by hand, you would look up the answer using a z-score table. If using technology, you can directly calculate the area to the left of +1.67.

I get an answer of about 0.9522. That is, there is a 95.22% chance of selecting a firm that earns less than 60 million.