On a particular section of a CPA exam for which the scores were normally distributed with a mean of 700 points and a standard deviation of 70 points.

Question

On a particular section of a CPA exam for which the scores were normally distributed with a mean of 700 points and a standard deviation of 70 points.   

 

What percent of the students scored lower than 640 on the exam?

 

 

What percent of the students scored between 630 and 730 on the exam?


What percent of the students scored between 600 and 700 on the exam?


What percent of the students scored above 700 on the exam?


If 900 people take the exam, how many people will score above 750 on the exam?

Answer 1

Heather,

 

Since you are given the mean and standard deviation and the data is normally distributed, this allows you to calculate a z value which corresponds to a percentage on the normal distribution range on a table. This can be found manually, through a calculator or via Excel. I'll show you how to find the z value and leave it to you to find the percentage.

 

First off the general formula for z=(X-m)/σ where m is the mean and σ is the standard deviation. A z lookup table will give you the percentage for all values from 0 up to X. This is important!

 

To find percentage of scores lower than 640, plug in X=640 to find z and then look up the percentage.

To find percentage of scores between 630 and 730, find z for X=630 and X=730 and find the difference.

To find percentage of scores between 600 and 700, find z for X=600 and X=700 and find the difference.

To find percentage of scores above 700, find z for X=700 and subtract the resulting percentage (in the form .XX) from 1. This is because all percentages total to one and we want all scores above 700, not below.

 

With a sample of 900 people, the z formula changes slightly because the standard deviation needs to be adjusted into something called standard error (SE). That's because standard deviation grows smaller as sample size grows bigger. Now z=(X-m)/SE where SE=σ/√n where n is the sample size. So, plug in n=900, X=750, m=700, σ=70 and find z and then the percentage. Remember we're looking for scores above 750 so you'll have to subtract the percentage from 1 to get the answer.

 

Hope that helps!