Question

There are two college entrance exams that are often taken by students, Exam A and Exam B. The composite score on Exam A is approximately normally distributed with mean 20.1 and standard deviation 5.1. The composite score on Exam B is approximately normally distributed with mean 1031 and standard deviation 215. Suppose you scored 24 on Exam A and 1167 on Exam B. Which exam did you score better on? Justify your reasoning using the normal model.

A.The score on Exam B is better, because the score is higher than the score for Exam A.
B.The score on Exam A is better, because the percentile for the Exam A score is higher.
C.The score on Exam A is better, because the difference between the score and the mean is lower than it is for Exam B.
D.The score on Exam B is better, because the percentile for the Exam B score is higher.

Answer 1

B.The score on Exam A is better, because the percentile for the Exam A score is higher.

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

Two exams. The exam that you did score better is the one in which you had a higher zscore.

The composite score on Exam A is approximately normally distributed with mean 20.1 and standard deviation 5.1.

This means that
`\mu = 20.1, \sigma = 5.1`.

You scored 24 on Exam A. So

`Z = (X - \mu)/(\sigma)`

`Z = (24 - 20.1)/(5.1)`

`Z = 0.76`

The composite score on Exam B is approximately normally distributed with mean 1031 and standard deviation 215.

This means that
`\mu = 1031, \sigma = 215`.

You scored 1167 on Exam B, s:

`Z = (X - \mu)/(\sigma)`

`Z = (1167 - 1031)/(215)`

`Z = 0.632`

You had a better Z-score on exam A, so you did better on that exam.

The correct answer is:

B.The score on Exam A is better, because the percentile for the Exam A score is higher.