Question

The results of a History test and Calculus test are normally distributed. The History test had a mean score of 68 with a standard deviation of 8. The Calculus test had a mean score of 70 with a standard deviation of 7.2. Joseph scored 82 on the Calculus test and Zoe scored 82 on the History test. Which student had a higher percentile rank in their class?

Answer 1

Zoe, at about the 96th percentile.

Explanation:

Problems of normally distributed samples are 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.

Zoe:

Zoe scored 82 on the History test. So X = 82.

The History test had a mean score of 68 with a standard deviation of 8. This means that
`\mu = 68, \sigma = 8`

Then, we find the z-score to find the percentile.

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

`Z = (82 - 68)/(8)`

`Z = 1.75`

`Z = 1.75` has a pvalue of 0.9599.

So Zoe was at abouth the 96th percentile.

Joseph:

Joseph scored 82 on the Calculus test. This means that
`X = 82`

The Calculus test had a mean score of 70 with a standard deviation of 7.2. This means that
`\mu = 70, \sigma = 7.2`

Z-score

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

`Z = (82 - 70)/(7.2)`

`Z = 1.67`

`Z = 1.67` has a pvalue of 0.9525.

Joseph scored in the 95th percentile, which is below Zoe.

So the correct answer is:

Zoe, at about the 96th percentile.