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

Answer:

Zoe, at about the 96th percentile.

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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.

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{82 - 68}{8}$

$Z = 1.75$

$Z = 1.75$ 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

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{82 - 70}{7.2}$

$Z = 1.67$

$Z = 1.67$ 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.