Question

The first Stats exam had a mean of 65 and a standard deviation of 10 points; the second had a mean of 80 and a standard deviation of 5 points. Derrick scored an 80 on both tests. Julie scored a 70 on the first test and a 90 on the second. They both totaled 160 points on the two exams, but Julie claims that her total is better. Explain.

Answer 1

Using z-scores, it is found that Julie's total is better as she scored above the mean in both tests.

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In a set with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

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

• It measures how many standard deviations X is above or below the mean.

• For the first test, $\mu = 65, \sigma = 10$.
• For the second test, $\mu = 80, \sigma = 5$.

Derrick:

• On the first test, scored 80, thus, the z-score is:

$Z = \frac{80 - 65}{10} = 1.5$

• On the second test, also 80, thus:

$Z = \frac{80 - 80}{5} = 0$

Julie:

• On the first test, scored 70, thus, the z-score is:

$Z = \frac{70 - 65}{10} = 0.5$

• On the second test, also 90, thus:

$Z = \frac{90- 80}{5} = 2$

Julie had positive z-scores in both tests, so she scored better.

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