Combining test scores. 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.

#### Solution

Let the first exam score be denoted by $X$ and second exam score be denoted by $Y$ .

Given that $X \sim N(65, 102)$ and $Y \sim N(80, 52)$.

The $Z$ score is defined as $Z=\frac{X-\mu}{\sigma}$.

#### Z-score for Derrick

Derrick scored an 80 on both tests.

Derrick first exam z score is $\frac{80-65}{10} = 1.5$.

Derrick second exam z score is $\frac{80-80}{5} = 0$.

Derrick's combined Z-score is $1.5 + 0 = 1.5$.

#### Z-score for Julie

Julie scored a 70 on the first test and a 90 on the second.

Julie first exam z score is $\frac{70-65}{10} = 0.5$.

Julie second exam z score is $\frac{90-80}{5} = 2$.

Julie's combined Z-score is $0.5 + 2 = 2.5$.

As Julie's combined z-score is more than Derrick's combined z-score, Julie's score is better than Derrick's

score.