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.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
