To compare the student's relative positions on the two tests, we can calculate her z-scores for both the calculus and history tests. A z-score measures how many standard deviations a score is above or below the mean. The formula for the z-score is:

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

Where:

• $X$ is the score,
• $\mu$ is the mean,
• $\sigma$ is the standard deviation.

1. Calculating the z-score for the calculus test:

For the calculus test:

• $X = 65$
• $\mu = 50$
• $\sigma = 10$

$z_{\text{calculus}} = \frac{65 - 50}{10} = \frac{15}{10} = 1.5$

So, the student's score is 1.5 standard deviations above the mean on the calculus test.

2. Calculating the z-score for the history test:

For the history test:

• $X = 30$
• $\mu = 25$
• $\sigma = 5$

$z_{\text{history}} = \frac{30 - 25}{5} = \frac{5}{5} = 1$

So, the student's score is 1 standard deviation above the mean on the history test.

Conclusion:

• On the calculus test, the student is 1.5 standard deviations above the mean.
• On the history test, the student is 1 standard deviation above the mean.

While the student performed better in terms of raw scores on the calculus test (65 vs 30), her relative position (compared to the rest of the class) was slightly higher in calculus, since she is further above the mean (1.5 standard deviations vs. 1 standard deviation).

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Would you like a deeper explanation of z-scores or any other details? Here are some related questions for further exploration:

1. How does a z-score help in understanding the distribution of scores?
2. How can we interpret z-scores in different contexts (e.g., grading vs. test performance)?
3. What happens when the z-score is negative?
4. How can we use z-scores to compare scores from different distributions?
5. What is the significance of a z-score greater than 2?

Tip: Z-scores are powerful tools for comparing scores from different tests, especially when the tests have different means and standard deviations.