Math Problem Statement

The first Stats exam had a mean of 60 and a standard deviation of 10 points; the second had a mean of 85 and a standard deviation of 5 points. Derrick scored an 85 on both tests. Julie scored a 70 on the first test and a 100 on the second. They both totaled 170 points on the two exams, but Julie claims that her total is better. Explain.

Solution

This question asks about comparing z-scores for two students based on their scores in two exams. Let’s break it down step by step:

Key details:

  • Exam 1: Mean = 60, Standard deviation = 10.
  • Exam 2: Mean = 85, Standard deviation = 5.
  • Derrick's scores: 85 on both exams.
  • Julie's scores: 70 on the first exam, 100 on the second exam.
  • Both students totalled 170 points across both exams.

Z-score formula:

z=Xμσz = \frac{X - \mu}{\sigma} Where:

  • XX is the score.
  • μ\mu is the mean.
  • σ\sigma is the standard deviation.

Step-by-step calculations:

Derrick's z-scores:

  1. First test: Derrick scored 85. z1=856010=2510=2.500z_1 = \frac{85 - 60}{10} = \frac{25}{10} = 2.500
  2. Second test: Derrick scored 85 again. z2=85855=0z_2 = \frac{85 - 85}{5} = 0

Julie's z-scores:

  1. First test: Julie scored 70. z1=706010=1010=1.000z_1 = \frac{70 - 60}{10} = \frac{10}{10} = 1.000
  2. Second test: Julie scored 100. z2=100855=155=3.000z_2 = \frac{100 - 85}{5} = \frac{15}{5} = 3.000

Total z-scores:

  • Derrick's total z-score: 2.500+0=2.5002.500 + 0 = 2.500
  • Julie's total z-score: 1.000+3.000=4.0001.000 + 3.000 = 4.000

Thus, Julie's claim is correct, as her total z-score (4.000) is higher than Derrick's (2.500).

The correct answer is:

  • B. Julie's claim is correct.

Would you like further details or have any questions? Here are 5 related questions to expand your understanding:

  1. How do z-scores help compare scores from different distributions?
  2. What would happen to the z-scores if the standard deviations were reversed between the two tests?
  3. Can you explain how the z-score reflects performance relative to the class average?
  4. How would you interpret a negative z-score?
  5. What is the significance of having a z-score of 0?

Tip: Z-scores allow comparison between scores from different tests or datasets by standardizing them based on the mean and standard deviation.

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Math Problem Analysis

Mathematical Concepts

Statistics
Z-Scores
Standard Deviation
Mean

Formulas

Z = (X - μ) / σ

Theorems

Z-Score Comparison

Suitable Grade Level

Grades 9-12