Math Problem Statement

Business Statistics is taught in three different departments at one institution. Three students taking exams in the course in different departments want to compare their scores. Zephrya scored 92 on the exam in a department where the overall mean was 72 with a standard deviation of 15. Marisa scored 75 on the exam in a department where the overall mean was 55 with a standard deviation of 20. Brady scored 90 on the exam in a department where the overall mean was 70 with a standard deviation of 10.  Which student performed the best, relative to the department in which they were taking the exam? 

                                                                                                                                                                                 

Business Statistics is taught in three different departments at one institution. Three students taking exams in the course in different departments want to compare their scores. Zephrya scored 92 on the exam in a department where the overall mean was 72 with a standard deviation of 15. Marisa scored 75 on the exam in a department where the overall mean was 55 with a standard deviation of 20. Brady scored 90 on the exam in a department where the overall mean was 70 with a standard deviation of 10.  Which student performed the best, relative to the department in which they were taking the exam? 

Zephrya

Marisa

Brady

It is impossible to compare the scores of these students since they took different exams in different departments. 

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Solution

To compare the students' performances relative to their departments, we should use z-scores, which measure how far a particular score is from the mean, in terms of standard deviations. The formula for calculating the z-score is:

z=Xμσz = \frac{X - \mu}{\sigma}

Where:

  • XX is the student's score,
  • μ\mu is the department mean score,
  • σ\sigma is the department's standard deviation.

Let's calculate the z-scores for each student:

Zephrya's z-score:

z=927215=2015=1.33z = \frac{92 - 72}{15} = \frac{20}{15} = 1.33

Marisa's z-score:

z=755520=2020=1.00z = \frac{75 - 55}{20} = \frac{20}{20} = 1.00

Brady's z-score:

z=907010=2010=2.00z = \frac{90 - 70}{10} = \frac{20}{10} = 2.00

Conclusion:

  • Zephrya's z-score is 1.33,
  • Marisa's z-score is 1.00,
  • Brady's z-score is 2.00.

Since Brady has the highest z-score, he performed the best relative to the department in which he was taking the exam.

Would you like more details on z-scores or any other part of this calculation?

Here are some questions to expand on this:

  1. What is the significance of a z-score in statistics?
  2. How can z-scores be used in grading systems?
  3. What is the importance of standard deviation in interpreting exam results?
  4. How do z-scores help in comparing results from different distributions?
  5. Can z-scores ever be negative? What would that indicate?

Tip: A higher z-score means a score is further above the mean, while a lower z-score (even negative) means it's below the mean.

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Math Problem Analysis

Mathematical Concepts

Statistics
Z-scores
Standard Deviation
Mean

Formulas

z = (X - μ) / σ

Theorems

Z-score calculation

Suitable Grade Level

Grades 10-12