Question

If the professor grades on a curve​ (for example, the professor could give​ A's to the top​ 10% of the​ class, regardless of the​ score), is a student better off with a grade of 83 on this exam or a grade of 75 on a different​ exam, where the mean is 67 and the standard deviation is 4​? Show your answer statistically and explain. A student is better off or worse off with a grade of 83 on this exam because the Z value for the grade of 83 is ___ and the Z value for the grade of 75 is ___. ​(Round to two decimal places as​ needed.)

a) Better off, 1.5, 2.0
b) Better off, 0.75, 1.25
c) Worse off, 1.5, 2.0
d) Worse off, 0.75, 1.25

Answer 1

After calculating the Z-scores, the student with a grade of 75 on an exam with a mean of 67 and standard deviation of 4 is better off with a Z-score of 2.0 compared to a Z-score of 1.625 for a grade of 83 on an exam with a mean of 70 and standard deviation of 8.

Step-by-step explanation:

To determine whether a student is better off with a grade of 83 on an exam or a grade of 75 on a different exam where the mean is 67 and the standard deviation is 4, we need to compare the Z-scores for both situations. The Z-score represents the number of standard deviations a value is from the mean. It can be calculated using the formula: Z = (X - μ) / σ, where X represents the value (student's score), μ is the mean, and σ is the standard deviation.

For a score of 83 on an exam where the mean is 70 and the standard deviation is 8 (from Matt's class), the Z-score is:
Z = (83 - 70) / 8 = 13 / 8 = 1.625

For a score of 75 on an exam where the mean is 67 and the standard deviation is 4 (from the provided scenario), the Z-score is:
Z = (75 - 67) / 4 = 8 / 4 = 2

Since being in the top 10% is equated to having a higher Z-score, the student with the Z-score of 2 (75 on the exam with a mean of 67) is better off, as they are further from the mean in terms of standard deviations compared to the student with a Z-score of 1.625 (83 on the exam with a mean of 70).