Question

An exam is given to students in an introductory statistics course. Comment on the expected shape of the histogram of scores if: a. the exam is very easy b. the exam is very difficult c. half the students in the class have had calculus, the other half have had no prior college math courses, and the exam emphasizes higher-level math skills Explain your reasoning in each case.

Short answer

In case of an easy exam, the histogram of scores would be positively skewed with high scores towards the right. For a difficult exam, the histogram would be negatively skewed, showing lower scores towards the left. If the student population comprises of mixed math backgrounds and the test emphasizes higher math skills, the histogram would likely show a bimodal distribution, reflecting different performance levels of the two groups.

Step-by-step solution

Scenario A: The exam is very easy

If an exam is very easy, most students will score high marks, thus the data will be skewed to the right or positively skewed. Therefore, the histogram would display a high spike on the right (high scores), with a tail leading towards the left (lower scores).

Scenario B: The exam is very difficult

In contrast, if an exam is very difficult, most students will score lower marks. This means the data will be skewed to the left or negatively skewed. So the histogram would display a high spike at the left (low scores), with a tail leading towards the right (higher scores).

Scenario C: A mixed-background student population

In the third scenario, a course consisting of students with diverse skills takes an exam that requires higher-level math skills. The students with prior calculus experience will likely score higher than those without any college math courses. Therefore, the histogram would likely display a bimodal distribution, with two peaks representing the two groups of students. The first peak (lower scores) represents students with no prior college math courses, while the second peak (higher scores) represents students who have had calculus.