Question

An exam is given to students in an introductory statistics course. What is likely to be true of the shape of the histogram of scores if: a. the exam is quite easy? b. the exam is quite 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 mathematical manipulation? Explain your reasoning in each case.

Short answer

a. For an easy exam, the histogram is likely to be negatively skewed as most students score high grades. b. For a difficult exam, the histogram will probably be positively skewed as most students score poor grades. c. In the case where half the students had calculus and half who didn't, the histogram is likely to display a bimodal distribution.

Step-by-step solution

Analysis of Easy Exam

If the exam is quite easy, then a majority of the students will score high grades. This means that majority of the data points (scores) will cluster towards the high end of the score range, leading to a histogram that skews to the left. This is also known as negatively skewed distribution.

Analysis of Difficult Exam

If the exam is quite difficult, then a majority of students would score poor grades. So, most of the data points will cluster towards the low end of the score range, resulting in a histogram that skews to the right. Known as a positively skewed distribution.

Analysis of Mixed Student Abilities

If half the students in the class have had calculus, the other half have had no prior college math courses, and the examination emphasizes mathematical manipulation, then most likely, the histogram will be a bimodal distribution, since there are two distinct groups of students – those who have had calculus (and will likely perform better), and those without prior college math courses (who might score lower).