Question

consider the exam scores for 59 students when 7 students scored between 90 and 100, 12 students scored between 80 and 89, 21 students scored between 70 and 79, and 19 students scored below 70, with their scores spread out to a low of 40. complete parts a through d

a. how many peaks would you expect for the distribution?

a. there would probably be two peaks, because even though each exam score could be anywhere between 0 and 100, the scores were clustered between 80 to 89 and between 70 to 79

b. there would probably be no peaks. the distribution of grades always tends to be uniform

c. there would probably be many peaks corresponding to the different exam scores that each student had

d. there would probably be one peak because there are no obvious reasons why the exam scores would form different groups

b. make a sketch of the distribution. choose the correct answer below

a.

b.

c.

d.

Explanation:

Part a

To determine the number of peaks, we analyze the score distribution. The scores are: 7 (90 - 100), 12 (80 - 89), 21 (70 - 79), 19 (below 70). There's no clear separation into distinct groups (e.g., two different cohorts with different score patterns). So, we expect one peak as there's no obvious reason for multiple groups. Option D states this, so it's correct.

From part a, we expect one peak (unimodal). Also, the number of students below 70 is 19, 70 - 79 has 21 (slightly more), 80 - 89 has 12 (less than 70 - 79), and 90 - 100 has 7 (least). So the distribution should be left - skewed? Wait, no. Wait, the lower scores (below 70) have 19, then 70 - 79 (21, a bit more), then 80 - 89 (12, less), 90 - 100 (7, less). Wait, actually, the peak should be around 70 - 79, and then the tail is towards the higher scores? Wait, no. Wait, the number of students: below 70 (19), 70 - 79 (21) – so the count increases from below 70 to 70 - 79, then decreases (12, 7). So the distribution is right - skewed? Wait, no, right - skewed has tail on right (higher values). Wait, the scores: lower scores (40 - 69) have 19, 70 - 79 (21), 80 - 89 (12), 90 - 100 (7). So the peak is at 70 - 79, and then as we go to higher scores (80 - 89, 90 - 100), the counts decrease. Also, the left side (below 70) has 19, which is less than 21 (70 - 79). So the distribution is left - skewed? Wait, no, left - skewed means tail on left (lower values). Wait, maybe I got it wrong. Wait, the key is unimodal. Let's look at the graphs:

• Graph A: bimodal (two peaks) – incorrect, we need unimodal.
• Graph B: unimodal, symmetric? No, but wait our data: below 70 (19), 70 - 79 (21), 80 - 89 (12), 90 - 100 (7). So the left side (below 70) has 19, then 70 - 79 (21) – so the left tail (below 70) is shorter than the peak's left? Wait, no. Wait, the number of students: the lowest scores (below 70) have 19, then 70 - 79 (21, more), then 80 - 89 (12, less), 90 - 100 (7, less). So the distribution should have a peak at 70 - 79, and then the counts decrease as we move to higher scores (right side) and also decrease as we move to lower scores (left side, but left side has 19 which is close to 21). Wait, maybe the correct graph is D? Wait, no, let's re - evaluate. Wait, the total number of students: 19 + 21+12 + 7=59, which matches. The peak is at 70 - 79. Then, the number of students below 70 (19) is less than at 70 - 79 (21), so the left side (below 70) is a bit lower than the peak. Then, as we go to higher scores (80 - 89, 90 - 100), the counts decrease. So the distribution is right - skewed? Wait, no, right - skewed has tail on right. Wait, the scores: the higher scores (90 - 100) have the least number of students, so the tail is on the right. So the distribution is right - skewed, unimodal. Looking at the graphs:

• Graph C: right - skewed, unimodal (tail on right)
• Graph D: left - skewed, unimodal (tail on left)

Wait, let's check the counts again. Below 70: 19, 70 - 79:21, 80 - 89:12, 90 - 100:7. So from below 70 to 70 - 79, the count increases (19 to 21). Then from 70 - 79 to 80 - 89, it decreases (21 to 12), then to 90 - 100 (12 to 7). So the left side (below 70) is a bit lower than the peak (70 - 79), and the right side has a long tail (since 12 and 7 are much lower than 21). So the distribution is right - skewed. So the correct graph should be the one that is unimodal and right - skewed, which is Graph C? Wait, no, maybe I made a mistake. Wait, the original data: the number of students below 70 is 19, 70 - 79 is 21 (so the peak is at 70 - 79), then 80 - 89 (12) and 90 - 100 (7). So the shape: starts at below 70 (19), goes up to 70 - 79 (21), then down to 80 - 89 (12), then down to 90 - 100 (7). So the left side (below 70) is a bit lower than the peak, and the right side has a tail. So the distribution is right - skewed. So the correct graph is the one with a peak and a right - skewed tail, which is Graph C? Wait, but let's check the options. T…

Answer:

D. There would probably be one peak because there are no obvious reasons why the exam scores would form different groups

Part b