Question

The Department of Mathematics and Computer Science of Dickinson College gives an exam each fall to freshmen that intend to take calculus; scores on the exam are used to determine into which level of calculus a student should be placed. The exam consists of 20 multiple-choice questions. Scores for the 213 students who took the exam in 1992 are tallied in the following table. Score 1 2 3 4 5 6 7 8 9 10

Count 1 1 5 7 12 13 16 15 17 32

Score 11 12 13 14 15 16 17 18 19 20

Count 17 21 12 16 8 4 7 5 4 0

The mean score on this exam is x = 10.221 . The standard deviation of the exam scores is s = 3.859. A histogram of this distribution follows:

a) Does this distribution appear to be roughly symmetric and mound-shaped?

b) Consider the question of how many scores fall within one standard deviation of the mean (denoted by x  s ). Determine the upper endpoint of this interval by adding the value of the standard deviation to that of the mean. Then determine the interval’s lower endpoint by subtracting the value of the standard deviation from that of the mean.

c) What proportion of the 213 scores fall within one standard deviation of the mean?

d) Determine how many of the 213 scores fall within two standard deviations of the mean, which turns out to be between 2.503 and 17.939. What proportion is this? 12

e) Determine how many of the 213 scores fall within three standard deviations of the mean, which turns out to be between –1.356 and 21.798. What proportion is this?

f) How do the values determined in c) to e) compare to the values learned for the empirical rule?

Answer 1

a)

Yes, the distribution appear to be roughly symmetric and mound-shaped.

b)

upper endpoint = x + s = 10.221 + 3.859 = 14.08

lower endpoint = x - s = 10.221 - 3.859 = 6.362

Number of scores fall within one standard deviation of the mean = 16 + 15 + 17 + 32 + 17 + 21 + 12 + 16

= 146

c)

proportion of the 213 scores fall within one standard deviation of the mean = 146 / 213 = 0.6854

d)

upper endpoint = x + 2s = 10.221 + 2 * 3.859 = 17.939

lower endpoint = x - 2s = 10.221 - 2 * 3.859 = 2.503

Number of scores fall within two standard deviation of the mean = 5 + 7 + 12 + 13 + 16 + 15 + 17 + 32 + 17 + 21 + 12 + 16 + 8 + 4 + 7

= 202

proportion of the 213 scores fall within two standard deviation of the mean = 202 / 213 = 0.9484

e)

Number of scores fall within three standard deviation of the mean (between –1.356 and 21.798) = All the scores = 213

proportion of the 213 scores fall within three standard deviation of the mean = 213 / 213 = 1

f)

By empirical rule,

proportion of the scores fall within one standard deviation of the mean is 0.68

proportion of the scores fall within two standard deviation of the mean is 0.95

proportion of the scores fall within three standard deviation of the mean is 0.997

The scores data is consistent with the empirical rule.