Loan Processing at Capital One Memo

Students at a major university are complaining of a serious housing crunch. Many of the university's students, they complain, have to commute too far to school because there is not enough housing near campus. The university officials respond with the following information: the mean distance commuted to school by students is miles, and the standard deviation of the distance commuted is miles.

The empirical rule states that

Figure 1

• approximately of the measurements in a bell-shaped distribution lie within standard deviation of the mean;
• approximately of the measurements in a bell-shaped distribution lie within standard deviations of the mean;
• approximately of the measurements in a bell-shaped distribution lie within standard deviations of the mean.
•
• We are asked to use Chebyshev's theorem to determine the minimum percentage of the students' commute distances that lie between and . To do this, we can express the values and in terms of their distance from the mean (in standard deviations) and then apply Chebyshev's theorem. We are told that the mean is and the standard deviation is . Note that and

or, equivalently, and .
This means that lies standard deviations below (since each standard deviation is units) and that lies standard deviations above . According to Chebyshev's theorem, at least of the commute distances lie within standard deviations of the mean . Thus, at least of the commute distances lie between and .
(b)
We are asked to use Chebyshev's theorem to determine the minimum percentage of the commute distances that lie between and . As in part (a) above, we can express and in terms of their distance from : and , so and .
Hence, lies standard deviations below , and lies standard deviations above . From Chebyshev's theorem, we know that at least of the commute distances lie within standard deviations of the mean . Thus, at least of the commute