Students at a major university are complaining of a serious housing crunch. Many
ID: 3126248 • Letter: S
Question
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 17.5 miles, and the standard deviation of the distance commuted is 3.7 miles.
Assuming that the university officials' information is correct, complete the following statements about the distribution of commute distances for students at this university.
(a) According to Chebyshev's theorem, at least ____ of the commute distances lie between 10.1 miles and 24.9 miles.
(b) According to Chebyshev's theorem, at least 36% of the commute distances lie between
miles
and
miles
. (Round your answer to 1 decimal place.)
(c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 99.7% of the commute distances lie between
miles
and
miles
.
(d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately _____of the commute distances lie between 10.1 miles and 24.9 miles.
Explanation / Answer
A)
Note that both 10.1 and 24.9 are 2 standard deviations above/below the mean.
Hence, k = 2 in
1 - 1/k^2 = 1-1/2^2 = 0.75 or 75% [ANSWER]
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b)
1 - 1/k^2 = 0.36
1/k^2 = 0.64
k = 1.25
Hence, it is between
u - 1.25*sigma = 17.5 - 1.25*3.7 = 12.9 mi [ANSWER]
and
u + 1.25*sigma = 17.5 + 1.25*3.7 = 22.1 mi [ANSWER]
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c)
It is within 3 standard deviations from the mean.
Hence, it is between
u - 3*sigma = 17.5 - 3*3.7 = 6.4 mi [ANSWER]
and
u + 3*sigma = 17.5 + 3*3.7 = 28.6 mi [ANSWER]
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d)
As they lie within 2 standard deviations from the mean, then it is around 95%. [ANSWER]