Students at a major university are complaining of a serious housing crunch. They complain that many students have to commute too far to school because is not enough housing near campus. University officials respond with the following information: the mean distance commuted to school by students is 18.0 miles, and the standard deviation of the distance commuted is 3.9 miles. Assuming that the university officials' information is correct, complete the following statements about the distribution of commute distances for students at university.
Students at a major university are complaining of a serious housing crunch. They complain that many students have to commute too far to school because is not enough housing near campus. University officials respond with the following information: the mean distance commuted to school by students is 18.0 miles, and the standard deviation of the distance commuted is 3.9 miles. Assuming that the university officials' information is correct, complete the following statements about the distribution of commute distances for students at university.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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Problem 1P
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![**Title:** Understanding Commute Distances Using Chebyshev’s Theorem
**Introduction:**
Students at a major university are experiencing a severe housing shortage. Many students argue that the lack of nearby housing forces them to commute excessively long distances to school. In response, university officials provide the following data: the average (mean) commute distance for students is 18.0 miles, with a standard deviation of 3.9 miles.
**Objective:**
Based on the information supplied by the university, apply Chebyshev’s theorem to determine the distribution of students' commute distances.
**Analysis:**
1. **Chebyshev’s Theorem:**
Chebyshev's theorem provides a way to understand the spread of a data set, regardless of its distribution. It states that at least \(\left(1 - \frac{1}{k^2}\right) \times 100\%\) of the data will fall within \(k\) standard deviations from the mean, where \(k > 1\).
2. **Calculations:**
(a) **Commuting Distance Between 10.2 Miles and 25.8 Miles:**
\[
\text{Mean} = 18.0 \text{ miles}
\]
\[
\text{Standard Deviation} = 3.9 \text{ miles}
\]
\[
\text{Range} = \text{Mean} \pm k \times \text{Standard Deviation}
\]
For 10.2 miles and 25.8 miles, calculate \(k\) as follows:
\[
18.0 - 10.2 = 7.8 \text{ and } 25.8 - 18.0 = 7.8
\]
\[
k = \frac{7.8}{3.9} = 2
\]
Therefore, at least \(\left(1 - \frac{1}{2^2}\right) \times 100\% = 75\%\) of commute distances fall within this range.
(b) **Commuting Distance Between 8.25 Miles and 27.75 Miles:**
\[
18.0 - 8.25 = 9.75 \text{ and } 27.75 - 18.0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee025788-43ba-44d1-a712-3a4f81fda024%2F30824af0-d7ba-4154-bdf1-a53638cb877c%2F75vkmq_processed.png&w=3840&q=75)
Transcribed Image Text:**Title:** Understanding Commute Distances Using Chebyshev’s Theorem
**Introduction:**
Students at a major university are experiencing a severe housing shortage. Many students argue that the lack of nearby housing forces them to commute excessively long distances to school. In response, university officials provide the following data: the average (mean) commute distance for students is 18.0 miles, with a standard deviation of 3.9 miles.
**Objective:**
Based on the information supplied by the university, apply Chebyshev’s theorem to determine the distribution of students' commute distances.
**Analysis:**
1. **Chebyshev’s Theorem:**
Chebyshev's theorem provides a way to understand the spread of a data set, regardless of its distribution. It states that at least \(\left(1 - \frac{1}{k^2}\right) \times 100\%\) of the data will fall within \(k\) standard deviations from the mean, where \(k > 1\).
2. **Calculations:**
(a) **Commuting Distance Between 10.2 Miles and 25.8 Miles:**
\[
\text{Mean} = 18.0 \text{ miles}
\]
\[
\text{Standard Deviation} = 3.9 \text{ miles}
\]
\[
\text{Range} = \text{Mean} \pm k \times \text{Standard Deviation}
\]
For 10.2 miles and 25.8 miles, calculate \(k\) as follows:
\[
18.0 - 10.2 = 7.8 \text{ and } 25.8 - 18.0 = 7.8
\]
\[
k = \frac{7.8}{3.9} = 2
\]
Therefore, at least \(\left(1 - \frac{1}{2^2}\right) \times 100\% = 75\%\) of commute distances fall within this range.
(b) **Commuting Distance Between 8.25 Miles and 27.75 Miles:**
\[
18.0 - 8.25 = 9.75 \text{ and } 27.75 - 18.0
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