Students At A Major University Are Complaining Of A Serious

Students At A Major University Are Complaining Of A Serious Housing Cr

Students at a major university are complaining of a serious housing crunch. Many students have to commute far from campus due to insufficient nearby housing. University officials report that the average distance students commute is a certain number of miles, with a specific standard deviation. Based on this information, we are asked to analyze the distribution of commute distances, assuming the given parameters are accurate. The questions involve applying Chebyshev's theorem and the empirical rule to determine the proportion of students within certain distance ranges.

Paper For Above instruction

The problem presents a scenario where university students experience a housing shortage, leading to long commutes. To analyze this problem statistically, we rely on the provided mean and standard deviation of commute distances and incorporate two key theorems: Chebyshev’s theorem and the empirical rule. These tools allow us to estimate the proportion of students living within certain distance ranges, considering both general and bell-shaped distributions.

Understanding the Parameters

The university's reports specify a mean commute distance (\(\mu\)) and a standard deviation (\(\sigma\)), although the actual numerical values are missing in the problem statement. These parameters are fundamental in understanding the distribution of the data. The mean provides the central tendency, while the standard deviation measures the dispersion of commute distances around the mean.

Part (a): Chebyshev’s Theorem for a Distance Range of 9.9 to 25.5 miles

Chebyshev’s theorem states that for any distribution—regardless of shape—the proportion of data within \(k\) standard deviations of the mean is at least \(\frac{1}{k^2}\). To apply Chebyshev’s theorem, we need to find how many standard deviations the interval from 9.9 to 25.5 miles encompasses, assuming we know \(\mu\) and \(\sigma\).

If these bounds are set at specific distances from the mean, we calculate \(k\) as:

\[

k = \frac{\text{distance from the mean to the boundary}}{\sigma}

\]

Once \(k\) is known, Chebyshev’s inequality guarantees that at least \(\frac{1}{k^2}\) of the data falls within this range.

Part (b): Chebyshev’s Theorem for a Distance Range of 7.95 to 27.45 miles

Similarly, for the interval from 7.95 to 27.45 miles, we calculate \(k\) in the same manner. Usually, this range is wider, implying a smaller value of \(k\) and thus a larger proportion of data within this interval. Chebyshev’s theorem provides a lower bound, which is at least \(\frac{1}{k^2}\).

Part (c): Empirical Rule for a Bell-Shaped Distribution

If the distribution of commute distances is approximately bell-shaped, the empirical rule applies. The empirical rule states that:

- About 68% of data lies within 1 standard deviation of the mean.

- About 95% within 2 standard deviations.

- About 99.7% within 3 standard deviations.

Given the interval from 9.9 to 25.5 miles, if it corresponds to approximately \(\mu \pm 2\sigma\), then approximately 95% of students’ commute distances fall within this range.

Part (d): Empirical Rule for 99.7% of Data

For the bell-shaped distribution, approximately 99.7% of the commute distances lie within \(\mu \pm 3\sigma\). If these bounds correspond to the given distances, the interval is from \(\mu - 3\sigma\) to \(\mu + 3\sigma\).

Conclusion

Without specific numerical values for \(\mu\) and \(\sigma\), precise calculations cannot be performed. However, the framework described allows estimation of the proportion of commute distances within specified ranges using Chebyshev’s theorem and the empirical rule, depending on whether the distribution shape assumption holds. These insights highlight how statistical tools can inform understanding of commuting behavior and housing needs for university students, guiding policy and resource allocation to address the housing crunch.

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