According To The US Census Bureau, The Mean Of The Commute T

According To The Us Census Bureau The Mean Of The Commute Time To Wor

According to the US Census Bureau, the mean commute time to work for residents of Boston, Massachusetts, is 27.3 minutes, with a standard deviation of 8.1 minutes. Utilizing Chebyshev's rule, this analysis aims to determine the minimum percentage of commuters whose commute times fall within specific deviations from the mean: within 2 standard deviations and within 1.5 standard deviations. Additionally, the reasoning behind employing Chebyshev's rule in this context will be discussed.

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Chebyshev’s inequality is a fundamental theorem in probability theory that provides a conservative estimate of the proportion of any data distribution that falls within a specified number of standard deviations from the mean. This rule applies to all data distributions, regardless of shape—be it symmetric, skewed, or otherwise non-normal—which makes it especially useful in real-world situations where the underlying distribution is unknown or non-normal. This analysis employs Chebyshev’s rule to estimate the minimum percentage of Boston commuters whose commute times are within set deviations from the average, providing critical insights for transportation planning and policy-making.

First, let’s interpret the parameters given: the mean commute time \(\mu = 27.3\) minutes, with a standard deviation \(\sigma = 8.1\) minutes. The goal is to determine the minimum proportion of commuters with commute times within specific multiples of \(\sigma\) from \(\mu\). Using Chebyshev's inequality, the core formula is:

\[

P(|X - \mu| \leq k\sigma) \geq 1 - \frac{1}{k^2}

\]

where \(\,k\,\) is the number of standard deviations from the mean.

Part A: Commute time within 2 standard deviations

Let’s calculate the minimum percentage of commuters with commute times within 2 standard deviations of the mean. Here, \(k = 2\). Applying Chebyshev’s inequality:

\[

P(|X - \mu| \leq 2\sigma) \geq 1 - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4} = 75\%

\]

Therefore, at least 75% of Boston residents have a commute time that is between \(27.3 - 2 \times 8.1 = 11.1\) minutes and \(27.3 + 2 \times 8.1 = 43.5\) minutes.

Part B: Commute time within 1.5 standard deviations

Next, for within 1.5 standard deviations, \(k = 1.5\), Chebyshev’s inequality provides:

\[

P(|X - \mu| \leq 1.5\sigma) \geq 1 - \frac{1}{(1.5)^2} = 1 - \frac{1}{2.25} \approx 1 - 0.4444 = 0.5556

\]

which translates to a minimum of approximately 55.56% of commuters having commute times between:

\[

27.3 - 1.5 \times 8.1 = 14.75 \text{ minutes}

\]

and

\[

27.3 + 1.5 \times 8.1 = 39.85 \text{ minutes}

\]

Part C: Why are we using Chebyshev's rule?

Chebyshev’s rule is especially advantageous when the distribution of data is unknown or non-normal, which is often the case with commute times that can be skewed due to various urban factors. Unlike the empirical rule, which is limited to symmetric, bell-shaped (normal) distributions, Chebyshev's inequality applies universally, providing a conservative estimate that holds regardless of the underlying distribution. This universality makes it a reliable tool for transportation analysts and urban planners to gauge the spread of commute times and identify potential areas for infrastructure improvements. Its ability to deliver minimum bounds ensures policymakers can make informed decisions even in uncertain or non-ideal data conditions.

Conclusion

Applying Chebyshev's rule, a minimum of 75% of Boston commuters have commute times within 2 standard deviations (11.1–43.5 minutes) of the mean, and approximately 55.56% are within 1.5 standard deviations (14.75–39.85 minutes). The employment of Chebyshev’s inequality here underscores its utility in environmental and urban planning contexts, offering reliable bounds where distributional assumptions are unconfirmed. Although more precise estimates could be obtained with distribution-specific rules like the empirical rule, Chebyshev's generality ensures robust management of uncertain data scenarios in transportation and city planning.

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