To solve this problem using Chebyshev's Theorem, we need to find the standard deviation (denoted as $\sigma$) of the times taken by the 2403 runners, based on the given percentage and time range.

Chebyshev's Theorem:

Chebyshev’s theorem states that for any distribution, at least $1 - \frac{1}{k^2}$ of the data lies within $k$ standard deviations from the mean, where $k$ is any positive number greater than 1.

In this case:

• You are told that at least 77.42% of the runners took between 59.7 and 77.9 minutes.
• Chebyshev's theorem formula is $1 - \frac{1}{k^2} \geq 0.7742$, where $k$ represents the number of standard deviations from the mean.

Step 1: Use Chebyshev's Formula to Find $k$

Set the formula $1 - \frac{1}{k^2} = 0.7742$.

Solve for $k$:

$1 - \frac{1}{k^2} = 0.7742$

$\frac{1}{k^2} = 1 - 0.7742 = 0.2258$

$k^2 = \frac{1}{0.2258}$

$k = \sqrt{\frac{1}{0.2258}} \approx \sqrt{4.429} \approx 2.105$

Thus, $k \approx 2.105$.

Step 2: Relate $k$ to the Given Data Range

The range of time given is from 59.7 minutes to 77.9 minutes, and the difference between these two values is:

$77.9 - 59.7 = 18.2 \text{ minutes}$

Since $k$ standard deviations cover the range from the mean, and $k = 2.105$, the total range covered is $2k \sigma$, where $\sigma$ is the standard deviation.

$2k \sigma = 18.2$

Substitute the value of $k$:

$2(2.105) \sigma = 18.2$

$4.21 \sigma = 18.2$

Step 3: Solve for $\sigma$

$\sigma = \frac{18.2}{4.21} \approx 4.32 \text{ minutes}$

Thus, the standard deviation $\sigma$ is approximately 4.32 minutes.

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Would you like further details or have any questions?

Here are 5 related questions:

1. What assumptions does Chebyshev’s theorem make about the distribution of data?
2. How would the calculation change if we were given a normal distribution instead of using Chebyshev’s theorem?
3. Can Chebyshev’s theorem be used for non-symmetric data distributions?
4. How does Chebyshev’s theorem compare to the Empirical Rule for normal distributions?
5. What is the meaning of $k$ in Chebyshev's theorem in practical terms?

Tip: Chebyshev's theorem is valuable for any dataset, regardless of the shape of the distribution, but provides more conservative estimates than rules for normal distributions.