Question 14

Using Chebysheff's theorm, you have concluded that at least 77.42% of the 2403 runners took between 59.7 and 77.9 minutes to complete the 10km race.  What was the standard deviation  of these 2403 runners?

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**.

---

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.

Using Chebysheff's theorem, you have concluded that at least 77.42% of the 2403 runners took between 59.7 and 77.9 minutes to complete the 10km race. What was the standard deviation of these 2403 runners?

Learn how to calculate the standard deviation of 2403 runners in a 10km race using Chebyshev's theorem. The problem involves determining the percentage of data within a given range and applying statistical concepts such as standard deviation. Suitable for students in Grades 10-12.

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