Explanation and Solution

We are working with a normal distribution for the 10 km finishing times:

• Mean ($\mu$) = 61 minutes
• Standard deviation ($\sigma$) = 9 minutes.

The standardized variable $z$ is given by: $z = \frac{x - \mu}{\sigma}$ where $x$ is the finishing time.

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Part c) Percentage of finishers with times between 40 and 65 minutes

We are asked for the percentage of finishers between $x = 40$ and $x = 65$. First, calculate the corresponding $z$-scores:

$z_1 = \frac{40 - 61}{9} = \frac{-21}{9} \approx -2.33$ $z_2 = \frac{65 - 61}{9} = \frac{4}{9} \approx 0.44$

The percentage of finishers between these times is equal to the area under the standard normal curve between $z_1 = -2.33$ and $z_2 = 0.44$. Using standard normal tables or a calculator:

• The cumulative probability for $z_1 = -2.33$ is approximately $0.0099$.
• The cumulative probability for $z_2 = 0.44$ is approximately $0.6700$.

The area between $z_1$ and $z_2$ is: $P(-2.33 \leq z \leq 0.44) = 0.6700 - 0.0099 = 0.6601$

Thus, 66.01% of finishers have times between 40 and 65 minutes.

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Part d) Percentage of finishers with times exceeding 82 minutes

For $x = 82$, calculate the corresponding $z$-score:

$z = \frac{82 - 61}{9} = \frac{21}{9} \approx 2.33$

The percentage of finishers exceeding 82 minutes corresponds to the area under the standard normal curve to the right of $z = 2.33$. From standard normal tables:

• The cumulative probability for $z = 2.33$ is approximately $0.9901$.

The area to the right is: $P(z > 2.33) = 1 - 0.9901 = 0.0099$

Thus, 0.99% of finishers have times exceeding 82 minutes.

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Summary of Results:

1. Mean and standard deviation of $x$: 61 and 9 respectively.
2. Mean and standard deviation of $z$: 0 and 1 respectively.
3. Percentage of finishers with times between 40 and 65 minutes: 66.01%.
4. Percentage of finishers with times exceeding 82 minutes: 0.99%.

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Would you like a visualization or further clarification?

Related Questions:

1. What is the percentage of finishers with times less than 50 minutes?
2. How would the results change if the mean or standard deviation were different?
3. Can we calculate the percentage of finishers with times between two other values (e.g., 30 and 90 minutes)?
4. What is the z-score corresponding to a time of 70 minutes?
5. How is the normal distribution useful for analyzing race times?

Tip:

To quickly calculate percentages or probabilities for a normal distribution, consider using z-tables or statistical software for accurate results.