Given that the finishing times follow a bell-shaped distribution (which likely refers to a normal distribution), we can apply the Empirical Rule (also known as the 68-95-99.7 rule) to estimate the percentage of runners who finished the race in less than 155 minutes.

Step 1: Calculate how many standard deviations 155 minutes is from the mean

The mean finishing time $\mu$ is 265 minutes, and the standard deviation $\sigma$ is 55 minutes. We calculate the z-score to determine how many standard deviations 155 minutes is from the mean:

$z = \frac{X - \mu}{\sigma} = \frac{155 - 265}{55} = \frac{-110}{55} = -2$

Step 2: Interpret the z-score using the Empirical Rule

• A z-score of -2 means 155 minutes is 2 standard deviations below the mean.
• According to the Empirical Rule:
  • About 68% of data falls within 1 standard deviation (between -1 and +1 z-scores),
  • About 95% of data falls within 2 standard deviations (between -2 and +2 z-scores).

Step 3: Determine the percentage below 155 minutes

Since 95% of the runners finished between 155 and 375 minutes (within 2 standard deviations of the mean), the remaining 5% of the distribution falls outside this range, with half of that 5% (or 2.5%) below 155 minutes.

Thus, 2.5% of runners finished the race in less than 155 minutes.

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Conclusion:

Approximately 2.5% of runners would have finished the race in less than 155 minutes.

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Would you like a detailed explanation of the Empirical Rule or z-scores, or any other clarification? Here are 5 questions related to this:

1. How does the Empirical Rule apply to other distributions?
2. How can we calculate percentages for non-standard z-scores?
3. What are the properties of a normal distribution?
4. How would we approach this problem using a z-table?
5. How can marathon finishing times differ from a perfect normal distribution?

Tip: In normal distributions, approximately 68% of the data falls within one standard deviation of the mean. This is a useful shortcut for estimating probabilities quickly.