Herschel finished his last race (the 2013 Fox Cities Marathon) in 221 minutes.  The average finishing time for the race was 265 minutes and the standard deviation was 55 minutes.Assuming finishing times for the 2013 Fox Cities Marathon were approximately bell-shaped, what percentage of runners would have finished the race in less than 155 minutes?

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.

Herschel finished his last race (the 2013 Fox Cities Marathon) in 221 minutes. The average finishing time for the race was 265 minutes and the standard deviation was 55 minutes. Assuming finishing times for the 2013 Fox Cities Marathon were approximately bell-shaped, what percentage of runners would have finished the race in less than 155 minutes?

Discover how to calculate the percentage of runners who finished the 2013 Fox Cities Marathon in under 155 minutes using the Empirical Rule. Learn about normal distribution, z-scores, and key statistics concepts. This problem is suitable for high school students studying grades 9-12.

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