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Let's work through each part of this problem based on the provided information.

### Given Information:
- **Women's finishing times**: Normally distributed with mean \(\mu_W = 259\) minutes and standard deviation \(\sigma_W = 32\) minutes.
- **Men's finishing times**: Normally distributed with mean \(\mu_M = 242\) minutes and standard deviation \(\sigma_M = 29\) minutes.

### (a) Standardized Score for Clay
Clay's finishing time is **289 minutes**. We need to calculate and interpret his standardized score (z-score) using the men’s distribution.

The formula for the z-score is:
\[
z = \frac{X - \mu}{\sigma}
\]
where:
- \(X\) is the observed value (Clay's finishing time),
- \(\mu\) is the mean of the distribution (men's mean),
- \(\sigma\) is the standard deviation of the distribution (men's standard deviation).

For Clay:
\[
z = \frac{289 - 242}{29} = \frac{47}{29} \approx 1.62
\]

**Interpretation:**  
Clay's finishing time of 289 minutes is approximately 1.62 standard deviations **above** the mean finishing time for men.

### (b) Proportion of Women with a Finishing Time Less Than Kathy's
Kathy's finishing time is **272 minutes**. We need to find the proportion of women who finished the marathon faster than Kathy, i.e., with a finishing time less than 272 minutes.

Calculate Kathy's z-score using the women’s distribution:
\[
z = \frac{272 - 259}{32} = \frac{13}{32} \approx 0.406
\]

Now, using the z-table or a standard normal distribution calculator, we find the proportion of the distribution that is less than this z-score. The cumulative probability for \(z = 0.406\) is approximately **0.658**.

**Interpretation:**  
Approximately 65.8% of women had a finishing time **less than** Kathy’s finishing time of 272 minutes.

### (c) Interpretation of Standard Deviation Differences
The standard deviation of finishing times for women (\(\sigma_W = 32\)) is greater than that for men (\(\sigma_M = 29\)).

**Interpretation:**  
A larger standard deviation indicates that the finishing times for women were more spread out around the mean compared to the men's finishing times. This means there is more variability in the marathon finishing times for women than for men.

Would you like more details or have any questions?  
Here are five related questions for further exploration:

1. How would Clay’s z-score change if he had finished in 260 minutes instead of 289 minutes?
2. How would you interpret Kathy's z-score if it were negative?
3. What does it imply if a runner has a z-score of 0?
4. How could the variability in marathon times impact strategies for training?
5. How would you calculate the percentage of men who finished in a range of times, say between 240 and 280 minutes?

**Tip:** Understanding z-scores is crucial in interpreting individual data points in the context of a normal distribution, allowing you to determine how typical or unusual an observation is within the given data set.

Learn how to calculate z-scores and interpret proportions for marathon finishing times in AP Statistics. This problem covers normal distribution, standardized scores (z-scores), and probability calculations. Suitable for AP Statistics students.

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