Suppose the mean shoe size for men is 9, with a standard deviation of 1.5.
If a slipper is designed to fit any size from 5 to 12, what percentage of
men will this slipper fit? Show your derivations.

To solve this problem, we need to find the percentage of men whose shoe sizes fall within the range of 5 to 12. We will assume that men's shoe sizes follow a normal distribution with a mean of 9 and a standard deviation of 1.5.

The steps to calculate this are as follows:

1. **Standardize the values (convert to z-scores):**

   The formula for the z-score is:

   \[
   z = \frac{x - \mu}{\sigma}
   \]

   where:
   - \(x\) is the shoe size,
   - \(\mu = 9\) is the mean,
   - \(\sigma = 1.5\) is the standard deviation.

   We need to calculate the z-scores for the boundaries, 5 and 12.

   - For \(x = 5\):

     \[
     z_1 = \frac{5 - 9}{1.5} = \frac{-4}{1.5} \approx -2.67
     \]

   - For \(x = 12\):

     \[
     z_2 = \frac{12 - 9}{1.5} = \frac{3}{1.5} = 2.0
     \]

2. **Find the cumulative probabilities:**

   Using the z-scores, we now find the cumulative probabilities from the standard normal distribution table (or use a calculator).

   - The cumulative probability for \(z_1 = -2.67\) is approximately 0.0038 (0.38%).
   - The cumulative probability for \(z_2 = 2.0\) is approximately 0.9772 (97.72%).

3. **Calculate the percentage within the range:**

   To find the percentage of men whose shoe sizes fall between 5 and 12, subtract the cumulative probability at \(z_1\) from the cumulative probability at \(z_2\):

   \[
   P(5 \leq x \leq 12) = 0.9772 - 0.0038 = 0.9734
   \]

   Therefore, approximately 97.34% of men will have shoe sizes between 5 and 12.

### Conclusion:
The slipper will fit about **97.34%** of men.

---

Would you like more details on any of the steps or concepts? 

Here are some related questions:
1. What if the standard deviation changes to 2? How does that affect the percentage?
2. How would this calculation change if the mean were 10 instead of 9?
3. What is the probability that a man's shoe size is less than 5?
4. How would you calculate the percentage for women, assuming a different mean and standard deviation?
5. How do you find z-scores if the data is not normally distributed?

**Tip:** Always check if a problem assumes normal distribution; many real-world variables, like shoe sizes, often do!

Suppose the mean shoe size for men is 9, with a standard deviation of 1.5. If a slipper is designed to fit any size from 5 to 12, what percentage of men will this slipper fit? Show your derivations.

Calculate what percentage of men have shoe sizes between 5 and 12, assuming a normal distribution with a mean of 9 and standard deviation of 1.5. Learn how to use z-scores and cumulative probabilities for a step-by-step solution. Suitable for students in Grades 10-12.

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