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The question involves calculating Z-scores for two individuals based on the heights of men and women in America. Let's break down and solve the problem step by step.

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### Given Information:
- **Men's height distribution:**
  - Mean \( \mu_m = 69.2 \) inches
  - Standard deviation \( \sigma_m = 2.68 \) inches
- **Women's height distribution:**
  - Mean \( \mu_w = 64.5 \) inches
  - Standard deviation \( \sigma_w = 2.58 \) inches

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### Part a: Z-score for a man who is 6 feet 3 inches tall (75 inches).
The Z-score formula is:

\[
Z = \frac{X - \mu}{\sigma}
\]

For the man:
- \( X = 75 \) inches (height of the man)
- \( \mu_m = 69.2 \) inches
- \( \sigma_m = 2.68 \) inches

Substitute the values:

\[
Z_m = \frac{75 - 69.2}{2.68}
\]

Let me calculate this for you.The Z-score for the man who is 6 feet 3 inches tall is approximately **2.1642** (to four decimal places).

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### Part b: Z-score for a woman who is 5 feet 11 inches tall (71 inches).
For the woman:
- \( X = 71 \) inches (height of the woman)
- \( \mu_w = 64.5 \) inches
- \( \sigma_w = 2.58 \) inches

Using the Z-score formula again:

\[
Z_w = \frac{71 - 64.5}{2.58}
\]

Let me calculate this for you as well.The Z-score for the woman who is 5 feet 11 inches tall is approximately **2.5194** (to four decimal places).

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### Part c: Who is relatively taller?
To determine who is relatively taller, we compare the Z-scores. The woman has a Z-score of **2.5194**, which is higher than the man's Z-score of **2.1642**. This means the woman is relatively taller compared to the average height of women, than the man is compared to the average height of men.

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Let me know if you would like more details on this!

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Here are 5 related questions:
1. How would the Z-score change if the man's height were 70 inches?
2. What does a negative Z-score imply in the context of height distribution?
3. How can Z-scores be used to compare different populations?
4. What percentage of men are shorter than a man with a Z-score of 1.5?
5. How would the standard deviation affect the Z-scores?

**Tip:** Z-scores allow for comparison between different distributions by standardizing the data in terms of how many standard deviations a value is from the mean.

The heights of adult men in America are normally distributed, with a mean of 69.2 inches and a standard deviation of 2.68 inches. The heights of adult women in America are also normally distributed, but with a mean of 64.5 inches and a standard deviation of 2.58 inches.

a. If a man is 6 feet 3 inches tall (75 inches), what is his z-score (to 4 decimal places)?
b. If a woman is 5 feet 11 inches tall (71 inches), what is her z-score (to 4 decimal places)?
c. Who is relatively taller?

Learn how to calculate the Z-score for a 6 feet 3 inches tall American man and a 5 feet 11 inches tall American woman. Understand how to compare their relative heights using Z-scores based on the normal distribution of adult heights in America.

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