In a certain​ city, the average​ 20- to​ 29-year old man is 69.6 inches​ tall, with a standard deviation of 3.1 inches, while the average​ 20- to​ 29-year old woman is 64.3 inches​ tall, with a standard deviation of 3.9 inches. Who is relatively​ taller, a​ 75-inch man or a​ 70-inch woman? Find the corresponding​ z-scores. Who is relatively​ taller, a​ 75-inch man or a​ 70-inch woman? Select the correct choice below and fill in the answer boxes to complete your choice.

To determine who is relatively taller, we need to calculate the z-scores for both the 75-inch man and the 70-inch woman. The z-score is calculated using the formula:

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

where:
- \(X\) is the value being measured (height in this case),
- \(\mu\) is the mean (average height for the respective group), and
- \(\sigma\) is the standard deviation.

### 1. Calculate the z-score for the 75-inch man:
- \(X = 75\) inches
- \(\mu_{\text{man}} = 69.6\) inches
- \(\sigma_{\text{man}} = 3.1\) inches

\[
z_{\text{man}} = \frac{75 - 69.6}{3.1} = \frac{5.4}{3.1} \approx 1.74
\]

### 2. Calculate the z-score for the 70-inch woman:
- \(X = 70\) inches
- \(\mu_{\text{woman}} = 64.3\) inches
- \(\sigma_{\text{woman}} = 3.9\) inches

\[
z_{\text{woman}} = \frac{70 - 64.3}{3.9} = \frac{5.7}{3.9} \approx 1.46
\]

### 3. Comparison of the z-scores:
- The z-score for the man is \(1.74\).
- The z-score for the woman is \(1.46\).

The z-score indicates how many standard deviations above (or below) the mean a particular measurement is. A higher z-score indicates that the individual is relatively taller compared to others in the same group.

### Conclusion:
The man, with a z-score of 1.74, is relatively taller compared to the woman, who has a z-score of 1.46.

### Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How would the results change if the standard deviation for the women was different?
2. What does a negative z-score represent in this context?
3. How would the z-scores compare if both heights were equal at 70 inches?
4. Can z-scores be used to compare heights across different age groups?
5. What if the standard deviations for both groups were equal?

**Tip:** Z-scores are a useful way to compare data points from different distributions, especially when the units or scales differ.

In a certain city, the average 20- to 29-year old man is 69.6 inches tall, with a standard deviation of 3.1 inches, while the average 20- to 29-year old woman is 64.3 inches tall, with a standard deviation of 3.9 inches. Who is relatively taller, a 75-inch man or a 70-inch woman? Find the corresponding z-scores. Who is relatively taller, a 75-inch man or a 70-inch woman? Select the correct choice below and fill in the answer boxes to complete your choice.

Discover how to determine who is relatively taller between a 75-inch man and a 70-inch woman by calculating their z-scores. This problem covers key statistical concepts such as mean, standard deviation, and z-scores. Ideal for students in Grades 9-12.

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