In a certain​ city, the average​ 20- to​ 29-year old man is 

72.572.5 

inches​ tall, with a standard deviation of 

3.13.1 

​inches, while the average​ 20- to​ 29-year old woman is 

64.164.1 

inches​ tall, with a standard deviation of 

3.83.8 

inches. Who is relatively​ taller, a​ 75-inch man or a​ 70-inch woman?

Question content area bottom

Part 1

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.

​(Round to two decimal places as​ needed.)

A.

The​ z-score for the 

manman​, 

enter your response here​, 

is larger than the​ z-score for the 

womanwoman​, 

enter your response here​, 

so 

hehe 

is relatively taller.

B.

The​ z-score for the 

manman​, 

enter your response here​, 

is smaller than the​ z-score for the 

womanwoman​, 

enter your response here​, 

so 

hehe 

is relatively taller.

C.

The​ z-score for the 

womanwoman​, 

enter your response here​, 

is smaller than the​ z-score for the 

manman​, 

enter your response here​, 

so 

sheshe 

is relatively taller.

D.

The​ z-score for the 

womanwoman​, 

enter your response here​, 

is larger than the​ z-score for the 

manman​, 

enter your response here​, 

so 

sheshe 

is relatively taller.

To determine who is relatively taller, we need to compute the **z-scores** for both the 75-inch man and the 70-inch woman. The z-score measures how many standard deviations a given value is from the mean. The formula for the z-score is:

\[
z = \frac{{X - \mu}}{{\sigma}}
\]
Where:
- \( X \) is the individual's height,
- \( \mu \) is the average height,
- \( \sigma \) is the standard deviation of heights.

### Step 1: Calculate the z-score for the man
For the man:
- \( X = 75 \, \text{inches} \)
- \( \mu = 72.5 \, \text{inches} \)
- \( \sigma = 3.1 \, \text{inches} \)

\[
z_{\text{man}} = \frac{75 - 72.5}{3.1} = \frac{2.5}{3.1} \approx 0.81
\]

### Step 2: Calculate the z-score for the woman
For the woman:
- \( X = 70 \, \text{inches} \)
- \( \mu = 64.1 \, \text{inches} \)
- \( \sigma = 3.8 \, \text{inches} \)

\[
z_{\text{woman}} = \frac{70 - 64.1}{3.8} = \frac{5.9}{3.8} \approx 1.55
\]

### Step 3: Compare the z-scores
- The z-score for the man is approximately \( 0.81 \).
- The z-score for the woman is approximately \( 1.55 \).

Since the z-score for the woman is larger, she is relatively taller compared to the average woman than the man is compared to the average man.

### Conclusion:
The correct choice is:

**D.** The z-score for the woman, 1.55, is larger than the z-score for the man, 0.81, so she is relatively taller.

---

Would you like more details or have any questions?

### Related Questions:
1. What does the z-score represent in the context of this problem?
2. How would the relative heights change if the standard deviation for men was larger?
3. If the man's height was 70 inches, how would his z-score compare to the woman's?
4. How do standard deviations affect the interpretation of relative height?
5. Can a negative z-score ever indicate that someone is "relatively taller"?

### Tip:
When comparing values from different distributions, z-scores standardize them by taking into account both the mean and the variability (standard deviation) of each distribution.

In a certain city, the average 20- to 29-year old man is 72.5 inches tall, with a standard deviation of 3.1 inches, while the average 20- to 29-year old woman is 64.1 inches tall, with a standard deviation of 3.8 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? (Round to two decimal places as needed.)

Compare the relative heights of a 75-inch man and a 70-inch woman using z-scores based on the average height and standard deviation for men and women aged 20-29. Learn how to calculate z-scores and determine who is relatively taller in this statistics problem, perfect for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation