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Chapter 3: Problem 10

Men versus Women The average 20 - to 29 -year-old man is 69.6 inches tall, with a standard deviation of 2.7 inches, while the average 20 - to 29 -year- old woman is 64.1 inches tall, with a standard deviation of 2.6 inches. Who is relatively taller, a 68 -inch man or a 62 -inch woman? (Source: Vital and Health Statistics, Advance Data, Oct. 2004

Short Answer

Expert verified
The 68-inch man is relatively taller than the 62-inch woman.

Step by step solution

01

- Understand the Problem

Identify who is relatively taller, a 68-inch tall man or a 62-inch tall woman, using the given average heights and standard deviations for men and women.
02

- Calculate the Z-score for the Man

To determine how many standard deviations a man is from the average, use the formula: \[ Z = \frac{X - \mu}{\sigma} \] where X is the man's height (68 inches), \(\mu\) is the mean height for men (69.6 inches), and \(\sigma\) is the standard deviation (2.7 inches). \[ Z_{man} = \frac{68 - 69.6}{2.7} = \frac{-1.6}{2.7} \approx -0.59 \]
03

- Calculate the Z-score for the Woman

To determine how many standard deviations a woman is from the average, use the same Z-score formula: \[ Z = \frac{X - \mu}{\sigma} \] where X is the woman's height (62 inches), \(\mu\) is the mean height for women (64.1 inches), and \(\sigma\) is the standard deviation (2.6 inches). \[ Z_{woman} = \frac{62 - 64.1}{2.6} = \frac{-2.1}{2.6} \approx -0.81 \]
04

- Compare the Z-scores

Compare the calculated Z-scores. The Z-score that is closer to zero indicates the individual who is relatively taller. \[-0.59 (man) > -0.81 (woman)\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Deviation
Standard deviation measures the amount of variation or dispersion in a set of values. It's a key concept in statistics. In our problem, it helps us understand how much heights vary around the average height for both men and women.
For men, the standard deviation is 2.7 inches. For women, it's 2.6 inches.
Smaller standard deviations mean values are closer to the mean, while larger ones indicate more spread out data.
Mean Height
The mean height represents the average height of a group. For 20- to 29-year-old men, the mean height is 69.6 inches. For women in the same age group, the mean height is 64.1 inches.
To understand how different an individual's height is from the average, we can compare their height to the mean height. This helps us understand relative heights better.
Relative Comparison
Relative comparison helps us see how one value stands in relation to others in the same group. In our exercise, we want to know who's relatively taller: a 68-inch man or a 62-inch woman.
We use Z-score for this. Z-score tells us how many standard deviations a value is from the average.
For example, a Z-score of -0.59 means the man is 0.59 standard deviations below the average male height. And a Z-score of -0.81 means the woman is 0.81 standard deviations below the average female height. Comparing these Z-scores reveals who's relatively taller.
Normal Distribution
Normal distribution is a bell-shaped curve that shows the spread of a set of data. Most values cluster around the mean, while fewer values can be found farther from the mean.
The heights of men and women in our exercise follow a normal distribution. This means these heights are symmetrically distributed around their respective mean heights with predictable patterns.
Understanding normal distribution helps us use Z-scores to compare data points like the heights in our problem.

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