The heights of adult men in America are normally distributed, with a mean of 69.1 inches and a standard deviation of 2.65 inches. The heights of adult women in America are also normally distributed, but with a mean of 64.3 inches and a standard deviation of 2.51 inches. a) If a man is 6 feet 3 inches tall, what is his z-score (to two decimal places)? z = b) If a woman is 5 feet 11 inches tall, what is her z-score (to two decimal places)? z =
The Correct Answer and Explanation is:
Problem Breakdown:
We are given two normal distributions: one for the heights of adult men and one for the heights of adult women in America. The task is to calculate the z-scores for a man who is 6 feet 3 inches tall and a woman who is 5 feet 11 inches tall.
Step 1: Z-Score Formula
The formula to calculate a z-score is: z=X−μσz = \frac{{X – \mu}}{{\sigma}}z=σX−μ
Where:
- XXX is the value for which you want to find the z-score.
- μ\muμ is the mean of the distribution.
- σ\sigmaσ is the standard deviation of the distribution.
a) Z-Score for the Man (6 feet 3 inches)
First, we need to convert the man’s height from feet and inches into inches only:
- 6 feet = 72 inches
- 3 inches = 3 inches
- Therefore, the man’s total height is 72+3=7572 + 3 = 7572+3=75 inches.
Now, we apply the z-score formula for the distribution of men’s heights:
- Mean height for men μ=69.1\mu = 69.1μ=69.1 inches
- Standard deviation for men σ=2.65\sigma = 2.65σ=2.65 inches
- The man’s height X=75X = 75X=75 inches
Substitute these values into the z-score formula: z=75−69.12.65=5.92.65≈2.23z = \frac{{75 – 69.1}}{{2.65}} = \frac{{5.9}}{{2.65}} \approx 2.23z=2.6575−69.1=2.655.9≈2.23
So, the z-score for the man is 2.23.
b) Z-Score for the Woman (5 feet 11 inches)
Next, we need to convert the woman’s height from feet and inches into inches only:
- 5 feet = 60 inches
- 11 inches = 11 inches
- Therefore, the woman’s total height is 60+11=7160 + 11 = 7160+11=71 inches.
Now, we apply the z-score formula for the distribution of women’s heights:
- Mean height for women μ=64.3\mu = 64.3μ=64.3 inches
- Standard deviation for women σ=2.51\sigma = 2.51σ=2.51 inches
- The woman’s height X=71X = 71X=71 inches
Substitute these values into the z-score formula: z=71−64.32.51=6.72.51≈2.67z = \frac{{71 – 64.3}}{{2.51}} = \frac{{6.7}}{{2.51}} \approx 2.67z=2.5171−64.3=2.516.7≈2.67
So, the z-score for the woman is 2.67.
Summary of Results:
- The z-score for the man who is 6 feet 3 inches tall is 2.23.
- The z-score for the woman who is 5 feet 11 inches tall is 2.67.
Explanation:
The z-score tells us how many standard deviations a particular data point (height in this case) is away from the mean of the distribution. A positive z-score indicates that the value is above the mean, while a negative z-score indicates it is below the mean.
For the man, a z-score of 2.23 means that his height is 2.23 standard deviations above the average height of men in America. Similarly, for the woman, a z-score of 2.67 means her height is 2.67 standard deviations above the average height of women in America.
These z-scores give us a way to compare how unusual or typical their heights are within the context of the two different height distributions for men and women.
