In a certain city, the average 20- to 29-year old man is 69.8
inches tall, with a standard deviation of 3.2
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?
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 formula is:
z = (x - μ) / σ
Where:
x = individual's height
μ = mean height
σ = standard deviation of height
For the 75-inch man:
z_man = (75 - 69.8) / 3.2
z_man = 1.625
For the 70-inch woman:
z_woman = (70 - 64.3) / 3.9
z_woman = 1.462
From the z-scores, we can see that the 75-inch man (z_man = 1.625) is relatively taller compared to the 70-inch woman (z_woman = 1.462).
To determine who is relatively taller, we need to compare the heights of the individuals to their respective gender averages.
First, let's calculate the z-scores for both the 75-inch man and the 70-inch woman.
For the 75-inch man:
Z-score = (X - μ) / σ
Z-score = (75 - 69.8) / 3.2
Z-score ≈ 1.63
For the 70-inch woman:
Z-score = (X - μ) / σ
Z-score = (70 - 64.3) / 3.9
Z-score ≈ 1.46
The z-score measures how many standard deviations an individual's height is away from the average height of their gender.
Since the z-score for the 75-inch man (1.63) is higher than the z-score for the 70-inch woman (1.46), the 75-inch man is relatively taller compared to their respective genders.