Answer:
70-inch woman is taller.
Step-by-step explanation:
Given data:
For men
age group = 20 - 29 year
Height = 72.5 inches
standard deviation = 3.2 inches
For women
age group = 20 - 29 year
Height = 64.5 inches
standard deviation = 3.8 inches
To find:
who is taller? a 75-inch man or a 70-inch woman?
To find that, firstly we find z- score for each gender.
Solve:
Z-score=(x-μ)/σ
where:
μ=mean
σ=std deviation
Z(men) = (75-72.5)/3.2
= 0.781
Z(women) = (70-64.5)/3.8
= 1.447
From the above, the z-score for the woman is relatively larger than that of the man, this means that the woman is taller.
We can use z scores to see which of the given values are relatively higher(taller) relative to their groups.
The 70-inch woman is relatively taller in women's group than 75-inch man in men's group (specified group).
If we've got a normal distribution, then we can convert it to standard normal distribution and its values will give us the z score.
If we have [tex]X \sim N(\mu, \sigma)[/tex] then it can be converted to standard normal distribution as
[tex]Z = \dfrac{X - \mu}{\sigma}, \\\\Z \sim N(0,1)[/tex]
Let X tracks height of men and Y tracks height of women, then we have:
[tex]X \sim N(72.5, 3.2)\\Y \sim N(64.5, 3.8)[/tex]
Converting them to standard deviation and getting the z score:
For 75 inch man:
[tex]Z = \dfrac{X - 72.5}{3.2} = \dfrac{75 - 72.5}{3.2} = \dfrac{2.5}{3.2} = 0.78125[/tex]
For 70 inch woman:
[tex]Z = \dfrac{Y - 64.5}{3.8} = \dfrac{70 - 64.5}{3.8} = \dfrac{5.2}{3.8} = 1.368[/tex]
Thus,
The 70-inch woman is relatively taller in women's group than 75-inch man in men's group (specified group).
Learn more about standard normal distribution here:
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