Question

the heights of adult men in america are normally distributed, with a mean of 69.8 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 64.5 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=
c) who is relatively taller?
○ the 6 foot 3 inch american man
○ the 5 foot 11 inch american woman

Explanation:

Step1: Convert height to inches

1 foot = 12 inches.
For a man: 6 feet 3 inches = \(6\times12 + 3=75\) inches.
For a woman: 5 feet 11 inches = \(5\times12+11 = 71\) inches.

Step2: Calculate z - score for man

The formula for z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
For men: \(\mu = 69.8\), \(\sigma=2.65\), \(x = 75\)
\(z=\frac{75 - 69.8}{2.65}=\frac{5.2}{2.65}\approx1.96\)

Step3: Calculate z - score for woman

For women: \(\mu = 64.5\), \(\sigma = 2.51\), \(x = 71\)
\(z=\frac{71-64.5}{2.51}=\frac{6.5}{2.51}\approx2.59\)

Step4: Compare z - scores

Since \(2.59>1.96\)

Answer:

a) \(z\approx1.96\)
b) \(z\approx2.59\)
c) The 5 foot 11 inch American woman