Question

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 64.6 inches and a standard deviation of 2.53 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 5 foot 11 inch american woman
the 6 foot 3 inch american man
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Explanation:

Step1: Convert heights to inches

1 foot = 12 inches. A man 6 feet 3 inches tall is \(6\times12 + 3=75\) inches. A woman 5 feet 11 inches tall is \(5\times12+11 = 71\) inches.

Step2: Calculate man's z - score

The z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(x\) is the value, \(\mu\) is the mean and \(\sigma\) is the standard deviation. For men, \(\mu = 69.1\) and \(\sigma=2.65\). So \(z_{man}=\frac{75 - 69.1}{2.65}=\frac{5.9}{2.65}\approx2.22\).

Step3: Calculate woman's z - score

For women, \(\mu = 64.6\) and \(\sigma = 2.53\). So \(z_{woman}=\frac{71-64.6}{2.53}=\frac{6.4}{2.53}\approx2.53\).

Step4: Compare z - scores

Since \(2.53>2.22\), the woman is relatively taller.

Answer:

a) \(z = 2.22\)
b) \(z = 2.53\)
c) The 5 foot 11 inch American woman