The heights of adult men in America are normally distributed, with a mean of 69.1 inches and a standard
deviation of 2.68 inches. The heights of adult women in America are also normally distributed, but with a
mean of 64.8 inches and a standard deviation of 2.55 inches.
a) If a man is 6 feet 3 inches tall, what is his z-score (to two decimal places)?

b) If a woman is 5 feet 11 inches tall, what is her -score (to two decimal places)?

c) Who is relatively taller?
The 5 foot 11 inch American woman
The 6 foot 3 inch American man

Question

The heights of adult men in America are normally distributed, with a mean of 69.1 inches and a standard
deviation of 2.68 inches. The heights of adult women in America are also normally distributed, but with a
mean of 64.8 inches and a standard deviation of 2.55 inches.
a) If a man is 6 feet 3 inches tall, what is his z-score (to two decimal places)?

b) If a woman is 5 feet 11 inches tall, what is her

-score (to two decimal places)?

c) Who is relatively taller?
The 5 foot 11 inch American woman
The 6 foot 3 inch American man

UpStudy AI · Accepted Answer

To find the z-score for a given height, we can use the formula:

$$ z = \frac{X - \mu}{\sigma} $$

where:
- $$ X $$ is the given height,
- $$ \mu $$ is the mean height,
- $$ \sigma $$ is the standard deviation.

Let's calculate the z-score for the man and the woman:

a) For the man:
Given height: 6 feet 3 inches = 75 inches
Mean height for men: 69.1 inches
Standard deviation for men: 2.68 inches

$$ z_{\text{man}} = \frac{75 - 69.1}{2.68} $$

b) For the woman:
Given height: 5 feet 11 inches = 71 inches
Mean height for women: 64.8 inches
Standard deviation for women: 2.55 inches

$$ z_{\text{woman}} = \frac{71 - 64.8}{2.55} $$

Now, let's calculate the z-scores for the man and the woman.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(75-69.1\right)}{2.68}$$
- step1: Remove the parentheses:
  $$\frac{75-69.1}{2.68}$$
- step2: Subtract the numbers:
  $$\frac{5.9}{2.68}$$
- step3: Convert the expressions:
  $$\frac{\frac{59}{10}}{\frac{67}{25}}$$
- step4: Multiply by the reciprocal:
  $$\frac{59}{10}\times \frac{25}{67}$$
- step5: Reduce the numbers:
  $$\frac{59}{2}\times \frac{5}{67}$$
- step6: Multiply the fractions:
  $$\frac{59\times 5}{2\times 67}$$
- step7: Multiply:
  $$\frac{295}{134}$$
Calculate or simplify the expression $$ (71 - 64.8) / 2.55 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(71-64.8\right)}{2.55}$$
- step1: Remove the parentheses:
  $$\frac{71-64.8}{2.55}$$
- step2: Subtract the numbers:
  $$\frac{6.2}{2.55}$$
- step3: Convert the expressions:
  $$\frac{\frac{31}{5}}{\frac{51}{20}}$$
- step4: Multiply by the reciprocal:
  $$\frac{31}{5}\times \frac{20}{51}$$
- step5: Reduce the numbers:
  $$31\times \frac{4}{51}$$
- step6: Multiply:
  $$\frac{31\times 4}{51}$$
- step7: Multiply:
  $$\frac{124}{51}$$
a) The z-score for the man is approximately 2.20.

b) The z-score for the woman is approximately 2.43.

c) The woman is relatively taller than the man.
a) The man's z-score is 2.20. b) The woman's z-score is 2.43. c) The 5 foot 11 inch American woman is relatively taller than the 6 foot 3 inch American man.

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