Questions: The heights of adult men in America are normally distributed, with a mean of 69.1 inches and a standard deviation of 2.64 inches. The heights of adult women in America have a mean of 64.2 inches and a standard deviation of 2.57 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

Solution Steps

To find the Z-score for a man who is 6 feet 3 inches tall, we first convert his height to inches: \[ X = 6 \times 12 + 3 = 75 \text{ inches} \] The mean height of adult men is \( \mu = 69.1 \) inches, and the standard deviation is \( \sigma = 2.64 \) inches. The Z-score is calculated using the formula: \[ z = \frac{X - \mu}{\sigma} = \frac{75 - 69.1}{2.64} = \frac{5.9}{2.64} \approx 2.23 \]

Next, we calculate the Z-score for a woman who is 5 feet 11 inches tall. Converting her height to inches gives: \[ X = 5 \times 12 + 11 = 71 \text{ inches} \] The mean height of adult women is \( \mu = 64.2 \) inches, and the standard deviation is \( \sigma = 2.57 \) inches. The Z-score is calculated as follows: \[ z = \frac{X - \mu}{\sigma} = \frac{71 - 64.2}{2.57} = \frac{6.8}{2.57} \approx 2.65 \]

Now we compare the Z-scores calculated:

• Z-score for the man: \( z \approx 2.23 \)
• Z-score for the woman: \( z \approx 2.65 \)

Since \( 2.65 > 2.23 \), the woman is relatively taller compared to her peers than the man is compared to his peers.

Final Answer