Questions: Use z scores to compare the given values. The tallest living man at one time had a height of 257 cm. The shortest living man at that time had a height of 118.9 cm. Heights of men at that time had a mean of 172.85 cm and a standard deviation of 7.91 cm. Which of these two men had the height that was more extreme? Since the z score for the tallest man is z= and the z score for the shortest man is z=, the man had the height that was more extreme. (Round to two decimal places.)

Solution Steps

To determine which man had the more extreme height, we need to calculate the $z$ scores for both the tallest and shortest men. The $z$ score is calculated using the formula:

\[ z = \frac{(X - \mu)}{\sigma} \]

where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. We will then compare the absolute values of the $z$ scores to determine which is more extreme.

1. Calculate the $z$ score for the tallest man.
2. Calculate the $z$ score for the shortest man.
3. Compare the absolute values of the $z$ scores to determine which is more extreme.

To find the $z$ score for the tallest man, we use the formula: \[ z = \frac{(X - \mu)}{\sigma} \] where \( X = 257 \) cm, \( \mu = 172.85 \) cm, and \( \sigma = 7.91 \) cm. Substituting the values, we get: \[ z_{\text{tallest}} = \frac{(257 - 172.85)}{7.91} = 10.64 \]

Similarly, to find the $z$ score for the shortest man, we use the same formula: \[ z = \frac{(X - \mu)}{\sigma} \] where \( X = 118.9 \) cm, \( \mu = 172.85 \) cm, and \( \sigma = 7.91 \) cm. Substituting the values, we get: \[ z_{\text{shortest}} = \frac{(118.9 - 172.85)}{7.91} = -6.82 \]

To determine which height is more extreme, we compare the absolute values of the $z$ scores: \[ |z_{\text{tallest}}| = 10.64 \] \[ |z_{\text{shortest}}| = 6.82 \]

Since \( 10.64 > 6.82 \), the height of the tallest man is more extreme.

Final Answer