The mean height of adult men is about 175 centimeters (cm) with wandard deviation 7 cm. The mean height of adult women is about 160 cm with standard deviation 6 cm.
If a man and a woman are both 166.5 cm tall, which person’s height is more tare with respect to other people of the same sex?
The correct answer and explanation is :
To determine whose height is more rare relative to their gender group, we use z-scores, which tell us how many standard deviations an individual value is from the mean.
Step 1: Use the Z-score formula
[
z = \frac{X – \mu}{\sigma}
]
Where:
- ( X ) is the individual’s height,
- ( \mu ) is the mean height for the group,
- ( \sigma ) is the standard deviation.
Step 2: Calculate Z-score for the man
- ( X = 166.5 \, \text{cm} )
- ( \mu = 175 \, \text{cm} )
- ( \sigma = 7 \, \text{cm} )
[
z_{\text{man}} = \frac{166.5 – 175}{7} = \frac{-8.5}{7} \approx -1.21
]
Step 3: Calculate Z-score for the woman
- ( X = 166.5 \, \text{cm} )
- ( \mu = 160 \, \text{cm} )
- ( \sigma = 6 \, \text{cm} )
[
z_{\text{woman}} = \frac{166.5 – 160}{6} = \frac{6.5}{6} \approx 1.08
]
Step 4: Compare the Z-scores
- The man has a Z-score of -1.21
- The woman has a Z-score of +1.08
The absolute value of the man’s Z-score is 1.21, and for the woman, it’s 1.08. A larger absolute Z-score means a value is further from the mean and hence more unusual or rare within the population.
Answer:
The man’s height is more rare with respect to other people of the same sex.
Explanation:
Z-scores standardize values across different groups, allowing fair comparison. Here, although both the man and the woman are 166.5 cm tall, their height’s rarity is measured by how far each is from their group’s average height relative to the group’s variability (standard deviation). The man is shorter than the average man, and the woman is taller than the average woman, but the man’s deviation is slightly greater. Therefore, his height is more unusual among men than the woman’s height is among women.