One of the tallest living men has a height of 263 cm. One of the tallest living women is 245 cm tall. Heights of men have a mean of 177 cm and a standard deviation of 8 cm. Heights of women have a mean of 163 cm and a standard deviation of 4 cm. Relative to the population of the same gender, who is taller? Explain.
women are taller.because The smaller the standard deviation, the more narrow the range between the lowest and highest scores.
In order to determine who is taller relative to the population of the same gender, we can convert the individual heights into standardized z-scores and compare them. The z-score represents the number of standard deviations an individual's height is from the population mean.
First, let's calculate the z-score for the tallest living man with a height of 263 cm:
z_male = (x_male - mean_male) / standard_deviation_male
= (263 cm - 177 cm) / 8 cm
z_male = 86 cm / 8 cm
= 10.75
Next, let's calculate the z-score for the tallest living woman with a height of 245 cm:
z_female = (x_female - mean_female) / standard_deviation_female
= (245 cm - 163 cm) / 4 cm
z_female = 82 cm / 4 cm
= 20.5
Since the z-score for the tallest living woman is higher than the z-score for the tallest living man, we can conclude that relative to the population of the same gender, the tallest living woman is taller. The z-score of 20.5 indicates that the height of the tallest living woman is approximately 20.5 standard deviations above the mean height of women, while the z-score of 10.75 for the tallest living man indicates that his height is approximately 10.75 standard deviations above the mean height of men.
To determine who is taller relative to their respective populations, we need to compare the heights of the individuals in terms of their standard deviations from the mean.
Let's start with the tallest living man who has a height of 263 cm. We know that the mean height for men is 177 cm, with a standard deviation of 8 cm. To measure how many standard deviations the tallest living man is from the mean, we can calculate the Z-score.
Z-score formula: Z = (X - μ) / σ
Here, X represents the individual's height, μ represents the mean height, and σ represents the standard deviation.
For the tallest living man:
Z = (263 - 177) / 8
Z ≈ 10.75
The Z-score of approximately 10.75 indicates that the tallest living man is roughly 10.75 standard deviations above the mean height for men in the population.
Now let's consider the tallest living woman who has a height of 245 cm. For women, the mean height is 163 cm with a standard deviation of 4 cm.
For the tallest living woman:
Z = (245 - 163) / 4
Z ≈ 20.5
The Z-score of approximately 20.5 indicates that the tallest living woman is about 20.5 standard deviations above the mean height for women in the population.
Comparing the Z-scores, we can see that the tallest living woman (Z ≈ 20.5) is significantly farther above the mean height for women than the tallest living man (Z ≈ 10.75) is above the mean height for men. Therefore, relative to their respective populations, the tallest living woman is taller.