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Answer :
Final answer:
Using the formula Z = (X - µ) / σ, the z-score for a woman who is 68 inches tall is 1.33, indicating she is 1.33 standard deviations above the average women's height. For a man who is 68 inches tall, the z-score is -0.41, indicating he is 0.41 standard deviations below the average men's height.
Explanation:
The subject of the question involves calculating the z-scores for heights of men and women, in which their heights are normally distributed with known means and standard deviations.
A z-score measures how many standard deviations an element is from the mean. You can calculate the z-score using the formula:
Z = (X - µ) / σ
where: Z is the z-score, X is the actual value, µ is the mean, and σ is the standard deviation.
Given that women's heights are normally distributed with a mean (µ) of 63.6 inches, and a standard deviation (σ) of 3.3 inches, applying the formula, a randomly selected woman who is 68 inches tall has a z-score:
Z = (68 - 63.6) / 3.3 ≈ 1.33 (rounded to two decimal places)
For the men's category, their heights are normally distributed with a mean (µ) of 69.2 inches, and a standard deviation (σ) of 2.9 inches, hence a randomly selected man who is 68 inches tall will have a z-score:
Z = (68 - 69.2) / 2.9 ≈ -0.41 (rounded to two decimal places)
The z-scores essentially show how far, in terms of standard deviations, a given height (for a man or a woman) is from the mean height for their respective gender.
Learn more about Z-scores & Normal Distribution here:
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