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In a survey of women in a certain country (ages 20–29), the mean height was 66.9 inches with a standard deviation of 2.86 inches. Answer the following questions about the specified normal distribution.

(a) What height represents the 95th percentile?

(b) What height represents the first quartile?

Answer :

Final answer:

The height representing the 95th percentile for women ages 20-29 is approximately 71.6 inches. The height representing the first quartile (25th percentile) is approximately 65.1 inches, calculated using a standard normal distribution and the given mean and standard deviation.

Explanation:

To determine the height representing the 95th percentile in a normal distribution, we need to use the standard normal distribution (Z-table). The mean height is given as 66.9 inches with a standard deviation of 2.86 inches. We can use the Z-score formula for the 95th percentile:

Z = (X - \\mu\) / \\sigma\

Looking up the Z-score that corresponds to the 95th percentile in a Z-table gives us a Z-score of approximately 1.645. We can then solve for X:

1.645 = (X - 66.9) / 2.86

X = 1.645 * 2.86 + 66.9

X ≈ 71.6 inches

So, the height that represents the 95th percentile is approximately 71.6 inches.

To find the height representing the first quartile or the 25th percentile, we find the Z-score for 25%, which is approximately -0.675:

-0.675 = (X - 66.9) / 2.86

X = -0.675 * 2.86 + 66.9

X ≈ 65.1 inches

Thus, the height that represents the first quartile is approximately 65.1 inches.

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