According to the National Health Center, the heights of 5-year-old girls are normally distributed with a mean of 40 inches and a standard deviation of 2.5 inches.

(a) In which percentile is a 5-year-old girl who is 46.5 inches tall?

(b) If a 5-year-old girl who is 46.5 inches tall grows up to be a woman at the same percentile of height, what height will she be? Assume women are distributed as [tex]N(64, 3)[/tex].

A 5-year-old girl who is 46.5 inches tall is in the 99.5th percentile for height. If she grows up to be a woman at the same percentile of height, she would be approximately 73.8 inches tall.

Explanation:

To solve part (a) of the question, we need to calculate the Z-score for a 5-year-old girl who is 46.5 inches tall. The Z-score is a measure of how many standard deviations an element is from the mean. We can calculate it using the formula Z = (X - μ) / σ, where X is the value in question, μ is the mean, and σ is the standard deviation.

So, in this case, Z = (46.5 - 40) / 2.5 = 2.6. Looking at a standard Normal Distribution table or using a Z-score calculator, we find that a Z-score of 2.6 corresponds approximately to the 99.5th percentile. So, a 5-year-old girl who is 46.5 inches tall is in the 99.5th percentile for height.

For part (b) of the question, we need to calculate the height that corresponds to the 99.5th percentile in the Normal Distribution of women's heights, N(64,3). We know that the percentile corresponds to a specific Z-score, which we have already calculated as 2.6. So, we solve for X in the Z-score formula, where μ is the mean height for women, and σ is the standard deviation for women's heights.

So, X = Zσ + μ = 2.6 × 3 + 64 = 73.8 inches.

In conclusion, if a 5-year-old girl who is 46.5 inches tall grows up to be a woman at the same percentile of height, she would be approximately 73.8 inches tall.

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