According to the National Health Center, the heights of 6-year-old girls are normally distributed with a mean of 45 inches and a standard deviation of 2 inches.

a) In which percentile is a 6-year-old girl who is 46.5 inches tall?

b) If a 6-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's heights are distributed as [tex]N(64, 2.5)[/tex].

A 6-year-old girl who is 46.5 inches tall is in the 77th percentile of height for her age. Assuming she remains in the same percentile, she will grow up to be about 66.875 inches tall.

Explanation:

A) To determine which percentile a 6-year-old girl who is 46.5 inches tall falls into, we first need to calculate her z-score (which measures how many standard deviations an element is from the mean). The formula for the z score is: Z = (X - μ) / σ. With X = 46.5, μ = 45, and σ = 2, the z score is 0.75, which corresponds to roughly the 77th percentile. B) To calculate her adult height given that she remains in the same percentile, we need to use the reverse of the Z score formula: X = Zσ + μ (where the new σ = 2.5 and the new μ = 64). For Z = the equivalent to the 77th percentile, which is 0.75, the woman's adult height will be about 66.875 inches.

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