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Chapter 6: Problem 56

According to the National Health Center, the heights of 5 -year-old boys are Normally distributed with a mean of 43 inches and a standard deviation of \(1.5\) inches. a. In which percentile is a 5 -year-old boy who is \(46.5\) inches tall? b. If a 5 -year-old boy who is \(46.5\) inches tall grows up to be a man at the same percentile of height, what height will he be? Assume adult men's heights (inches) are distributed as \(N(69,3)\).

Short Answer

Expert verified
a. The 5-year-old boy who is 46.5 inches tall is in the 99th percentile. b. If he grows at the same percentile, he would be 76 inches tall as an adult man.

Step by step solution

01

Finding the Z-score for the boy's height

The Z-score is a measurement of how many standard deviations an element is from the mean. It can be calculated by the formula Z = (X - 碌) / 蟽, where X is the value, 碌 is the mean and 蟽 is the standard deviation. For the boy who is 46.5 inches tall, the Z-score is (46.5 - 43) / 1.5 = 2.33.
02

Calculating the corresponding percentile for the Z-score

We can find the percentile that corresponds to a Z-score of 2.33 using a standard Normal distribution table or using a percentile calculator. The table gives the area to the left of the Z-score under the standard normal curve, which represents the percentile. In this case, it is about 0.99 or 99%.
03

Predicting the boy's future height

The future height of this 5-year-old boy, assuming he will grow to become a man at the same percentile of height, can be found by using the information about the distribution of adult men's heights: N(69,3). Here, we reverse the Z-score calculation to find the height corresponding to Z=2.33. The formula X = Z蟽 + 碌 gives the future height as 2.33*3 + 69 = 76 inches.

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