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The heights of twelve month old boys are normally distributed with a mean of 29.8 inches and a standard deviation of 1.2 inches. About twenty-one percent of twelve month old boys are shorter than what height? Report your answer to the nearest tenths place.

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Answer:

About twenty-one percent of twelve month old boys are shorter than 28.8 inches.

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

\mu = 29.8, \sigma = 1.2

About twenty-one percent of twelve month old boys are shorter than what height?

This is the value of X when Z has a pvalue of 0.21. So it is X when Z = -0.805.

Z = \frac{X - \mu}{\sigma}

-0.805 = \frac{X - 29.8}{1.2}

X - 29.8 = -0.805*1.2

X = 28.8

About twenty-one percent of twelve month old boys are shorter than 28.8 inches.

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Holli
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