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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $47$ and standard deviation $13$ , the bottom $90\%$ of the values are those less than ___.

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Find values of normal variables

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Q. Complete the statement. Round your answer to the nearest thousandth.

\newline

In a population that is normally distributed with mean

47

and standard deviation

13

, the bottom

90\%

of the values are those less than ___.

Find z-score for $90$ %: Find the z-score for $90$ \%. Using a z-table or calculator, the z-score for $90$ \% is approximately $1.282$ .
Use z-score formula: Use the z-score formula to solve for $X$ .
mean ( $\mu$ ): $47$
standard deviation ( $\sigma$ ): $13$
z-score ( $Z$ ): $1.282$
$Z = \frac{X - \mu}{\sigma}$
$1.282 = \frac{X - 47}{13}$
Multiply by standard deviation: Multiply both sides by the standard deviation. $\newline$ $1.282 \times 13 = X - 47$ $\newline$ $16.666 = X - 47$
Add mean to solve for X: Add the mean to both sides to solve for X. $\newline$ $X = 16.666 + 47$ $\newline$ $X = 63.666$

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