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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $78$ and standard deviation $13$ , the bottom $20\%$ of the values are those less than ___.

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Find values of normal variables

Full solution

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

\newline

In a population that is normally distributed with mean

78

and standard deviation

13

, the bottom

20\%

of the values are those less than ___.

Find z-score for $20$ \%: Find the z-score for $20$ \%. Using a z-table or calculator, the z-score for $20$ \% is approximately $-0.842$ .
Use z-score formula: Use the z-score formula to solve for $X$ .
mean ( $\mu$ ): $78$
standard deviation ( $\sigma$ ): $13$
z-score ( $Z$ ): $-0.842$
$Z = \frac{X - \mu}{\sigma}$
$-0.842 = \frac{X - 78}{13}$
Multiply to isolate $X$ : Multiply both sides by the standard deviation to isolate $(X - \mu)$ . $\newline$ $-0.842 \times 13 = X - 78$ $\newline$ $-10.946 = X - 78$
Add mean to solve: Add the mean to both sides to solve for $X$ . $X = -10.946 + 78$ $X = 67.054$
Round the answer: Round the answer to the nearest thousandth. $X \approx 67.054$

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