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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $34$ and standard deviation $6$ , the bottom $70\%$ 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

34

and standard deviation

6

, the bottom

70\%

of the values are those less than ___.

Find z-score for $70$ \%: Find the z-score for $70\%$ . $\newline$ Using a z-table or calculator, the z-score for $70\%$ is approximately $0.524$ .
Use z-score formula: Use the z-score formula to solve for $X$ .
mean ( $\mu$ ): $34$
standard deviation ( $\sigma$ ): $6$
z-score ( $Z$ ): $0.524$
$Z = \frac{X - \mu}{\sigma}$
$0.524 = \frac{X - 34}{6}$
Multiply to isolate $X$ : Multiply both sides by the standard deviation to isolate $(X - \mu)$ . $0.524 \times 6 = X - 34$ $3.144 = X - 34$
Add mean to solve: Add the mean to both sides to solve for $X$ . $X = 3.144 + 34$ $X = 37.144$
Round the answer: Round the answer to the nearest thousandth. $X \approx 37.144$

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