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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $19$ and standard deviation $12$ , the bottom $10\%$ 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

19

and standard deviation

12

, the bottom

10\%

of the values are those less than ___.

Find z-score for $10$ %: Find the z-score for $10\%$ . Using a z-table or calculator, the z-score for $10\%$ is approximately $-1.282$ .
Use z-score formula: Use the z-score formula to solve for $X$ . $\mu$ : $19$ $\sigma$ : $12$ $Z$ : $-1.282$ $Z = \frac{X - \mu}{\sigma}$ $-1.282 = \frac{X - 19}{12}$
Multiply to isolate $X$ : Multiply both sides by the standard deviation to isolate $(X - \mu)$ . $\newline$ $-1.282 \times 12 = X - 19$
Perform the multiplication: Perform the multiplication. $\newline$ $-15.384 = X - 19$
Add mean to solve for X: Add the mean to both sides to solve for X. $\newline$ $X = -15.384 + 19$
Calculate X value: Calculate the value of X. $\newline$ $X = 3.616$
Round to nearest thousandth: Round the answer to the nearest thousandth. $\newline$ $X \approx 3.616$ (rounded to three decimal places)

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