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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $48$ and standard deviation $19$ , the bottom $30\%$ 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

48

and standard deviation

19

, the bottom

30\%

of the values are those less than ___.

Find z-score for $30$ %: We need to find the z-score that corresponds to the bottom $30\%$ of the normal distribution.
Use z-table for $-0.52$ : Using a z-table, we find that the z-score for the bottom $30\%$ is approximately $-0.52$ .
Apply z-score formula: Now we use the z-score formula: $z = \frac{X - \text{mean}}{\text{standard deviation}}$ , where $X$ is the value we are looking for.
Rearrange formula for $X$ : We rearrange the formula to solve for $X$ : $X = z \times \text{standard deviation} + \text{mean}$ .
Plug in values: Plug in the values: $X = -0.52 \times 19 + 48$ .
Calculate X: Calculate the value of $X$ : $X = -9.88 + 48$ .
Round to nearest thousandth: $X = 38.12$ , but we need to round to the nearest thousandth.
Final answer: The final answer, rounded to the nearest thousandth, is $38.120$ .

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