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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $41$ and standard deviation $4$ , 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

41

and standard deviation

4

, the bottom

20\%

of the values are those less than ___.

Find z-score for bottom $20\%$ : First, we need to find the z-score that corresponds to the bottom $20\%$ of a normal distribution.
Use z-score formula: Using a z-table, we find that the z-score for the bottom $20\%$ is approximately $-0.842$ .
Rearrange formula for $X$ : Now, we use the z-score formula: $z = \frac{X - \text{mean}}{\text{standard deviation}}$ , where $X$ is the value we're looking for.
Plug in values: We rearrange the formula to solve for $X$ : $X = z \times \text{standard deviation} + \text{mean}$ .
Calculate X value: Plug in the values: $X = -0.842 \times 4 + 41$ .
Calculate X value: Plug in the values: $X = -0.842 \times 4 + 41$ .Calculate the value of X: $X = -3.368 + 41$ .
Calculate X value: Plug in the values: $X = -0.842 \times 4 + 41$ .Calculate the value of X: $X = -3.368 + 41$ . $X = 37.632$ , which we round to the nearest thousandth as $37.632$ .

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