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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $66$ and standard deviation $20$ , the bottom $40\%$ 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

66

and standard deviation

20

, the bottom

40\%

of the values are those less than ___.

Find z-score for $40$ \%: First, we need to find the z-score that corresponds to the bottom $40\%$ of a normal distribution.
Calculate z-score: Using a z-table, we find that the z-score for the bottom $40\%$ is approximately $-0.25$ .
Use z-score formula: Now, we use the z-score formula: $z = \frac{X - \mu}{\sigma}$ , where $X$ is the value we're looking for, $\mu$ is the mean, and $\sigma$ is the standard deviation.
Solve for X: We rearrange the formula to solve for X: $X = z \cdot \sigma + \mu$ .
Plug in values: Plug in the values: $X = -0.25 \times 20 + 66$ .
Calculate X: Calculate the value: $X = -5 + 66$ .
Final result: $X = 61$ .

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