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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $63$ and standard deviation $7$ , 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

63

and standard deviation

7

, the bottom

30\%

of the values are those less than ___.

Calculate z-score for $30$ th percentile: To find the value that separates the bottom $30\%$ from the rest, we need to use the z-score formula for the $30$ th percentile.
Find z-score using formula: The z-score for the $30^{\text{th}}$ percentile can be found using a z-table or a calculator with inverse normal function. Let's assume the z-score for the $30^{\text{th}}$ percentile is approximately $-0.52$ .
Use z-score formula: Now we use the z-score formula: $X = \mu + z\sigma$ , where $X$ is the value we're looking for, $\mu$ is the mean, $z$ is the z-score, and $\sigma$ is the standard deviation.
Plug in values: Plug in the values: $X = 63 + (-0.52)(7)$ .
Calculate result: Calculate the value: $X = 63 - 3.64$ .
Round to nearest thousandth: $X = 59.36$ , but we need to round to the nearest thousandth.
Final answer: The final answer, rounded to the nearest thousandth, is $59.360$ .

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