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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $36$ and standard deviation $16$ , 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

36

and standard deviation

16

, the bottom

20\%

of the values are those less than ___.

Find z-score for $20$ %: To find the value that corresponds to the bottom $20\%$ of the population, we need to use the z-score table to find the z-score that corresponds to a cumulative probability of $0.20$ .
Lookup z-score in table: Looking up the z-score for a cumulative probability of $0.20$ in the z-score table, we find that the z-score is approximately $-0.84$ .
Use z-score formula: Now we use the z-score formula to find the value: $X = \mu + z\sigma$ , where $X$ is the value we are looking for, $\mu$ is the mean, $z$ is the z-score, and $\sigma$ is the standard deviation.
Plug in values: Plug in the values: $X = 36 + (-0.84)(16)$ .
Calculate value: Calculate the value: $X = 36 - 13.44$ .
Round to nearest thousandth: $X = 22.56$ , but we need to round to the nearest thousandth.
Round to nearest thousandth: $X = 22.56$ , but we need to round to the nearest thousandth.Rounded to the nearest thousandth, the value is $22.560$ .

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