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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $43$ and standard deviation $5$ , the bottom $80\%$ 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

43

and standard deviation

5

, the bottom

80\%

of the values are those less than ___.

Find z-score for $80$ %: We need to find the z-score that corresponds to the bottom $80\%$ of a normal distribution.
Use z-table to find z-score: Using a z-table, we find that the z-score for $80\%$ is approximately $0.84$ .
Calculate $X$ using z-score formula: Now we use the z-score formula: $z = \frac{X - \text{mean}}{\text{standard deviation}}$ . We rearrange it to solve for $X$ : $X = z \times \text{standard deviation} + \text{mean}$ .
Plug in values and solve: Plug in the values: $X = 0.84 \times 5 + 43$ .
Calculate final value: Calculate the value: $X = 4.2 + 43$ .
Round to nearest thousandth: $X = 47.2$ .
Round to nearest thousandth: $X = 47.2$ .Round the answer to the nearest thousandth: The value is $47.200$ .

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\newline

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