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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $72$ and standard deviation $8$ , the bottom $10\%$ 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

72

and standard deviation

8

, the bottom

10\%

of the values are those less than ___.

Find z-score for bottom $10\%$ : We need to find the z-score that corresponds to the bottom $10\%$ of a normal distribution.
Use z-table to find $-1.28$ : Using a z-table, we find that the z-score for the bottom $10\%$ is approximately $-1.28$ .
Apply z-score formula: Now we use the z-score formula: $z = \frac{X - \text{mean}}{\text{standard deviation}}$ , where $X$ is the value we are looking for.
Solve for X: Rearrange the formula to solve for X: $X = z \times \text{standard deviation} + \text{mean}$ .
Substitute values in formula: Plug in the values: $X = -1.28 \times 8 + 72$ .
Calculate X: Calculate the value of X: $X = -10.24 + 72$ .
Round to nearest thousandth: $X = 61.76$ , but we need to round to the nearest thousandth.
Final answer: The final answer, rounded to the nearest thousandth, is $61.760$ .

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\newline

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