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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $98$ and standard deviation $13$ , the bottom $90\%$ 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

98

and standard deviation

13

, the bottom

90\%

of the values are those less than ___.

Find z-score: We need to find the z-score that corresponds to the bottom $90\%$ of a normal distribution.
Use z-table: Using a z-table, we find that the z-score for the bottom $90\%$ is approximately $1.28$ .
Apply z-score formula: Now we use the z-score formula: $z = \frac{X - \text{mean}}{\text{standard deviation}}$ . We need to solve for $X$ , which represents the value we're looking for.
Rearrange formula: Rearrange the formula to solve for $X$ : $X = z \times \text{standard deviation} + \text{mean}$ .
Plug in values: Plug in the values: $X = 1.28 \times 13 + 98$ .
Calculate X: Calculate the value of X: $X = 16.64 + 98$ .
Correct z-score: $X = 114.64$ , but we made a mistake, the z-score for the bottom $90\%$ is not $1.28$ , it's actually $-1.28$ because we are looking for the value below the mean, not above it.

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