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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $87$ and standard deviation $11$ , the bottom $70\%$ 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

87

and standard deviation

11

, the bottom

70\%

of the values are those less than ___.

Find Z-score for $70$ %: To find the value that separates the bottom $70\%$ from the rest, we need to use the z-score table to find the z-score that corresponds to a cumulative probability of $0.70$ .
Lookup Z-score in table: Looking up $0.70$ in the z-score table, we find that the closest z-score is approximately $0.52$ .
Calculate X using formula: Now, we use the z-score formula to find the value $X$ : $Z = \frac{X - \text{mean}}{\text{standard deviation}}$ . Rearranging the formula to solve for $X$ gives us $X = Z \times \text{standard deviation} + \text{mean}$ .
Plug in values for X: Plugging in the values, we get $X = 0.52 \times 11 + 87$ .
Round to nearest thousandth: Calculating this, $X = 5.72 + 87$ .
Round to nearest thousandth: Calculating this, $X = 5.72 + 87$ .So, $X = 92.72$ . But we need to round to the nearest thousandth, which doesn't change the value in this case since it's already at the thousandth place.

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