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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $64$ and standard deviation $13$ , the bottom $40\%$ 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

64

and standard deviation

13

, the bottom

40\%

of the values are those less than ___.

Find Z-Score: We need to find the z-score that corresponds to the bottom $40\%$ of a normal distribution.
Use Z-Table: Using a z-table, we find that the z-score for the bottom $40\%$ is approximately $-0.25$ .
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.
Plug in Values: Plugging in the values we get: $-0.25 = \frac{(X - 64)}{13}$ .
Isolate X: Multiplying both sides by $13$ to isolate $X$ , we get: $-0.25 \times 13 = X - 64$ .
Calculate Left Side: Calculating the left side: $-0.25 \times 13 = -3.25$ .
Solve for X: Adding $64$ to both sides to solve for $X$ , we get: $X = 64 - 3.25$ .
Calculate Final Value: Calculating the final value: $X = 60.75$ .

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