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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $50$ and standard deviation $20$ , the bottom $20\%$ 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

50

and standard deviation

20

, the bottom

20\%

of the values are those less than ___.

Find z-score for $20$ %: We need to find the z-score that corresponds to the bottom $20\%$ of a normal distribution.
Use z-table: Using a z-table, we find that the z-score for the bottom $20\%$ is approximately $-0.842$ .
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.
Plug in values: Plugging in the values we get: $-0.842 = \frac{X - 50}{20}$ .
Isolate X: Multiplying both sides by $20$ to isolate $X$ , we get: $-0.842 \times 20 = X - 50$ .
Do multiplication: Now we do the multiplication: $-16.84 = X - 50$ .
Solve for X: Adding $50$ to both sides to solve for $X$ , we get: $X = 50 - 16.84$ .
Calculate final value: Finally, we calculate $X$ : $X = 33.16$ .

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\newline

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