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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $54$ and standard deviation $17$ , 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

54

and standard deviation

17

, the bottom

20\%

of the values are those less than ___.

Find z-score: We need to find the z-score that corresponds to the bottom $20\%$ of a normal distribution.
Use z-table: Looking up the z-score for the bottom $20\%$ in a z-table, we find that it is approximately $-0.84$ .
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.
Solve for X: Plugging in the values we get: $-0.84 = \frac{X - 54}{17}.$
Calculate left side: Multiplying both sides by $17$ to solve for $X$ gives us: $-0.84 \times 17 = X - 54$ .
Isolate X: Calculating the left side: $-0.84 \times 17 = -14.28$ .
Calculate right side: Adding $54$ to both sides to isolate $X$ gives us: $-14.28 + 54 = X$ .
Round answer: Calculating the right side gives us: $39.72 = X$ .
Round answer: Calculating the right side gives us: $39.72 = X$ .We round our answer 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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