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Complete the statement. Round your answer to the nearest thousandth. $\newline$ In a population that is normally distributed with mean $43$ and standard deviation $3$ , the bottom $30\%$ 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

43

and standard deviation

3

, the bottom

30\%

of the values are those less than ______.

Determine z-score for bottom $30$ %: Determine the z-score that corresponds to the bottom $30$ % of a normal distribution. $\newline$ To find the z-score for the bottom $30$ %, we can use a z-table or a statistical calculator. The z-score tells us how many standard deviations away from the mean a particular value is. For the bottom $30$ %, we look for the z-score that corresponds to a cumulative probability of $0.30$ .
Look up z-score for $30$ th percentile: Look up the z-score for the $30$ th percentile. $\newline$ Using a z-table or statistical software, we find that the z-score corresponding to the $30$ th percentile is approximately $-0.5244$ . This means that the value we are looking for is $-0.5244$ standard deviations below the mean.
Use z-score formula to solve for X: Use the z-score formula to solve for the unknown value (X). $\newline$ The z-score formula is $Z = \frac{X - \mu}{\sigma}$ , where $Z$ is the z-score, $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. We can rearrange this formula to solve for $X$ : $X = Z \cdot \sigma + \mu$ .
Calculate X using formula: Plug the values into the formula to calculate X. $\newline$ Using the z-score of $-0.5244$ , the mean ( $\mu$ ) of $43$ , and the standard deviation ( $\sigma$ ) of $3$ , we get: $\newline$ $X = (-0.5244) \times 3 + 43$ $\newline$ $X = -1.5732 + 43$ $\newline$ $X = 41.4268$
Round answer to nearest thousandth: Round the answer to the nearest thousandth. $\newline$ Rounding $41.4268$ to the nearest thousandth gives us $41.427$ .

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