Complete the statement. Round your answer to the nearest thousandth.In a population that is normally distributed with mean 43 and standard deviation 3, the bottom 30% of the values are those less than ______.
Q. Complete the statement. Round your answer to the nearest thousandth.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.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 30th percentile: Look up the z-score for the 30th percentile.Using a z-table or statistical software, we find that the z-score corresponding to the 30th 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).The z-score formula is Z=σX−μ, where Z is the z-score, X is the value, μ is the mean, and σ is the standard deviation. We can rearrange this formula to solve for X: X=Z⋅σ+μ.
Calculate X using formula: Plug the values into the formula to calculate X.Using the z-score of −0.5244, the mean (μ) of 43, and the standard deviation (σ) of 3, we get:X=(−0.5244)×3+43X=−1.5732+43X=41.4268
Round answer to nearest thousandth: Round the answer to the nearest thousandth.Rounding 41.4268 to the nearest thousandth gives us 41.427.
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