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Complete the statement. Round your answer to the nearest thousandth.\newlineIn a population that is normally distributed with mean 4747 and standard deviation 2525, the bottom 20%20\% of the values are those less than ___.

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Full solution

Q. Complete the statement. Round your answer to the nearest thousandth.\newlineIn a population that is normally distributed with mean 4747 and standard deviation 2525, the bottom 20%20\% of the values are those less than ___.
  1. Find z-score for 2020\%: Find the z-score for 2020\%. Using a z-table or calculator, the z-score for 2020\% is approximately −0.842-0.842.
  2. Use z-score formula: Use the z-score formula to solve for XX.
    mean (μ\mu): 4747
    standard deviation (σ\sigma): 2525
    z-score (ZZ): −0.842-0.842
    Z=X−μσZ = \frac{X - \mu}{\sigma}
    −0.842=X−4725-0.842 = \frac{X - 47}{25}
  3. Multiply by standard deviation: Multiply both sides by the standard deviation to isolate (X−μ)(X - \mu).\newline−0.842×25=X−47-0.842 \times 25 = X - 47\newline−21.05=X−47-21.05 = X - 47
  4. Add mean to solve for X: Add the mean to both sides to solve for X.\newlineX=−21.05+47X = -21.05 + 47\newlineX=25.95X = 25.95

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