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Complete the statement. Round your answer to the nearest thousandth.\newlineIn a population that is normally distributed with mean 4141 and standard deviation 44, the bottom 20%20\% of the values are those less than ___.

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Q. Complete the statement. Round your answer to the nearest thousandth.\newlineIn a population that is normally distributed with mean 4141 and standard deviation 44, the bottom 20%20\% of the values are those less than ___.
  1. Find z-score for bottom 20%20\%: First, we need to find the z-score that corresponds to the bottom 20%20\% of a normal distribution.
  2. Use z-score formula: Using a z-table, we find that the z-score for the bottom 20%20\% is approximately −0.842-0.842.
  3. Rearrange formula for XX: Now, we use the z-score formula: z=X−meanstandard deviationz = \frac{X - \text{mean}}{\text{standard deviation}}, where XX is the value we're looking for.
  4. Plug in values: We rearrange the formula to solve for XX: X=z×standard deviation+meanX = z \times \text{standard deviation} + \text{mean}.
  5. Calculate X value: Plug in the values: X=−0.842×4+41X = -0.842 \times 4 + 41.
  6. Calculate X value: Plug in the values: X=−0.842×4+41X = -0.842 \times 4 + 41.Calculate the value of X: X=−3.368+41X = -3.368 + 41.
  7. Calculate X value: Plug in the values: X=−0.842×4+41X = -0.842 \times 4 + 41.Calculate the value of X: X=−3.368+41X = -3.368 + 41. X=37.632X = 37.632, which we round to the nearest thousandth as 37.63237.632.

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