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Complete the statement. Round your answer to the nearest thousandth.\newlineIn a population that is normally distributed with mean 5454 and standard deviation 1717, 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 5454 and standard deviation 1717, the bottom 20%20\% of the values are those less than ___.
  1. Find z-score: We need to find the z-score that corresponds to the bottom 20%20\% of a normal distribution.
  2. Use z-table: Looking up the z-score for the bottom 20%20\% in a z-table, we find that it is approximately −0.84-0.84.
  3. Apply z-score formula: Now we use the z-score formula: z=X−meanstandard deviationz = \frac{X - \text{mean}}{\text{standard deviation}}. We need to solve for XX, which represents the value we're looking for.
  4. Solve for X: Plugging in the values we get: −0.84=X−5417.-0.84 = \frac{X - 54}{17}.
  5. Calculate left side: Multiplying both sides by 1717 to solve for XX gives us: −0.84×17=X−54-0.84 \times 17 = X - 54.
  6. Isolate X: Calculating the left side: −0.84×17=−14.28-0.84 \times 17 = -14.28.
  7. Calculate right side: Adding 5454 to both sides to isolate XX gives us: −14.28+54=X-14.28 + 54 = X.
  8. Round answer: Calculating the right side gives us: 39.72=X39.72 = X.
  9. Round answer: Calculating the right side gives us: 39.72=X39.72 = 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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