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Monte

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6. Normal curve

Mean and SD, then any value. More than, less than, between, the top or bottom cutoff, several in a row, and averages of n.

  1. z for x = 5401(x − μ) / σ = (540 − 500) / 40. How many SDs from the mean.
  2. P(X < 540)0.8413Φ(1)
  3. P(X > 540)0.15871 − Φ(1)
  4. Top 5% starts at565.7941μ + zσ with z = 1.64: 500 + 1.64 × 40. Work backwards: area first, then z, then x.
  5. Bottom 5% ends at434.2059500 − 1.64 × 40
  6. All 3 below 5400.59560.8413^3. Independent, so multiply.
  7. All 3 above 5400.0040.1587^3
  8. At least one of 3 above 5400.40441 − 0.8413^3
  9. Standard error of the average10σ / √n = 40 / √16. Averages wobble less than single values.
  10. z for an average of 5404(540 − 500) / 10
  11. P(average of 16 < 540)1Φ(4)
  12. P(average of 16 > 540)< 0.00011 − Φ(4)
  13. Empirical rule check68% within 460 to 540; 95% within 420 to 580; 99.7% within 380 to 620Plus or minus 1, 2, and 3 SDs.