Finding Probabilities and Values from a Normal Model · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Reading the bell curve both ways

Mathematics · Probability · ages 17-18
Name ______________________   Date ____________

  1. For a continuous normal model, P(X < 12) = 0.2 and P(X < 18) = 0.7. Which expression finds P(12 < X < 18)?

    • 0.7 + 0.2
    • 0.7 - 0.2
    • 1 - 0.7
    • 0.2 / 0.7

  2. X is normal with mean 70 and positive standard deviation 9. What is P(X < 70)?

    Answer: ______________

  3. X is normal with mean 54 and standard deviation 8. A left-cumulative table gives Φ(2) = 0.9772. Find P(X < 70).

    Answer: ______________

  4. A normal model has mean 50 and standard deviation 3. What is the z-score of 59?

    Answer: ______________

  5. A normal score model has mean 100 and standard deviation 15. Using the empirical rule, below which value does approximately the lower 2.5% lie?

    • 85
    • 70
    • 55

  6. In a normal model, the values corresponding to z = -1 and z = 1 are 40 and 52. What is the standard deviation?

    • 12
    • 4
    • 6
    • 24

  7. X is normal with mean 64 and standard deviation 8. A left-cumulative table gives z ≈ -0.675 for left area 0.25. Find the lower-quartile cutoff using that z.

    Answer: ______________

  8. A normal mark model has mean 70 and standard deviation 4. Using the empirical rule, what is the approximate cutoff above which the upper 2.5% lies?

    Answer: ______________

  9. Using the empirical rule, approximately what share of a normal model lies more than two standard deviations above its mean?

    • 5%
    • 25%
    • 2.5%

  10. A normal mark model has mean 60 and standard deviation 8. The upper 10% cutoff has z approximately 1.28. What is the cutoff to the nearest whole mark?

    • about 70
    • about 68
    • about 76
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Reading the bell curve both ways W1-mt_naXMRySvos-s1

  1. 0.7 - 0.2 · The left area below 18 includes the interval and the area below 12. Subtract 0.2 from 0.7 to isolate the interval.
  2. 0.5 · 70 is the model mean, so its z-score is 0. A symmetric normal model has half its area below its mean.
  3. 0.9772 · The boundary has z = (70 - 54) / 8 = 2. Its left probability is Φ(2) = 0.9772.
  4. 3 · z = (59 - 50) / 3 = 9 / 3 = 3.
  5. 70 · The lower 2.5% begins about two standard deviations below μ: 100 - 2 × 15 = 70.
  6. 6 · The raw gap is 52 - 40 = 12 and the z gap is 1 - (-1) = 2. Thus σ = 12 / 2 = 6.
  7. 58.6 · x = 64 + (-0.675) × 8 = 64 - 5.4 = 58.6.
  8. 78 · The upper 2.5% begins about two standard deviations above μ: 70 + 2 × 4 = 78.
  9. 2.5% · About 95% lies within two σ, leaving about 5% outside. Symmetry gives half, about 2.5%, to the upper tail.
  10. about 70 · x = μ + zσ = 60 + 1.28 × 8 = 70.24, which rounds to 70.
Worksheet · LightMySky