Normal distribution: 7 exam questions solved step by step

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What this lesson covers

  • A normal probability is an area under the curve, found from the mean and standard deviation.
  • Working backwards from a probability finds an unknown value, mean or standard deviation.
Key idea

7 questions, one method

  • 1. Sketch the curve, shade the area.
  • 2. Forwards: z, then the area.
  • 3. Backwards: area to the left, then z.
  • 4. Round at the end, in context.
Your turn

Question 1 · Achieved

  • Hawke's Bay apple weights are normal:
  • mean 172 g, sd 18 g.
  • Apples under 150 g go to juice.
  • Find P(an apple goes to juice).
Question 1 · Achieved
Worked example

Q1: P(X < 150)

  • Below the mean: z < 0.
  • A small left tail.
  1. X ~ N(172, 18²)grams
  2. P(X < 150)
  3. z = (150 − 172) ÷ 18
  4. = −22 ÷ 18 = −1.2222below: negative
  5. P(X < 150) = 0.110812
  6. = 0.11084 d.p.
  7. about 11% go to juice
Your turn

Question 2 · Achieved

  • The same apples: mean 172 g, sd 18 g.
  • The heaviest 10% are sold as premium.
  • Find the lightest premium weight.
Question 2 · Achieved
Worked example

Q2: the top 10%

  • Backwards: area to z.
  • Use the area to the left.
  1. P(X > H) = 0.1
  2. P(X < H) = 0.9area to the left
  3. z = 1.2816inverse normal
  4. H = 172 + 1.2816 × 18
  5. = 172 + 23.0688 = 195.0688
  6. about 195.07 gabove the mean
  7. 172 − 23.0688 = 148.93wrong tail
Your turn

Question 3 · Merit

  • Avocado weights are normal:
  • mean 215 g, sd 22 g.
  • Find P(an avocado is 190 g to 250 g).
Question 3 · Merit
Worked example

Q3: P(190 < X < 250)

  • Two values, two z-values.
  • Left area minus left area.
  1. X ~ N(215, 22²)
  2. P(190 < X < 250)
  3. z = (190 − 215) ÷ 22 = −1.1364
  4. z = (250 − 215) ÷ 22 = 1.5909
  5. P(X < 250) = 0.944185
  6. P(X < 190) = 0.127902
  7. 0.944185 − 0.127902 = 0.816283
  8. = 0.81634 d.p.
Your turn

Question 4 · Merit

  • Times to finish a long day walk are
  • normal with mean 6.5 hours.
  • 12% take longer than 8 hours.
  • Find the standard deviation.
Question 4 · Merit
Worked example

Q4: find σ

  • Area to the left: 0.88.
  • Substitute, solve for σ.
  1. μ = 6.5, σ unknown
  2. P(X > 8) = 0.12
  3. P(X < 8) = 0.88area to the left
  4. z = 1.1750inverse normal
  5. 1.1750 = (8 − 6.5) ÷ σ
  6. σ = 1.5 ÷ 1.1750
  7. = 1.2766 hours4 d.p.
  8. about 77 minutes
  9. σ = 1.2766: P(X > 8) = 0.1200
Your turn

Question 5 · Merit, in context

  • An e-scooter range is normal, sd 3.2 km.
  • Only 4% may fall under 30 km.
  • What mean range is needed?
Question 5 · Merit, in context
Worked example

Q5: find μ

  • 30 is below: z < 0.
  • Solve for μ.
  1. σ = 3.2, μ unknown
  2. P(X < 30) = 0.04
  3. z = −1.7507below: negative
  4. −1.7507 = (30 − μ) ÷ 3.2
  5. −1.7507 × 3.2 = 30 − μ
  6. −5.6022 = 30 − μ
  7. μ = 30 + 5.6022 = 35.6022
  8. a mean range of about 35.6 km
  9. μ = 35.6022: P(X < 30) = 0.0400
Your turn

Question 6 · Excellence

  • The apples again: mean 172 g, sd 18 g.
  • An apple is not for juice (over 150 g).
  • Find P(it is over 205 g).
Question 6 · Excellence
Worked example

Q6: given over 150 g

  • "Given" shrinks the total.
  • Over 205 is over 150.
  1. P(X > 205 | X > 150)
  2. = P(X > 205) ÷ P(X > 150)over 205 is over 150
  3. z = (205 − 172) ÷ 18 = 1.8333
  4. P(X > 205) = 0.033377
  5. P(X > 150) = 1 − 0.110812from question 1
  6. = 0.889188the reduced total
  7. 0.033377 ÷ 0.889188 = 0.037536
  8. = 0.03754 d.p.
  9. about 1 in 27 of these apples
Your turn

Question 7 · Excellence

  • The apples: 11% go to juice (Q1).
  • A bag holds 8 apples picked at random.
  • Find P(at least 2 in the bag
  • would go to juice).
Question 7 · Excellence
Worked example

Q7: normal, then binomial

  • Normal gives p.
  • Binomial counts the bag.
  1. p = P(X < 150) = 0.110812from question 1
  2. Y ~ Bin(8, 0.110812)8 apples, independent
  3. P(Y ≥ 2) = 1 − P(Y ≤ 1)
  4. P(Y = 0) = 0.889188⁸ = 0.390795
  5. P(Y = 1) = 8 × p × (1 − p)⁷
  6. = 0.389612
  7. 1 − (0.390795 + 0.389612)
  8. = 1 − 0.780407 = 0.219593
  9. = 0.21964 d.p.
Summary

The method that solved all seven

  • Sketch, z or inverse normal, solve, check, say it in context.
StepHow
Sketchshade the area you want
Forwardsz = (x − μ) ÷ σ, then area
Backwardsarea to the left gives z
Unknownz = (x − μ) ÷ σ, solve
Given thatdivide by the reduced total
Two stagesnormal p into a binomial

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