Mathematics Standard • Year 12 • Module 8 • Lesson 6

Normal Distribution — Skill Drill

Build fluency in the 68-95-99.7 (empirical) rule, identifying intervals in terms of μ ± kσ and reading percentages off the normal curve.

Build · Skill Drill

1. Quick recall

Answer each in the space provided. 1 mark each

Q1.1 The 68-95-99.7 rule:
About ____ % within μ ± 1σ About ____ % within μ ± 2σ About ____ % within μ ± 3σ.

Q1.2 A normal distribution is symmetric. So if 95% sits within μ ± 2σ, then ____ % sits ABOVE μ + 2σ (one tail).

Q1.3 In a normal distribution, mean = ____ = ____ (which three statistics all coincide?).

Stuck? Revisit lesson § Key Ideas — 68-95-99.7 rule, Mean = Median = Mode.

2. Worked example — exam marks (μ = 72, σ = 10)

Find: (a) % scoring between 62 and 82, (b) % scoring above 92, (c) the minimum mark for the top 2.5%.

Step 1 — Convert each bound to "μ ± kσ".

62 = 72 − 10 = μ − 1σ. 82 = 72 + 10 = μ + 1σ. 92 = 72 + 20 = μ + 2σ.

Step 2 — Apply the 68-95-99.7 rule.

(a) Between μ − 1σ and μ + 1σ → 68%.
(b) Above μ + 2σ: total outside μ ± 2σ = 5%; by symmetry, half is above. → 2.5%.
(c) "Top 2.5%" means above μ + 2σ. Minimum mark = 72 + 2(10) = 92 marks.

Conclusion. (a) 68% (b) 2.5% (c) 92 marks.

3. Faded example — newborn weights (μ = 3.4 kg, σ = 0.4 kg)

Find the percentage of newborns weighing between 2.6 kg and 4.2 kg. 3 marks

Step 1 — Convert each bound:

2.6 = 3.4 − ____ = μ − ____ σ. 4.2 = 3.4 + ____ = μ + ____ σ.

Step 2 — Identify which interval: Between μ − ____ σ and μ + ____ σ.

Step 3 — Read off the rule: The 68-95-99.7 rule says ____ % sit within this interval.

Conclusion: About ____ % of newborns weigh between 2.6 kg and 4.2 kg.

Stuck? Both bounds are an equal number of σ from the mean — that's the clue.

4. Graduated practice — empirical rule

Foundation — one-step (4 questions)

Q	Problem	Answer
4.1 1	μ = 100, σ = 15. What is μ + 2σ?
4.2 1	μ = 50, σ = 5. What is μ − 1σ?
4.3 1	For a normal distribution, what % of values sit within ±2σ of the mean?
4.4 1	For a normal distribution, what % sit BELOW μ − 1σ?

Standard — typical HSC difficulty (6 questions)

4.5 Adult height (cm): μ = 170, σ = 8. (a) What % are taller than 178 cm? (b) What % are shorter than 154 cm? 2 marks

4.6 Test marks: μ = 65, σ = 12. What percentage scored between 41 and 89? 2 marks

4.7 Battery life (hours): μ = 200, σ = 20. What is the minimum value for the top 16% of batteries? (Hint: 16% is half of the 32% outside μ ± 1σ.) 2 marks

4.8 IQ scores: μ = 100, σ = 15. (a) In a group of 1,000 people, about how many have IQ above 130? (b) What % score above 145? 2 marks

4.9 Reaction times (seconds): μ = 0.50, σ = 0.05. What percentage of reaction times are between 0.40 and 0.60? 2 marks

4.10 A normal distribution has 68% of values between 80 and 96. Find μ and σ. 2 marks

Extension — combine intervals (2 questions)

4.11 Exam marks: μ = 70, σ = 8. Find the percentage of students scoring (a) between 62 and 78, (b) between 78 and 86, (c) below 54. 3 marks

4.12 Heights of Year 12 girls (cm) are normally distributed with μ = 165 and σ = 7. In a cohort of 800, about how many girls would you expect to be (a) between 158 and 172, (b) taller than 179, (c) shorter than 151? 3 marks

Stuck on 4.11(b)? "78 to 86" = μ + 1σ to μ + 2σ. Use (95% − 68%)/2 to get the percentage in that "shell" on one side.

5. Self-check the easy 3

For 4.1 I added 2σ (= 30) to μ (= 100), getting 130 — not multiplied.

For 4.3 I used 95% (the μ ± 2σ rule), NOT 68% or 99.7%.

For 4.4 I used (100 − 68)/2 = 16% (one tail of the μ ± 1σ split), NOT 32%.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Empirical rule

68% within μ ± 1σ. 95% within μ ± 2σ. 99.7% within μ ± 3σ.

Q1.2 — Tail above μ + 2σ

(100 − 95)/2 = 2.5%.

Q1.3 — Centre statistics

Mean = median = mode at the centre of the normal curve.

Q3 — Newborn weights faded

2.6 = 3.4 − 0.8 = μ − 2σ. 4.2 = 3.4 + 0.8 = μ + 2σ. Between μ ± 2σ → 95% of newborns weigh between 2.6 kg and 4.2 kg.

Q4.1 – Q4.4 — Foundation

4.1: μ + 2σ = 100 + 30 = 130. 4.2: μ − 1σ = 50 − 5 = 45. 4.3: 95%. 4.4: (100 − 68)/2 = 16%.

Q4.5 — Adult heights

(a) 178 = 170 + 8 = μ + 1σ. Above μ + 1σ = (100 − 68)/2 = 16%.
(b) 154 = 170 − 16 = μ − 2σ. Below μ − 2σ = (100 − 95)/2 = 2.5%.

Q4.6 — Marks between 41 and 89

41 = 65 − 24 = μ − 2σ. 89 = 65 + 24 = μ + 2σ. Between μ ± 2σ → 95%.

Q4.7 — Top 16% of batteries

Top 16% sits above μ + 1σ. Minimum value = 200 + 20 = 220 hours.

Q4.8 — IQ

(a) Above 130 = above μ + 2σ → 2.5%. In 1,000 people: 0.025 × 1,000 = 25 people.
(b) Above 145 = above μ + 3σ → (100 − 99.7)/2 = 0.15%.

Q4.9 — Reaction times

0.40 = 0.50 − 0.10 = μ − 2σ. 0.60 = 0.50 + 0.10 = μ + 2σ. Between μ ± 2σ → 95%.

Q4.10 — 68% between 80 and 96 → find μ, σ

68% corresponds to μ ± 1σ. So 80 = μ − σ and 96 = μ + σ. Add: 2μ = 176 → μ = 88. Subtract: 2σ = 16 → σ = 8.

Q4.11 — Three intervals

(a) 62 = μ − 1σ; 78 = μ + 1σ → 68%.
(b) 78 to 86 = μ + 1σ to μ + 2σ → one "shell" on the right = (95 − 68)/2 = 13.5%.
(c) 54 = 70 − 16 = μ − 2σ. Below μ − 2σ = (100 − 95)/2 = 2.5%.

Q4.12 — 800 girls

(a) 158 = 165 − 7 = μ − 1σ; 172 = 165 + 7 = μ + 1σ → 68%. Count = 0.68 × 800 = 544 girls.
(b) Taller than 179 = above μ + 2σ → 2.5%. Count = 0.025 × 800 = 20 girls.
(c) Shorter than 151 = below μ − 2σ → 2.5%. Count = 0.025 × 800 = 20 girls.