Mathematics Standard • Year 12 • Module 5 • Lesson 10
The Empirical Rule, Build
Scaffolded Practice for The Empirical Rule. Work through each self-contained problem, show your reasoning, then open the worked answers to check your method.
1. Core skill
A normal distribution has mean μ=50 and standard deviation σ=5. Complete the intervals: about 68% lies between ___ and ___; about 95% between ___ and ___; about 99.7% between ___ and ___. 3 marks
2. Use the idea
For the same distribution, estimate the percentage below 55 and the percentage above 60. Show how symmetry is used. 3 marks
3. Check your reasoning
A value is 3 standard deviations above the mean. Use the empirical rule to estimate the percentage of observations above it and say whether it is unusual. 2 marks
Self-check
For each problem, check that your answer includes the requested variable, units or contextual interpretation.
1. Core skill (3 marks)
Within 1σ: 50±5, so 45 to 55. Within 2σ: 50±10, so 40 to 60. Within 3σ: 50±15, so 35 to 65.
2. Use the idea (3 marks)
55 is μ+1σ, so 68% lies from 45 to 55 and half of the remaining 32% lies below 45: 50%+34%=84% below 55. Above 60=μ+2σ is half of the 5% outside ±2σ, or about 2.5%.
3. Check your reasoning (2 marks)
About 0.3% lies outside ±3σ, split equally between the two tails, so about 0.15% is above μ+3σ. It is very unusual under the model.