Mathematics Standard • Year 12 • Module 5 • Lesson 10
The Empirical Rule, Apply
Contextual Problem Set 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
The fill volume of bottles is approximately normal with μ=120 mL and σ=10 mL. Give the interval containing about 95% of fills and estimate the proportion below 100 mL. 3 marks
2. Use the idea
Estimate the percentage of bottles between 110 mL and 130 mL. Then estimate the percentage between 120 mL and 140 mL. 3 marks
3. Check your reasoning
A bottle has volume 151 mL. Calculate how many standard deviations above the mean it is and use the empirical rule to assess 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)
About 95% lies within 2σ: 120±20, or 100 to 140 mL. 100 mL is μ−2σ; about 2.5% lies below it.
2. Use the idea (3 marks)
110 to 130 mL is μ±1σ, so about 68%. 120 to 140 mL is from μ to μ+2σ: 50%−2.5%=47.5%.
3. Check your reasoning (2 marks)
z-distance=(151−120)/10=3.1 standard deviations above the mean. The empirical rule places only about 0.15% above μ+3σ, so 151 mL is very unusual under this model.