Mathematics Standard • Year 12 • Module 5 • Lesson 12
Normal Distribution Applications & Module Review, Build
Scaffolded Practice for Normal Distribution Applications & Module Review. Work through each self-contained problem, show your reasoning, then open the worked answers to check your method.
1. Core skill
A normal model for battery life has μ=500 hours and σ=20 hours. Calculate the z-score for 540 hours and identify whether the value is below or above the mean. 2 marks
2. Use the idea
For a standard normal variable, a table gives P(Z<1.28)=0.8997. In a batch of 2,000 items, estimate how many fall below z=1.28. 2 marks
3. Check your reasoning
Choose the better tool: (a) approximate percentage within 2 standard deviations of a normal mean; (b) percentage below a value 1.28 standard deviations above the mean using a z-table. Explain. 2 marks
Self-check
For each problem, check that your answer includes the requested variable, units or contextual interpretation.
1. Core skill (2 marks)
z=(540−500)/20=2. The battery life is 2 standard deviations above the mean.
2. Use the idea (2 marks)
Expected proportion is 0.8997, so 0.8997×2,000=1,799.4, approximately 1,799 items (about 1,800).
3. Check your reasoning (2 marks)
(a) Use the empirical rule: about 95%. (b) Use the z-table: P(Z<1.28)=0.8997, about 89.97%. The z-table gives a more specific cumulative probability.