Mathematics Standard • Year 12 • Module 5 • Lesson 12
Normal Distribution Applications & Module Review, Master
Mastery Challenge 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 has μ=100 and σ=15. For x=130, calculate z and use the empirical rule to estimate the percentage below x. 3 marks
2. Use the idea
For the same model, a table gives P(Z<−1)=0.1587. Estimate the percentage below x=85 and the number below 85 in a population of 1,200. 3 marks
3. Check your reasoning
An analyst reports x=130 as the 95th percentile based on the empirical rule. Correct the statement and distinguish empirical-rule approximation from an exact z-table result. 2 marks
Self-check
For each problem, check that your answer includes the requested variable, units or contextual interpretation.
1. Core skill (3 marks)
z=(130−100)/15=2. About 95% lies between −2σ and +2σ; 2.5% lies above +2σ. Therefore approximately 97.5% lies below x.
2. Use the idea (3 marks)
z=(85−100)/15=−1. The percentage is 15.87%. Count≈0.1587×1,200=190.44, about 190 people/items.
3. Check your reasoning (2 marks)
130 is +2σ, which is approximately the 97.5th percentile, not the 95th. The empirical rule gives approximate landmark percentages; a z-table gives a more precise cumulative probability for a specified z-score.