Mathematics Standard • Year 12 • Module 5 • Lesson 12
Normal Distribution Applications & Module Review, Apply
Contextual Problem Set 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 factory fills packets with mass X approximately normal, μ=250 g and σ=10 g. Find the z-score for a 270 g packet and estimate the percentage above 270 g using the empirical rule. 3 marks
2. Use the idea
A different packet has mass 235 g. A standard normal table gives P(Z<−1.50)=0.0668. Estimate the number below 235 g in a production run of 3,000 packets. 3 marks
3. Check your reasoning
A quality rule rejects the bottom 10% of packets. The z-table gives z=−1.28 at the 10th percentile. Find the mass cutoff and explain the decision. 3 marks
Self-check
For each problem, check that your answer includes the requested variable, units or contextual interpretation.
1. Core skill (3 marks)
z=(270−250)/10=2. About 95% lies within ±2σ, leaving 5% in both tails; symmetry gives about 2.5% above +2σ.
2. Use the idea (3 marks)
z=(235−250)/10=−1.50. Expected count=0.0668×3,000=200.4, about 200 packets.
3. Check your reasoning (3 marks)
x=μ+zσ=250+(−1.28)(10)=237.2 g. Reject packets below about 237.2 g (with the exact inclusion rule set by the factory); this cutoff corresponds to the bottom 10% under the model.