2021 HSC Mathematics Extension 1
Question 14(d):
Normal distribution
At a certain factory, the proportion of faulty items produced by a machine is , which is considered to be acceptable. To confirm that the machine is working to this standard, a sample of size is taken and the sample proportion is calculated.
It is assumed that is approximately normally distributed with and .
Production by this machine will be shut down if .
The sample size is to be chosen so that the chance of shutting down the machine unnecessarily is less than .
Find the approximate sample size required, giving your answer to the nearest thousand.
How to recognise this question
Question type: normal distribution
- The command and notation point to normal distribution.
- The requested response is a worked solution.
How to handle it: standardise to a z-score and use the matching standard-normal probability.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Unnecessarily shutting down means when the true . Require .
- For a normal model, a one-sided upper tail begins at about (using the conventional for in this course).
- Set .
- So .
- Square after dividing by : wait more cleanly from MG:
- .
- Derivation: .
- Nearest thousand: .
- NESA: 3 marks full; 2 for recognising with in terms of ; 1 for in terms of or a correct normal sketch/probability statement.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
True , shut down if , want when . Find using , nearest hundred.
Question 2
Show that the exam calculation gives before rounding.
Question 3
Why is the threshold placed at for a unnecessary-shutdown risk?