2024 HSC Mathematics Extension 1
Question 8:
Binomial distribution
A local council is proposing to ban dog-walking on the beach. It is known that the proportion of households that have a dog is .
The local council wishes to poll households about this proposal.
Let be the random variable representing the proportion of households polled that have a dog.
What is the smallest sample size, , for which the standard deviation of is less than ?
How to recognise this question
Question type: binomial distribution
- The command and notation point to binomial distribution.
- The requested response is a multiple choice.
How to handle it: identify the random variable and its parameters before using the matching probability formula.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- For a sample proportion, .
- Here , so and .
- Require . Square both sides: .
- Thus .
- The smallest integer strictly above is . Check: at the SD is still slightly above ; at it drops below.
- Answer: B. (NESA multiple-choice key: B.)
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 1 total mark 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
If , the smallest integer with is
Question 2
For and bound , which inequality correctly finds ?
Question 3
With , the product equals