2023 HSC Mathematics Extension 1
Question 11(f):
Binomial distribution
A recent census found that of Australians were born overseas. A sample of randomly selected Australians was surveyed.
Let be the sample proportion of surveyed people who were born overseas. A normal distribution is to be used to approximate .
(i) Show that the variance of the random variable is .
(ii) Use the standard normal distribution and the supplied normal table to approximate , giving your answer correct to two decimal places.
How to recognise this question
Question type: binomial distribution
- The command and notation point to binomial distribution.
- The word “hence” means the later part is intended to reuse the earlier result.
How to handle it: identify the random variable and its parameters before using the matching probability formula.
Watch out: Do not restart the second part with unrelated numbers or discard the result proved in the first part.
Step-by-step answer
- (i) Population proportion , so , and .
- For a sample proportion, .
- NESA (i): 2 marks correct; 1 for obtaining or equivalent merit.
- (ii) .
- .
- ; .
- From the standard normal table, (two decimal places).
- NESA (ii): 2 marks correct; 1 for correct -score.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 4 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
A population has proportion . A random sample of size is taken and is the sample proportion.
(i) Show that .
(ii) Hence use the standard normal distribution to approximate , giving your answer correct to two decimal places.
Question 2
A population has proportion . A random sample of size is taken and is the sample proportion.
(i) Show that .
(ii) Hence approximate using the standard normal table, correct to two decimal places.
Question 3
A population has proportion . A random sample of size is taken and is the sample proportion.
(i) Show that .
(ii) Hence approximate using the standard normal table, correct to two decimal places.