2023HSCShort answer

Q11 (f) (ii)

2 marks

Question

A recent census found that 30% of Australians were born overseas.

A sample of 900 randomly selected Australians was surveyed.

Let $\hat p$ be the sample proportion of surveyed people who were born overseas.

A normal distribution is to be used to approximate $P(\hat p\leq0.31)$ .

(ii) Use the standard normal distribution and the information on page 16 to approximate $P(\hat p\leq0.31)$ , giving your answer correct to two decimal places.

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	2
Obtains the correct $z$ -score, or equivalent merit	1

Sample answer:

P(\hat p\le0.31)\approx P\left(Z\le\frac{0.31-0.3}{\sqrt{7/30\,000}}\right)

=P(X\le0.6546\ldots)\approx P(Z\le0.65)=0.7422\text{ from table}=0.74\text{ to 2 decimal places}

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Compare this with your working, not only your final sentence.

Syllabus alignment

Statistical Analysis

The Binomial DistributionModel Bernoulli trials with binomial random variables and use binomial probabilities and summary measures to analyse chance processes.

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