2023HSCShort answer

Q11 (f) (i)

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 $P(\hat p\leq0.31)$ .

(i) Show that the variance of the random variable $\hat p$ is

\frac{7}{30\,000}.

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	2
Obtains $p=0.3$ , or equivalent merit	1

Sample answer:

$P(\text{born overseas})=0.3$

Let $p=0.3$ . Therefore $1-p=0.7$ .

$n=900$

\text{Distribution of }\hat p\text{ is approximately normal with mean }\mu=0.3\text{ and}

\sigma^2=\frac{0.3\times0.7}{900}=\frac{0.21}{900}=\frac{7}{30\,000}

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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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