Mathematical Methods — Question 16

QCAA 2024, Paper 2 · 4 marks

Q16 · 2024 · Technology-activeComplex familiar4 marks

QUESTION 16 (4 marks)

At council meetings in a particular town, new proposals are only discussed if more than 80% of the community are in favour of the proposal. To discover community opinion on a new bus route proposal, the council conducted several surveys, each with a sample size of 120. The distribution of the sample proportions from the surveys had a standard deviation of 0.04. Make a justified decision as to whether the new bus route proposal would be discussed at a council meeting.

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WORKED SOLUTION

QCAA guide · typeset solution

4 marks

ANSWER

No. The information implies

p\approx0.2592

p\approx0.7408

, both below 0.80.

Worked solution

For a sample proportion,

\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n}}.

With

n=120

and

\sigma_{\hat p}=0.04

0.04=\sqrt{\frac{p(1-p)}{120}},

p(1-p)=0.192.

Thus

p^2-p+0.192=0,

giving

p\approx0.2592\quad\text{or}\quad p\approx0.7408.

Neither value is more than 0.80, so the proposal will not be discussed.

correctly substitutes the given information into the standard deviation formula for sample proportion

[1 mark]

determines a possible value for the population proportion

[1 mark]

determines a second possible population proportion value

[1 mark]

makes a justified decision regarding the proposal

[1 mark]

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Mathematical Methods — Question 16

QUESTION 16 (4 marks)

QCAA guide · typeset solution

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