Mathematical Methods — Question 15

QCAA 2024, Paper 1 · 4 marks

Q15 · 2024 · Technology-freeComplex familiar4 marks

QUESTION 15 (4 marks)

A survey was conducted to understand whether people support a new policy. Using a

z

-score of 2, the approximate confidence interval for the population proportion of people who support the policy was calculated as

\left(\frac3{10},\frac7{10}\right)

Determine the margin of error.

[1 mark]

Determine the number of people surveyed.

[3 marks]

WORKED SOLUTION

4 marks

ANSWER

\frac15

; b)

25

people.

Worked solution

a) The confidence interval is

(\hat p-E,\hat p+E)

. Thus

2E=\frac7{10}-\frac3{10}=\frac4{10},

E=\frac15.

b) From the midpoint,

\hat p=\frac12

. Using the upper confidence limit,

\frac7{10}=\frac12+2\sqrt{\frac{(1/2)(1-1/2)}{n}}.

Hence

\frac15=2\sqrt{\frac{1/4}{n}}=\frac1{\sqrt n},

n=25

. Therefore 25 people were surveyed.

correctly determines the margin of error

[1 mark]

determines the value of the sample proportion

\hat p

[1 mark]

substitutes

\hat p

and z-score into the confidence interval formula

[1 mark]

determines the number of people surveyed

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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