Mathematical Methods — Question 13

QCAA 2025, Paper 2 · 4 marks

Q13 · 2025 · Technology-activeSimple familiar4 marks

QUESTION 13 (4 marks)

A school investigated how many hours students sleep per night. To obtain data, a random sample of students was surveyed. The results are shown.

Bar chart showing hours of sleep per night for the sampled students

Use the data to determine the sample proportion of students who had seven or more hours of sleep per night.

[3 marks]

Use your result from Question 13a) to determine an 80% approximate confidence interval for the proportion of students who get seven or more hours of sleep per night.

[1 mark]

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

4 marks

ANSWER

\hat p=\frac{52}{86}\approx0.605

; b)

(0.537,0.672)

Worked solution

a) The total number of students is

15+19+20+27+5=86.

The number with seven or more hours of sleep is

20+27+5=52.

Therefore

\hat p=\frac{52}{86}=0.605.

b) Using a GDC interval function or equivalent method gives the 80% approximate confidence interval

(0.537,0.672).

correctly determines the total number of students in the sample

[1 mark]

correctly determines the total number of students having seven hours of sleep or more

[1 mark]

determines the sample proportion of students having seven hours of sleep or more

[1 mark]

determines the 80% approximate confidence interval

[1 mark]

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Mathematical Methods — Question 13

QUESTION 13 (4 marks)

QCAA guide · typeset solution

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