Mathematical Methods — Question 12

QCAA 2022, Paper 2 · 4 marks

Q12 · 2022 · Technology-activeSimple familiar4 marks

QUESTION 12 (4 marks)

Suppose that the distance travelled by vehicles in a year can be modelled by a normal distribution. In 2021, vehicles travelled a mean of 13 700 km with a standard deviation of 3400 km.

Determine the probability that a vehicle chosen at random travelled less than 12 000 km in 2021.

[2 marks]

Determine the value of

x

where 60% of vehicles travelled less than

x

km in 2021.

[2 marks]

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

4 marks

ANSWER

0.3085

; b)

x=14\,561.38

km.

Worked solution

a) Using the normal distribution with

\mu=13\,700

and

\sigma=3400

P(X<12\,000)=0.3085.

b) Solve

P(X<x)=0.60

for the same distribution, giving

x=14\,561.38\text{ km}.

Official normal-distribution diagrams for parts a and b

correctly uses an appropriate mathematical representation

[1 mark]

correctly determines the probability

[1 mark]

correctly uses an appropriate mathematical representation

[1 mark]

correctly determines the value of

x

[1 mark]

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Mathematical Methods — Question 12

QUESTION 12 (4 marks)

QCAA guide · typeset solution

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