Mathematical Methods — Question 18

QCAA 2022, Paper 2 · 3 marks

Q18 · 2022 · Technology-activeComplex unfamiliar3 marks

QUESTION 18 (3 marks)

The intelligence quotient (IQ) of individuals in a population is normally distributed, with a mean of 100 and a standard deviation of 16. Nine individuals are chosen at random from the population. Determine the probability that no more than two of the individuals have an IQ of at least 120.

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WORKED SOLUTION

QCAA guide · typeset solution

3 marks

ANSWER

0.9391

Worked solution

For one person,

P(IQ\ge120)=0.1057.

The number among nine with IQ at least 120 can therefore be modelled using a binomial distribution with

n=9

and

p=0.1057

. Using a GDC,

P(X\le2)=0.9391.

Official normal-distribution diagram for P(IQ ≥ 120)

correctly determines the probability of IQ

\ge120

[1 mark]

correctly recognises context is suitable for modelling as a binomial

[1 mark]

determines the required probability

[1 mark]

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Mathematical Methods — Question 18

QUESTION 18 (3 marks)

QCAA guide · typeset solution

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