Mathematical Methods — Question 11

QCAA 2020, Paper 2 · 4 marks

Q11 · 2020 · Technology-activeSimple familiar4 marks

QUESTION 11 (4 marks)

A sugar company samples the packets of sugar it produces and finds that 5% of packets are underweight. Consider a batch of 20 packets.

Determine how many packets of sugar the company can expect to be underweight in a batch of 20 packets.

[1 mark]

Determine the variance of the batch.

[1 mark]

Determine the probability that at most one of the packets of sugar in the batch is underweight.

[2 marks]

WORKED SOLUTION

4 marks

ANSWER

a) 1 packet; b) 0.95; c) 0.7358

Worked solution

5\%\times20=1

\operatorname{Var}(X)=np(1-p)=20(0.05)(0.95)=0.95

c) With

X\sim\operatorname{Bin}(20,0.05)

P(X\le1)=0.7358

correctly determines the expected number of the underweight packets

[1 mark]

correctly determines the variance of the underweight packets

[1 mark]

correctly uses an appropriate mathematical representation to communicate approach

[1 mark]

correctly determines the probability

[1 mark]

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

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