Mathematical Methods — Question 11

QCAA 2022, Paper 2 · 7 marks

Q11 · 2022 · Technology-activeSimple familiar7 marks

QUESTION 11 (7 marks)

A salesperson has a 20% probability of making a sale to each customer who enters the store. Each sale is independent of all other sales.

Determine the mean number of sales on a day where 25 customers enter the store.

[2 marks]

Determine the standard deviation of the number of sales on a day where 25 customers enter the store.

[2 marks]

Determine the minimum number of customers who would have to enter the store to have an 88% chance or more of making at least one sale.

[3 marks]

WORKED SOLUTION

7 marks

ANSWER

a) 5; b) 2; c) 10 customers.

Worked solution

a) Mean number of sales:

np=25(0.2)=5.

b) Standard deviation:

\sqrt{np(1-p)}=\sqrt{25(0.2)(0.8)}=2.

1-\binom n0(0.2)^0(0.8)^n\ge0.88.

Hence

1-0.8^n\ge0.88

, so

0.8^n\le0.12

and

n\ge\log_{0.8}(0.12)\approx9.50179.

Therefore the minimum number of customers is 10.

correctly substitutes into formula for mean

[1 mark]

correctly determines the mean

[1 mark]

correctly substitutes into formula for standard deviation

[1 mark]

correctly determines the standard deviation

[1 mark]

correctly determines the required inequation

[1 mark]

correctly determines the unknown in the inequation

[1 mark]

determines the minimum number of customers

[1 mark]

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

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