21M.2.SL.TZ1.3

pestleMathematicsAASLPaper 221M· sl-4-9-normal-distribution-and-calculations, sl-4-6-combined-mutually-exclusive-conditional-independence-prob-diagramssource ↗

A company produces bags of sugar whose masses, in grams, can be modelled by a normal distribution with mean $1000$ and standard deviation $3.5$ . A bag of sugar is rejected for sale if its mass is less than $995$ grams.

Find the probability that a bag selected at random is rejected.

[2]

Estimate the number of bags which will be rejected from a random sample of $100$ bags.

[1]

Given that a bag is not rejected, find the probability that it has a mass greater than $1005$ grams.

[3]

Markscheme / solution

Note: In this question, do not penalise incorrect use of strict inequality signs.

Let $X =$ mass of a bag of sugar

evidence of identifying the correct area (M1)

$P (X < 995) = 0.0765637 \dots$

$= 0.0766$ A1

[2 marks]

Note: In this question, do not penalise incorrect use of strict inequality signs.

Let $X =$ mass of a bag of sugar

$0.0766 \times 100$

$\approx 8$ A1

Note: Accept $7.66$ .

[1 mark]

Note: In this question, do not penalise incorrect use of strict inequality signs.

Let $X =$ mass of a bag of sugar

recognition that $P (X > 1005 X \geq 995)$ is required (M1)

$\frac{P (X \geq 995 \cap X > 1005)}{P (X \geq 995)}$

$\frac{P (X > 1005)}{P (X \geq 995)}$ (A1)

$\frac{0.0765637 \dots}{1 - 0.0765637 \dots} (= \frac{0.0765637 \dots}{0.923436 \dots})$

$= 0.0829$ A1

[3 marks]

Examiners’ report

[N/A]