22M.1.SL.TZ1.8

pestleMathematicsAISLPaper 122M· sl-4-9-normal-distribution-and-calculationssource ↗

A factory produces bags of sugar with a labelled weight of $500 g$ . The weights of the bags are normally distributed with a mean of $500 g$ and a standard deviation of $3 g$ .

A bag that weighs less than $495 g$ is rejected by the factory for being underweight.

Write down the percentage of bags that weigh more than $500 g$ .

[1]

Find the probability that a randomly chosen bag is rejected for being underweight.

[2]

A bag that weighs more than $k$ grams is rejected by the factory for being overweight. The factory rejects $2 %$ of bags for being overweight.

Find the value of $k$ .

[3]

Markscheme / solution

$50 %$ A1

Note: Do not accept $0.5$ or $\frac{1}{2}$ .

[1 mark]

$0.0478 (0.0477903 \dots, 4.78 %)$ A2

[2 marks]

$P (X < k) = 0.98$ OR $P (X > k) = 0.02$ (M1)

Note: Award (M1) for a sketch with correct region identified.

$506 g (506.161 \dots)$ A2

[3 marks]

Examiners’ report

Many candidates made no attempt at this question, suggesting poor preparation, although it has appeared frequently in past sessions.

Few candidates recognized that a normal distribution is symmetric.

Of the few candidates who were able to interpret the question correctly and use their calculator to identify the numerical digits satisfying the probability requirements, a number were out by a factor of 10.

Few correct answers were seen. Of those that tried to show a diagram, there was often not enough detail to award a method mark (e.g. the shaded area was numerically identified as 2%).