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Date	May 2022	Marks available	4	Reference code	22M.1.AHL.TZ1.9
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 1
Command term	Find	Question number	9	Adapted from	N/A

Question

A company produces bags of sugar with a labelled weight of $1 kg$ . The weights of the bags are normally distributed with a mean of $1 kg$ and a standard deviation of $100 g$ . In an inspection, if the weight of a randomly chosen bag is less than $950 g$ then the company fails the inspection.

Find the probability that the company fails the inspection.

[2]

A statistician in the company suggests it would be fairer if the company passes the inspection when the mean weight of five randomly chosen bags is greater than $950 g$ .

Find the probability of passing the inspection if the statistician’s suggestion is followed.

[4]

Markscheme

let $X$ be the weight of sugar in the bag

$P (X < 950) = 0.308537 \dots \approx 0.309$ (M1)A1

[2 marks]

METHOD 1

let $\bar{X}$ be the mean weight of $5$ bags of sugar

$E (\bar{X}) = 1000$ (A1)

use of $Var (\bar{X}) = \frac{σ^{2}}{n}$ (M1)

$Var (\bar{X}) = \frac{100^{2}}{5} (= 2000)$ (A1)

$\bar{X} ~ N (1000, 2000)$

$P (\bar{X} > 950) = 0.868223 \dots \approx 0.868 (86.8 %)$ A1

METHOD 2

let $T$ be the total weight of $5$ bags of sugar

$E (T) = 5000$ (A1)

use of $Var (X_{1} + X_{2}) = Var (X_{1}) + Var (X_{2})$ for independent random variables (M1)

$Var (T) = 5 \times 100^{2} (= 50 000)$ (A1)

$T ~ N (5000, 50 000)$

$P (T > 4750) = 0.868223 \dots \approx 0.868 (86.8 %)$ A1

[4 marks]

Examiners report

Part (a) was straightforward, and a good number of candidates showed their knowledge in achieving a correct answer. Candidates are advised to not use calculator notation, as examiners cannot be familiar with all variations of GDC syntax; instead, correct mathematical notation and/or a written commentary will ensure the method is communicated to the examiner. Rounding errors once again caused problems for some. Good answers to part (b) were much less common and this was a challenging question for many. A few understood how to use the central limit theorem to find the sampling distribution of the sample mean and a few used the mean and variance of the sum of independent random variables.

[N/A]

Syllabus sections

Topic 4—Statistics and probability » AHL 4.14—Linear transformation of a single RV, E(X) and VAR(X), unbiased estimators

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Topic 4—Statistics and probability » AHL 4.15—Central limit theorem

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