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Date	November Example question	Marks available	2	Reference code	EXN.2.AHL.TZ0.6
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	6	Adapted from	N/A

Question

The masses in kilograms of melons produced by a farm can be modelled by a normal distribution with a mean of $2.6 kg$ and a standard deviation of $0.5 kg$ .

Find the probability that two melons picked at random and independently of each other will

One year due to favourable weather conditions it is thought that the mean mass of the melons has increased.

The owner of the farm decides to take a random sample of $16$ melons to test this hypothesis at the $5 %$ significance level, assuming the standard deviation of the masses of the melons has not changed.

Unknown to the farmer the favourable weather conditions have led to all the melons having $10 %$ greater mass than the model described above.

Find the probability that a melon selected at random will have a mass greater than $3.0 kg$ .

[2]

both have a mass greater than $3.0 kg$ .

[2]

b.i.

have a total mass greater than $6.0 kg$ .

[2]

b.ii.

Write down the null and alternative hypotheses for the test.

[1]

Find the critical region for this test.

[4]

Find the mean and standard deviation of the mass of the melons for this year.

[3]

Find the probability of a Type II error in the owner’s test.

[2]

Markscheme

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

Let $X$ represent the mass of a melon

$P (X > 3.0) = 0.212 (0.2118 \dots)$ (M1)A1

[2 marks]

$0.2118 \dots \times 0.2118 \dots$ (M1)

$= 0.0449 (0.04488 \dots)$ A1

[2 marks]

b.i.

Let $T$ represent the total mass

$E (T) = 5.2$ A1

$Var (T) = 0 . 5^{2} + 0 . 5^{2} = 0.5$ (M1)A1

$T ~ N (5.2, 0.5)$

$P (T > 6.0) = 0.129 (0.1289 \dots)$ A1

[4 marks]

b.ii.

Let $μ$ be the mean mass of the melons produced by the farm.

$H_{0} : μ = 2.6 kg$ , $H_{1} : μ > 2.6 kg$ only A1

Note: Accept $H_{0} :$ The mean mass of melons produced by the farm is equal to $2.6 kg$
$H_{1} :$ The mean mass of melons produced by the farm is greater than $2.6 kg$

Note: Award A0 if $2.6 kg$ does not appear in the hypothesis.

[1 mark]

Under $H_{0}$ $\bar{X} ~ N (2.6, \frac{0 . 5^{2}}{16})$ A1

$P (\bar{X} > a) = 0.05$ (M1)

$a = 2.81 (2.805606 \dots)$ (A1)

Critical region is $\bar{X} > 2.81$ A1

[4 marks]

Let $W$ represent the new mass of the melons

$E (W) = 1.1 \times 2.6 = 2.86$ A1

Standard deviation of $W = 1.1 \times 0.5$ (M1)

$= 0.55$ A1

Note: award M1A0 for $Var (W) = 1 . 1^{2} \times 0 . 5^{2} = 0.3025$

[3 marks]

$P (Type II error) =$

$P (\bar{X} < 2.81 μ = 2.86, σ = \frac{0.55}{4})$ (M1)

$= 0.346 (0.346204 \dots)$ A1

Note: Accept $0.358$ from use of the three‐figure answer to part (d)

[2 marks]

Examiners report

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

Syllabus sections

Topic 4—Statistics and probability » SL 4.9—Normal distribution and calculations

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Find a matrix $P$ , with integer elements, such that $T = P D P^{- 1}$ , where $D$ is a diagonal matrix.
21M.2.AHL.TZ2.4d:
Hence, show that the long-term transition matrix $T^{\infty}$ is given by $T^{\infty} = (\begin{matrix} \frac{10}{17} & \frac{10}{17} \\ \frac{7}{17} & \frac{7}{17} \end{matrix})$ .
21M.2.AHL.TZ2.4e:
Hence, or otherwise, determine the expected ratio of the number of patients Doctor Black would have compared to Doctor Green in the long term.
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The sacks of potatoes are transported in crates. There are $10$ sacks in each crate and the weights of the sacks of potatoes are independent of each other.

Find the probability that the total weight of the sacks of potatoes in a crate exceeds $500 kg$ .
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Find the value of $a$ and the value of $b$ , giving your answers correct to one decimal place.
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21N.2.SL.TZ0.5a.i:
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21N.2.SL.TZ0.5a.ii:
a dart lands more than $15 cm$ from $O$ .
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Find the probability that Arianne does not score a point on a turn of three darts.
21N.2.SL.TZ0.5d.ii:
Find the probability that Arianne scores at least $5$ points and less than $8$ points.
21N.2.SL.TZ0.5d.iii:
Given that Arianne scores at least $5$ points, find the probability that Arianne scores less than $8$ points.
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a dart lands less than $13 cm$ from $O$ .
21N.2.AHL.TZ0.3a.ii:
a dart lands more than $15 cm$ from $O$ .
21N.2.AHL.TZ0.3c:
Find the probability that Arianne does not score a point on a turn of three darts.
21N.2.AHL.TZ0.3d.i:
Find Arianne’s expected score in the competition.
21N.2.AHL.TZ0.3d.ii:
Find the probability that Arianne scores at least $5$ points in the competition.
21N.2.AHL.TZ0.3d.iii:
Find the probability that Arianne scores at least $5$ points and less than $8$ points.

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Topic 4—Statistics and probability

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