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Date	May Example question	Marks available	3	Reference code	EXM.1.AHL.TZ0.59
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 0
Command term	Calculate	Question number	59	Adapted from	N/A

Question

The heights, $x$ $x$ metres, of the 241 new entrants to a men’s college were measured and the following statistics calculated.

$\sum x = 412.11, \sum x^{2} = 705.5721$ $\sum {x = 412.11,\,\,\sum {{x^2} = 705.5721} }$

The Head of Mathematics decided to use a $χ^{2}$ ${\chi ^2}$ test to determine whether or not these heights could be modelled by a normal distribution. He therefore divided the data into classes as follows.

Calculate unbiased estimates of the population mean and the population variance.

[3]

a.

State suitable hypotheses.

[1]

b.i.

Calculate the value of the $χ^{2}$ ${\chi ^2}$ statistic and state your conclusion using a 10% level of significance.

[11]

b.ii.

Markscheme

$¯ x = \frac{412.11}{241} = 1.71$ $\bar x = \frac{{412.11}}{{241}} = 1.71$ A1

$s^{2} = \frac{705.5721}{240} - \frac{{412.11}^{2}}{240 \times 241} = 0.0036$ ${s^2} = \frac{{705.5721}}{{240}} - \frac{{{{412.11}^2}}}{{240 \times 241}} = 0.0036$ M1A1

[3 marks]

a.

H₀: Data can be modelled by a normal distribution

H₁: Data cannot be modelled by a normal distribution A1

[1 mark]

b.i.

The expected frequencies are

A1A1A1A1A1A1

$χ^{2} = \frac{5^{2}}{8.04} + \frac{34^{2}}{30.19} + \dots + \frac{12^{2}}{16.10} - 241 = 3.30 / 3.29$ ${\chi ^2} = \frac{{{5^2}}}{{8.04}} + \frac{{{{34}^2}}}{{30.19}} + \ldots + \frac{{{{12}^2}}}{{16.10}} - 241 = 3.30\,{\text{/}}\,3.29$ M1A1

Degrees of freedom = 3 A1

Critical value = 6.251 or p-value = 0.35 A1

The data can be modelled by a normal distribution. R1

[11 marks]

b.ii.

Examiners report

[N/A]

a.

[N/A]

b.i.

[N/A]

b.ii.

Syllabus sections

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

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Draw a diagram that shows this information.
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Find the mean and standard deviation of the caffeine content of coffee.
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Sketch a diagram to represent this information.
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Find the value of $x$ $x$ .
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${\text{P}}\left( {X < \mu + 5\,\left| {\,X > \mu - 5} \right.} \right)$ .
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State a conclusion for this test. Justify your answer.
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Find the probability that an apple from the tree has a weight greater than $90$ grams.
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Given that $p = 0.2$ find the probability of making a Type II error.
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Calculate the probability that a student chosen at random spent at least 90 minutes preparing for the test.
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Find a prediction for the ratio of the number of patients Doctor Black will have, compared to Doctor Green, after two years.
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Find the probability that a bag selected at random weighs more than $1 kg$ .
21M.2.SL.TZ1.4b.ii:
$6.75 m$ and $7 m$ .
21M.1.AHL.TZ1.14b:
Find the standard deviation of the weights of these bags of apples.
21M.2.SL.TZ2.4d:
It is found that $12$ of the cats weigh more than $x kg$ . Estimate the value of $x$ .
21M.2.AHL.TZ2.4c:
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.
21N.1.AHL.TZ0.14c:
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 $500 kg$ .
21N.2.SL.TZ0.5d.i:
Find the probability that Arianne scores at least $5$ points in the competition.
EXN.2.AHL.TZ0.3b:
Find the value of $a$ and the value of $b$ , giving your answers correct to one decimal place.
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Given that Arianne scores at least $5$ points, find the probability that Arianne scores less than $8$ points.
21N.2.SL.TZ0.5a.i:
a dart lands less than $13 cm$ from $O$ .
21N.2.SL.TZ0.5a.ii:
a dart lands more than $15 cm$ from $O$ .
21N.2.SL.TZ0.5c:
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.
21N.2.AHL.TZ0.3a.i:
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.
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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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