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Date	May 2022	Marks available	2	Reference code	22M.2.SL.TZ1.2
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	2	Adapted from	N/A

Question

The number of hours spent exercising each week by a group of students is shown in the following table.

The median is $4.5$ hours.

Find the value of $x$ .

[2]

Find the standard deviation.

[2]

Markscheme

EITHER

recognising that half the total frequency is $10$ (may be seen in an ordered list or indicated on the frequency table) (A1)

$5 + 1 + 4 = 3 + x$ (A1)

$\sum f = 20$ (A1)

THEN

$x = 7$ A1

[2 marks]

METHOD 1

$1.58429 \dots$

$1.58$ A2

METHOD 2

EITHER

$σ^{2} = \frac{5 \times {(2 - 4.3)}^{2} + 1 \times {(3 - 4.3)}^{2} + 4 \times {(4 - 4.3)}^{2} + 3 \times {(5 - 4.3)}^{2} + 7 \times {(6 - 4.3)}^{2}}{20} (= 2.51)$ (A1)

$σ^{2} = \frac{5 \times 2^{2} + 1 \times 3^{2} + 4 \times 4^{2} + 3 \times 5^{2} + 7 \times 6^{2}}{20} - 4 . 3^{2} (= 2.51)$ (A1)

THEN

$σ = \sqrt{2.51} = 1.58429 \dots$

$= 1.58$ A1

[2 marks]

Examiners report

Most candidates attempted both parts, with varying levels of success, particularly in part (b).

In part (a), the most successful approach seen was from candidates who made an ordered list to visualize the given data set, which enabled them to recognise either the number of sixes required for the median to lie at 4.5, or the total frequency. The most common error was to mistake the median for the mean, which led to a non-integer value of $x = 9.67$ .

Part (b) proved to be more challenging, with many candidates either not taking into account the frequency of the exercise time when generating the summary statistics or treating frequency as an additional variable and using two-variable statistics on their GDC. With both, this led to $σ = 1.41$ being the most common wrong answer seen. A few candidates gave the sample standard deviation rather than the population standard deviation. A number of candidates attempted to use the standard deviation formula but were usually not successful. This formula is not in the course, although it can be obtained in the HL section of the formula booklet.

[N/A]

Syllabus sections

Topic 4—Statistics and probability » SL 4.3—Mean, median, mode. Mean of grouped data, standard deviation. Quartiles, IQR

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Write down, for $N$ , the mid-interval value of the modal class.
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State the alternative hypothesis.
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Find the value of x.
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Write down the number of degrees of freedom.
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Calculate the expected number of students that spent at least 90 minutes preparing for the test.
17N.2.SL.TZ0.T_1a:
State whether $N$ is a discrete or a continuous variable.
17N.2.SL.TZ0.T_1f.ii:
Write down the ${\chi ^2}$ statistic.
16N.2.SL.TZ0.T_1d:
Write down the value of $r$ , the Pearson’s product–moment correlation coefficient, for these data.
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Write down the ratio of teenagers to non-teenagers in its simplest form.
17N.1.SL.TZ0.T_1a.i:
For the students in this group find the mean age;
20N.1.SL.TZ0.T_7d.i:
Using your answers to parts (b) and (c), find the height of Flower $A$ .
19M.2.SL.TZ1.T_3g:
Calculate the expected number of teenagers that prefer cats.
17M.2.SL.TZ1.T_5a.i:
Calculate the mean test grade of the students;
20N.1.SL.TZ0.T_7a:
Find the height of Flower null.
16N.1.SL.TZ0.T_2a:
Giving your answer to one decimal place, write down the value of

(i) the median level of Vitamin C content of the oranges in the sample;

(ii) the lower quartile;

(iii) the upper quartile.
17M.2.SL.TZ1.T_5e:
Given that the first student chosen at random scored a grade 5 or higher, find the probability that both students scored a grade 6.
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Find the probability that this student scored a grade 5 or higher.
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Write down the probability that this flight arrived on time.
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Use your graphic display calculator to estimate the standard deviation of $N$ .
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Find the expected frequency of students choosing the Science category and obtaining 31 to 40 correct answers.
17N.2.SL.TZ0.T_1e.i:
Write down the null hypothesis for this test;
17N.2.SL.TZ0.T_1e.ii:
Write down the number of degrees of freedom.
19M.2.SL.TZ1.T_3h:
Use your graphic display calculator to find the $p$ -value for this test.
18N.2.SL.TZ0.T_3a.ii:
Find the interquartile range.
18N.2.SL.TZ0.T_3c.i:
Write down the modal class.
18N.2.SL.TZ0.T_3c.ii:
Write down the mid-interval value of the modal class.
18N.2.SL.TZ0.T_3e:
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Use the cumulative frequency graph to estimate the number of students whose final examination result was below this grade boundary.
18M.2.SL.TZ2.T_2a:
Find the number of buses that travelled a distance between 15000 and 20000 kilometres.
16N.2.SL.TZ0.T_1a:
On graph paper, draw a scatter diagram for these data. Use a scale of 2 cm to represent 5 hours on the $x$ -axis and 2 cm to represent 10 points on the $y$ -axis.
16N.2.SL.TZ0.T_1b:
(i) ${\bar x}$ , the mean number of hours spent on social media;

(ii) ${\bar y}$ , the mean number of IB Diploma points.
16N.2.SL.TZ0.T_1f:
Draw the regression line, from part (e), on your scatter diagram.
16N.2.SL.TZ0.T_1g:
Use the given equation of the regression line to estimate the number of IB Diploma points that this girl obtained.
17M.2.SL.TZ1.T_5a.ii:
Calculate the standard deviation.
17M.2.SL.TZ1.T_5c:
Find the interquartile range.
17M.2.SL.TZ1.T_5f.i:
Calculate the probability that a student chosen at random spent at least 90 minutes preparing for the test.
19M.2.SL.TZ1.T_3i:
State the conclusion for this test. Give a reason for your answer.

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