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Date	May 2022	Marks available	2	Reference code	22M.2.AHL.TZ1.3
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	3	Adapted from	N/A

Question

A Principal would like to compare the students in his school with a national standard. He decides to give a test to eight students made up of four boys and four girls. One of the teachers offers to find the volunteers from his class.

The marks out of $40$ , for the students who took the test, are:

$25, 29, 38, 37, 12, 18, 27, 31 .$

For the eight students find

The national standard mark is $25.2$ out of $40$ .

Two additional students take the test at a later date and the mean mark for all ten students is $28.1$ and the standard deviation is $8.4$ .

For further analysis, a standardized score out of $100$ for the ten students is obtained by multiplying the scores by $2$ and adding $20$ .

For the ten students, find

Name the type of sampling that best describes the method used by the Principal.

[1]

the mean mark.

[2]

b.i.

the standard deviation of the marks.

[1]

b.ii.

Perform an appropriate test at the $5 %$ significance level to see if the mean marks achieved by the students in the school are higher than the national standard. It can be assumed that the marks come from a normal population.

[5]

State one reason why the test might not be valid.

[1]

their mean standardized score.

[1]

e.i.

the standard deviation of their standardized score.

[2]

e.ii.

Markscheme

quota A1

[1 mark]

$27.125 \approx 27.1$ (M1)A1

[2 marks]

b.i.

$8.29815 \dots \approx 8.30$ A1

[1 mark]

b.ii.

(let $μ$ be the national mean)

$H_{0} : μ = 25.2$

$H_{1} : μ > 25.2$ A1

Note: Accept hypotheses in words if they are clearly expressed and ‘population mean’ or ‘school mean’ is referred to. Do not accept $H_{0} : μ = μ_{0}$ unless $μ_{0}$ is explicitly defined as “national standard mark” or given as $25.2$ .

recognizing $t$ -test (M1)

$p$ -value $= 0.279391 \dots$ A1

$0.279391 \dots > 0.05$ R1

Note: The R1 mark is for the comparison of their $p$ -value with $0.05$ .

insufficient evidence to reject the null hypothesis (that the mean for the school is $25.2$ ) A1

Note: Award the final A1 only if the null hypothesis is also correct (e.g. $μ_{0} = 25.2$ or (population) mean $= 25.2$ ) and the conclusion is consistent with both the direction of the inequality and the alternative hypothesis.

[5 marks]

EITHER

the sampling process is not random R1

For example:

the school asked for volunteers

the students were selected from a single class

the quota might not be representative of the student population R1

For example:

the school may have only $4$ boys and $400$ girls.

Note: Do not accept ‘the sample is too small’.

[1 mark]

$(28.1 \times 2 + 20 =) 76.2$ A1

[1 mark]

e.i.

$8.4 \times 2$ (A1)

$= 16.8$ A1

[2 marks]

e.ii.

Examiners report

The most common answer to this question was ‘convenience sampling’. Though it is a convenience sample because four boys and four girls were required the most appropriate response was ‘quota sampling’.

Some candidates still try to calculate a mean and standard deviation by hand. This is not expected.

b.i.

[N/A]

b.ii.

This was surprisingly poorly answered given that statistical testing forms a large part of the course. Candidates need to give the null and alternative hypotheses, find a p-value, compare this to the significance level and write their conclusion, in context of the question; examination questions may ask for each element individually or the question may say “Perform the test” wherein it is expected that each individual element will be clearly stated (as the test is incomplete if any are omitted). Many candidates had the null hypothesis as an inequality. The easiest way to write the null hypothesis is $H_{0} : μ=25.2$ , but it could also be stated in words so long as it is clear that the population mean is being referred to rather than the sample mean. For example, H₀: The mean score of the whole school is equal to 25.2.

The answer that the sample was self-selecting or unrepresentative was the expected response. The sample being small was also accepted if the additional reason of therefore ‘not able to assume a normal population’ was also given. In general, a small sample can be valid (though will probably not be reliable).

Some candidates missed the point of this question, that it was concerned with transformations of the mean and standard deviation, and instead tried to work out the actual values for the extra two candidates.

e.i.

[N/A]

e.ii.

Syllabus sections

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

Show 143 related questions

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Draw a box-and-whisker diagram, for these students’ ages, on the following grid.
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22M.2.SL.TZ2.1a.ii:
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22M.2.SL.TZ2.1c:
Determine the $90 th$ percentile of the reaction times from the cumulative frequency graph.
22M.2.SL.TZ2.1d.i:
Write down the value of $a$ .
22M.2.SL.TZ2.1e:
Write down the modal class from the table.
19M.2.SL.TZ1.T_3i:
State the conclusion for this test. Give a reason for your answer.
22M.2.SL.TZ1.3a.i:
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22M.1.SL.TZ2.7b:
With reference to the box and whisker diagrams, state one aspect that may support the researcher’s opinion and one aspect that may counter it.
21N.1.AHL.TZ0.14a:
Find the probability that a sack is under its labelled weight.
21N.1.AHL.TZ0.14b:
Find the lower quartile of the weights of the sacks of potatoes.
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Find μ, the expected value of X.
18N.2.AHL.TZ0.H_10a.ii:
Using this distribution model, find the standard deviation of $X$ .
18N.2.AHL.TZ0.H_10b.i:
an estimate for the mean number of emails received per working day.
18M.2.SL.TZ2.T_2f:
Find the value of m.
18M.1.SL.TZ1.S_2a:
Find the value of the interquartile range.
18M.2.SL.TZ1.T_2d.ii:
Write down the associated p-value.
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Write down, for $N$ , the mid-interval value of the modal class.
18M.2.SL.TZ2.T_2d:
Write down the percentage of buses that travelled a distance greater than the upper quartile.
16N.2.SL.TZ0.T_1h:
Write down a reason why this estimate is not reliable.
17M.2.SL.TZ1.T_5a.ii:
Calculate the standard deviation.
17M.2.SL.TZ1.T_5b:
Find the median test grade of the students.
19M.2.SL.TZ1.T_3f:
Write down the number of degrees of freedom for this test.
19M.1.SL.TZ2.T_14a.i:
On the following diagram, shade the region representing the probability that the price of a kilogram of tomatoes, chosen at random, will be higher than 3.22 euro.
19M.1.SL.TZ2.T_14a.ii:
Find the price that is two standard deviations above the mean price.
19M.1.SL.TZ2.T_14c:
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Find the highest price that a seller can charge and still receive a free permit.
19M.1.SL.TZ2.T_6c:
The health inspector visits two school cafeterias. She inspects the same number of meals at each cafeteria. The data is shown in the following box-and-whisker diagrams.

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State, giving a reason, which school cafeteria has more meals that do not meet the requirement.
17N.1.SL.TZ0.T_1a.ii:
For the students in this group write down the median age.
19M.2.SL.TZ1.T_3a:
Write down the total number of people, from this group, who are pet owners.
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Find P(X > μ).
19M.2.SL.TZ1.T_3b:
Write down the modal number of pets.
SPM.1.SL.TZ0.3b:
Find the value of $k$ .
17M.1.SL.TZ2.T_7c:
Find the value of $x$ and of $y$ .
18M.2.SL.TZ1.T_2d.i:
Write down the χ² statistic.
18M.2.SL.TZ2.T_2b.ii:
Use the cumulative frequency curve to find the lower quartile.
18M.2.SL.TZ1.T_2a:
State the alternative hypothesis.
17N.2.SL.TZ0.T_1c.i:
Use your graphic display calculator to estimate the mean of $N$ ;
17N.2.SL.TZ0.T_1f.ii:
Write down the ${\chi ^2}$ statistic.
17M.1.SL.TZ2.S_8a.ii:
Write down the number of employees who worked 50 hours or less.
19M.2.SL.TZ1.T_3g:
Calculate the expected number of teenagers that prefer cats.
18M.2.SL.TZ1.T_2c:
Write down the number of degrees of freedom.
18N.2.AHL.TZ0.H_10a.i:
Using this distribution model, find ${\text{P}}\left( {X < 60} \right)$ .
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.
19M.2.SL.TZ1.T_3c.iii:
For these data, write down the upper quartile.
19M.2.SL.TZ1.T_3d:
Write down the ratio of teenagers to non-teenagers in its simplest form.
16N.2.SL.TZ0.T_1f:
Draw the regression line, from part (e), on your scatter diagram.
18M.1.SL.TZ1.S_2b:
One student sent k text messages, where k > 11 . Given that k is an outlier, find the least value of k.
17M.2.SL.TZ1.T_5c:
Find the interquartile range.
17N.2.SL.TZ0.T_1e.ii:
Write down the number of degrees of freedom.
17N.2.SL.TZ0.T_1b.i:
Write down, for $N$ , the modal class;
18N.2.AHL.TZ0.H_10e:
Now suppose that Archie received exactly 20 emails in total in a consecutive two day period. Show that the probability that he received exactly 10 of them on the first day is independent of λ.
17N.2.SL.TZ0.T_1a:
State whether $N$ is a discrete or a continuous variable.
17N.2.SL.TZ0.T_1c.ii:
Use your graphic display calculator to estimate the standard deviation of $N$ .
19N.1.SL.TZ0.T_13a:
State whether the weight of the eggs is a continuous or discrete variable.
18N.2.AHL.TZ0.H_10c:
Give one piece of evidence that suggests Willow’s Poisson distribution model is not a good fit.
17M.1.SL.TZ2.T_7a:
Write down an equation, in terms of $x$ and $y$ , for the total number of times the die was rolled.
SPM.1.SL.TZ0.3a:
State whether the data is discrete or continuous.
18M.1.SL.TZ1.T_6c.i:
Write down the mid-interval value for the 100 < x ≤ 150 group.
16N.2.SL.TZ0.S_8b:
During week 2, the students worked on a major project and they each spent an additional five hours browsing the Internet. For week 2, write down

(i) the mean;

(ii) the standard deviation.
18M.1.SL.TZ2.T_12a:
Write down the mid-interval value for 10 ≤ t < 15.
18M.1.SL.TZ2.T_12b.i:
Write down the total number of customers in terms of k.
19M.2.SL.TZ1.T_3e.ii:
State the alternative hypothesis.
16N.2.AHL.TZ0.H_11d:
(i) Hence show that $X$ has two modes ${m_1}$ and ${m_2}$ .

(ii) State the values of ${m_1}$ and ${m_2}$ .
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.
17N.2.SL.TZ0.T_1f.i:
Write down the $p$ -value for the test;
17N.2.SL.TZ0.T_1e.i:
Write down the null hypothesis for this test;
16N.2.AHL.TZ0.H_11b:
Find the values of the constants $a$ and $b$ .
16N.2.AHL.TZ0.H_11a:
Show that ${\text{P}}(X = 3) = 0.001$ and ${\text{P}}(X = 4) = 0.0027$ .
17N.2.SL.TZ0.T_1g:
State the result of the test. Give a reason for your answer.
18M.2.SL.TZ2.T_2c:
Hence write down the interquartile range.
17M.1.SL.TZ2.T_7b:
Using the mean score, write down a second equation in terms of $x$ and $y$ .
18M.2.SL.TZ2.T_2b.i:
Use the cumulative frequency curve to find the median distance.
18M.2.SL.TZ2.T_2b.iii:
Use the cumulative frequency curve to find the upper quartile.
20N.1.SL.TZ0.S_8a:
Find the value of $a$ .
20N.2.SL.TZ0.S_9d:
Find the probability that Fiona will arrive on time.
20N.1.SL.TZ0.T_3b:
Use your graphic display calculator to find an estimate of the standard deviation of the weights of mangoes from this harvest.
20N.1.SL.TZ0.T_3c:
On the grid below, draw a histogram for the data in the table.
20N.1.SL.TZ0.T_7b:
Using this information, write down an equation in $p$ and $q$ .
20N.1.SL.TZ0.T_7d.ii:
Using your answers to parts (b) and (c), find the height of Flower $D$ .
17N.2.SL.TZ0.T_1d:
Find the expected frequency of students choosing the Science category and obtaining 31 to 40 correct answers.
20N.1.SL.TZ0.S_8b:
Write down the value of the median distance in kilometres (km).
20N.1.SL.TZ0.S_8d:
Find $m$ .
20N.2.SL.TZ0.S_9c:
Find the probability that the bus journey takes less than $45$ minutes.
20N.2.SL.TZ0.S_9e:
This year, Fiona will go to school on $183$ days.

Calculate the number of days Fiona is expected to arrive on time.
16N.1.SL.TZ0.T_2b:
Draw a box-and-whisker diagram on the grid below to represent the Vitamin C content, in milligrams, for this sample.
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.
SPM.1.SL.TZ0.3c:
It was not possible to ask every person in the school, so the Headmaster arranged the student names in alphabetical order and then asked every 10th person on the list.

Identify the sampling technique used in the survey.
21M.1.SL.TZ1.2b:
the mean.
19M.2.SL.TZ1.T_3e.i:
State the null hypothesis.
16N.2.SL.TZ0.S_8a:
Find the mean number of hours spent browsing the Internet.
21M.1.SL.TZ1.2a:
the mode.
21M.1.SL.TZ1.2c:
the standard deviation.
16N.2.AHL.TZ0.H_11e:
Determine the minimum value of $x$ such that the probability Kati receives at least one free gift is greater than 0.5.
18M.2.SL.TZ1.T_2e:
State, with a reason, whether you would reject the null hypothesis.
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_5f.ii:
Calculate the expected number of students that spent at least 90 minutes preparing for the test.
20N.2.SL.TZ0.S_9b:
Find $σ$ .
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_3c.i:
For these data, write down the median number of pets.
19M.2.SL.TZ1.T_3c.ii:
For these data, write down the lower quartile.
20N.1.SL.TZ0.S_8c:
Find the value of $b$ .
16N.2.SL.TZ0.T_1c:
Plot the point $(\bar x,{\text{ }}\bar y)$ on your scatter diagram and label this point M.
20N.1.SL.TZ0.T_3a:
Write down the modal group for these data.
20N.1.SL.TZ0.S_8e:
The first $150$ athletes that completed the race won a prize.

Given that an athlete took between $22$ and $m$ minutes to complete the $5 km$ race, calculate the probability that they won a prize.
17M.2.SL.TZ1.T_5a.i:
Calculate the mean test grade of the students;
17M.2.SL.TZ1.T_5d:
Find the probability that this student scored a grade 5 or higher.
18M.2.SL.TZ1.T_2b:
Calculate the expected frequency of flights travelling at most 500 km and arriving slightly delayed.
18M.1.AHL.TZ2.H_3a:
Find the value of p.
16N.2.SL.TZ0.S_8d:
(i) Find the number of students who spent between 25 and 30 hours browsing the Internet.

(ii) Given that 10% of the students spent more than k hours browsing the Internet, find the maximum value of $k$ .
16N.2.SL.TZ0.T_1e:
Write down the equation of the regression line $y$ on $x$ for these eight male students.
18M.2.SL.TZ1.T_2f:
Write down the probability that this flight arrived on time.
18M.2.SL.TZ1.T_2h:
Two flights are chosen at random from those which were slightly delayed.

Find the probability that each of these flights travelled at least 5000 km.
16N.2.AHL.TZ0.H_11c:
Deduce that $\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}} = \frac{{0.9(n - 1)}}{{n - 3}}$ for $n > 3$ .
20N.1.SL.TZ0.T_7c:
Write down a second equation in $p$ and $q$ .
19N.1.SL.TZ0.T_13d:
The mean weight of these eggs is 64.9 grams, correct to three significant figures.

Use the table and your answer to part (c) to find the smallest possible number of eggs that could be within one standard deviation of the mean.
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.
20N.2.SL.TZ0.S_9a:
Find the probability that it will take Fiona between $15$ minutes and $30$ minutes to walk to the bus stop.
16N.2.SL.TZ0.S_8c:
During week 3 each student spent 5% less time browsing the Internet than during week 1. For week 3, find

(i) the median;

(ii) the variance.
21N.1.SL.TZ0.9c.i:
Find the number of students who obtained a grade $5$ .
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.1.SL.TZ0.9a:
Find the median of the scores obtained.
21N.1.SL.TZ0.9b:
Find an expression for $a$ in terms of $b$ .
21N.1.SL.TZ0.9c.ii:
Find the minimum score needed to obtain a grade $5$ .
20N.1.SL.TZ0.T_7a:
Find the height of Flower $B$ .

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Topic 4—Statistics and probability

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