4.1 Collecting data and sampling questions for IB Maths AI

32 questions on Collecting data and sampling, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Applications and Interpretation
Part of 4 Statistics and probability
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	13	1	14
Paper 2	13	4	17
Paper 3HL only	0	1	1
Every paper	26	6	32

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AI-00190Paper 1 · 6 marks

The Ridgeway Cycling Club has 340 members. The club secretary, Tomasz, emails an online questionnaire to every member and 45 members reply. The questionnaire records, for each member who replies:

the number of punctures they had last year
the distance they cycled last week, in km
their time, in minutes, for the club's 10 km time trial.

(a)
For each of the three variables listed, state whether the data are discrete or continuous.
3 marks
(b)
Identify the population and the sample in this survey.
2 marks
(c)
Give one reason why the 45 replies may not be representative of the whole club.
1 mark

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Question 2 · AI-00198Paper 2 · 18 marks

Lindiwe's farm has three apple orchards. The numbers of apple trees in each orchard are shown in the following table.

Orchard	Hillside	Riverbank	Valley
Number of trees	1840	1150	1010

Lindiwe wants to check 80 trees for disease.

(a)
She decides to use a stratified sample. Find the number of trees she should check in each orchard.
3 marks
(b)
Explain why a stratified sample may be more suitable than a simple random sample of 80 trees.
1 mark
(c)
Lindiwe also weighs apples. State whether the mass of an apple is discrete or continuous data.
1 mark
(d)
The masses, in grams, of 12 apples picked from Riverbank are:
$142, 156, 161, 148, 170, 152, 165, 139, 158, 147, 163, 236$
Find
(i) the mean mass;
(ii) the standard deviation of the masses.
3 marks
(e)
(i) Find the lower quartile, the median and the upper quartile of the masses.
(ii) Show that the mass of 236 g is an outlier.
6 marks
(f)
Draw a box-and-whisker diagram to represent the masses of the 12 apples, using a horizontal scale from 120 g to 240 g.
3 marks
(g)
Lindiwe checks her notebook and confirms that the 236 g apple was weighed correctly. State, with a reason, whether she should remove it from the data.
1 mark

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Question 3 · AI-01643Paper 3 · HL only · 27 marks

The purpose of this question is to plan and analyse a survey of the lengths of trout in two lakes.

Aroha is a fisheries scientist. She is planning a survey of rainbow trout in Lake Moana. Previous surveys show that the lengths of trout in the lake have a standard deviation of 6.2 cm, and that the distribution of lengths is positively skewed. You may assume that this standard deviation has not changed.

The lake has three regions. Of the trout in the lake, 45 % live in the northern region, 30 % in the central region and 25 % in the southern region.

(a)
Aroha will measure a stratified sample of 60 trout, stratified by region.
Find the number of trout she should measure from each region.
2 marks
(b)
Let $\overset{ˉ}{L}$ be the mean length, in cm, of a random sample of 60 trout from the lake, and let $μ$ be the mean length of all the trout in the lake.
Explain why $\overset{ˉ}{L}$ can be modelled by a normal distribution, and state this distribution in terms of $μ$ .
2 marks
(c)
Find the probability that $\overset{ˉ}{L}$ is within 1.5 cm of $μ$ .
2 marks
(d)
Find the smallest sample size for which the probability that the sample mean is within 1.5 cm of $μ$ is at least 0.98.
3 marks
(e)
Aroha measures a random sample of 60 trout. Their mean length is 34.6 cm.
Find a 95 % confidence interval for $μ$ .
2 marks
(f)
Ten years earlier, the mean length of trout in Lake Moana was 32.5 cm.
Comment on whether Aroha's interval suggests that the mean length has changed.
2 marks
(g)(i)
Aroha also studies trout in Lake Kōwhai, where the standard deviation of lengths is unknown and the lengths are normally distributed. She measures the lengths, in cm, of a random sample of 10 trout.
Trout
1
2
3
4
5
6
7
8
9
10
Length (cm)
31.2
35.8
28.4
33.1
30.6
36.9
29.7
34.3
32.5
27.9
Find unbiased estimates of the mean and the variance of the lengths of trout in Lake Kōwhai.
2 marks
(g)(ii)
Find a 90 % confidence interval for the mean length of trout in Lake Kōwhai.
2 marks
(g)(iii)
Explain why a $t$ -distribution, and not a normal distribution, is used for this interval.
1 mark
(h)
Trout in Lake Kōwhai can only be fished commercially if their mean length is greater than 30 cm.
Use Aroha's sample to test, at the 5 % significance level, whether the mean length of trout in Lake Kōwhai is greater than 30 cm.
5 marks
(i)(i)
The price, $V$ euros (EUR), paid for a trout from Lake Moana of length $L$ cm is given by $V = 0.9 L - 6$ . Assume that $L$ has mean 34.6 cm and standard deviation 6.2 cm.
Find the mean and the standard deviation of $V$ .
2 marks
(i)(ii)
A box contains 25 trout from Lake Moana, whose lengths are independent.
Find the mean and the standard deviation of the total price paid for the box.
2 marks

Trout	1	2	3	4	5	6	7	8	9	10
Length (cm)	31.2	35.8	28.4	33.1	30.6	36.9	29.7	34.3	32.5	27.9

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Question 4 · AI-00191Paper 1 · 6 marks

Kowhai High School has 1072 students in five year groups, as shown in the following table.

Year group	9	10	11	12	13
Number of students	214	198	247	231	182

The principal, Mrs Ngata, wants to survey 60 students about the school's new timetable. She decides to use a stratified sample, with year group as the strata.

(a)
Find the number of students that should be chosen from each year group.
3 marks
(b)
Describe how the students from Year 11 could then be selected so that every Year 11 student has the same chance of being chosen.
2 marks
(c)
State one advantage of using a stratified sample rather than a simple random sample of 60 students from the whole school.
1 mark

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Question 5 · AI-00199Paper 2 · 14 marks

The council of the town of Dunmore wants to find out how long residents take to travel to work. A council officer stands at the central bus station one morning and asks 50 people waiting there how long their journey to work takes.

(a)
Name the sampling technique used by the officer.
1 mark
(b)
Explain why this sample is likely to be biased.
1 mark
(c)
The council has a list of all 18 400 adult residents. Describe how a simple random sample of 120 residents could be chosen from this list.
2 marks
(d)
The council takes a simple random sample of 120 residents. The time, $t$ minutes, that each takes to travel to work is shown in the following table.
Time ( $t$ minutes)
$0 < t \leq 10$
$10 < t \leq 20$
$20 < t \leq 30$
$30 < t \leq 40$
$40 < t \leq 50$
$50 < t \leq 60$
Number of residents
14
31
38
22
11
4
Write down the modal class.
1 mark
(e)
Use the mid-interval values to find an estimate of
(i) the mean time;
(ii) the standard deviation of the times.
4 marks
(f)
Explain why your answer to part (e)(i) is only an estimate.
1 mark
(g)
Find the class that contains the median time.
2 marks
(h)
The council's target is that no more than 15% of residents travel for more than 40 minutes. Determine whether the sample suggests that the target is being met.
2 marks

Time ( $t$ minutes)	$0 < t \leq 10$	$10 < t \leq 20$	$20 < t \leq 30$	$30 < t \leq 40$	$40 < t \leq 50$	$50 < t \leq 60$
Number of residents	14	31	38	22	11	4

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Question 6 · AI-00192Paper 1 · 6 marks

Amaia wants to estimate the mean time spent on homework each week by the 900 students at her school. She considers four different ways of choosing a sample.

Method A: ask the first 30 students who arrive at the school library one morning.
Method B: take the alphabetical list of all 900 students, choose a random number from 1 to 30, then choose that student and every 30th student after them on the list.
Method C: number the students from 1 to 900 and use a random number generator to choose 30 different numbers.
Method D: ask students in the corridor until she has asked 10 students from each of Grades 10, 11 and 12.

(a)
Name the sampling technique used in each method.
4 marks
(b)
State which method is most likely to overestimate the mean homework time. Give a reason for your answer.
2 marks

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