State what is meant by a 'population' in the context of statistical sampling.
(Total for Question 1 is 1 mark)
2
State what is meant by a 'sample'.
(Total for Question 2 is 1 mark)
3
State what is meant by a 'sampling frame'.
(Total for Question 3 is 1 mark)
4
A quality inspector picks a random starting point on a production line and then selects every 20th item after that. State the name of this sampling method.
(Total for Question 4 is 1 mark)
5
A researcher stands outside a supermarket and interviews the first 40 shoppers who agree to stop and answer her questions. State the name of this sampling method.
(Total for Question 5 is 1 mark)
6
A local newspaper prints a coupon and asks readers to cut it out, fill it in, and post it back if they want to take part in a survey. State the name of this sampling method.
(Total for Question 6 is 1 mark)
7
A factory produces 464 identical components during a shift. A systematic sample of 8 components is required. Calculate the sampling interval that should be used.
(Total for Question 7 is 2 marks)
8
A college has 936 students enrolled. A systematic sample of 24 students is required. Calculate the sampling interval that should be used.
(Total for Question 8 is 2 marks)
9
A company has 250 staff: 150 full-time and 100 part-time. A stratified sample of 20 staff, proportional to employment type, is required. Calculate the number of full-time staff that should be included in the sample.
(Total for Question 9 is 2 marks)
10
A hospital has 320 patients across three wards: Ward A has 140 patients, Ward B has 100 patients and Ward C has 80 patients. A stratified sample of 40 patients, proportional to ward size, is required. Calculate the number of patients that should be sampled from Ward C.
(Total for Question 10 is 2 marks)
11
A cinema chain has three branches with 900, 600 and 300 regular customers respectively. A stratified sample of 60 customers, proportional to branch size, is required. Calculate the number of customers that should be sampled from the branch with 600 regular customers.
(Total for Question 11 is 2 marks)
12
In a certain town, 45% of residents are male and 55% are female. A market researcher plans to interview a quota sample of 80 residents, in these proportions. Calculate the number of male residents she should aim to interview.
(Total for Question 12 is 2 marks)
13
A stable has 60 sheep, numbered 00 to 59. To select a simple random sample, two-digit random numbers were generated in this order: 07, 63, 42, 91, 18, 29. Any number greater than 59 must be rejected. Identify, in order, the first four valid sheep numbers selected.
(Total for Question 13 is 2 marks)
14
A company has 640 staff across two sites: Site A has 400 staff and Site B has 240 staff. A stratified sample of 40 staff, proportional to site size, is required. Of the staff sampled from Site A, 60% work in Operations and the rest work in Support. Find the number of Operations staff at Site A that should be included in the sample.
(Total for Question 14 is 3 marks)
15
A company with 2000 employees at several sites wants to estimate average commuting time. A researcher instead surveys only the 50 employees who work in the head office building, since they are the easiest for her to reach. Identify the sampling method used, and explain why the resulting sample could give a biased estimate of the average commuting time for all 2000 employees.
(Total for Question 15 is 3 marks)
16
A school has 1200 students across three year groups: Year 9 has 440 students, Year 10 has 400 students and Year 11 has 360 students. A stratified sample of 60 students, proportional to year group size, is required. Calculate the number of Year 10 students that should be included in the sample, and describe how the individual Year 10 students should then be chosen so that they form a simple random sample within that year group.
(Total for Question 16 is 3 marks)
17
A conveyor belt produces 980 identical parts during a shift. A systematic sample of 30 parts is required. Calculate the sampling interval, to the nearest whole number, and hence find how many parts can actually be sampled using this interval before the shift ends.
(Total for Question 17 is 3 marks)
18
A charity has 730 volunteers across four regions: North (210), South (180), East (160) and West (180). A stratified sample of 50 volunteers, proportional to region size, is required, with each region's value rounded to the nearest whole number. Calculate the number of volunteers that should be sampled from each region, and explain why the four rounded values might not sum to exactly 50.
(Total for Question 18 is 4 marks)
19
A university with 24000 students wants to estimate the proportion of students who cycle to campus. Method 1 selects a simple random sample of 400 students from the central enrolment database (which lists every student) and emails them a survey. Method 2 involves standing at the three main bike racks on campus for one morning and interviewing everyone who arrives by bike, until 400 people have been interviewed. Identify one source of bias in Method 2 that Method 1 does not share, and one practical problem with Method 1 that Method 2 does not share.
(Total for Question 19 is 4 marks)
Mark scheme · S1D Statistics: Sampling: Fluency and Exam Drill
Question 1
B1 the whole set of individuals or items that information is wanted about, oe
Answer: The whole set of individuals or items about which information is wanted.
Question 2
B1 a subset of the population, selected for study, oe
Answer: A subset of the population that is actually selected and studied.
Question 3
B1 a list of all the members of the population, from which the sample is selected, oe
Answer: A list of all the members of the population, from which the sample is selected.
M1 correctly rejects 63 and 91 as outside the range 00 to 59
A1 cao: sheep 07, 42, 18 and 29 selected, in this order
Answer: Sheep 07, 42, 18 and 29.
Question 14
M1 400 / 640 x 40 (= 25, the Site A sample size)
M1 0.6 x 25 (method to find the Operations count)
A1 15 cao
Answer: 15
Question 15
B1 opportunity sampling (convenience sampling)
B1 identifies that head office employees may differ systematically from staff at other sites, e.g. tending to live nearer that particular location
B1 explains the consequence: commuting times at head office may not be typical of the whole company, so the sample is unlikely to be representative and the estimate could be biased, oe
Answer: Opportunity sampling. Head office staff may live closer to that one site than staff based elsewhere, so their commuting times are unlikely to be typical of the whole company; the estimate from this sample is therefore likely to be biased.
Question 16
M1 400 / 1200 x 60
A1 20 cao
B1 use simple random sampling (e.g. a random number generator) to choose 20 students from within Year 10, so every Year 10 student has an equal chance of selection
Answer: 20 students; select them using simple random sampling (e.g. a random number generator) from within Year 10, so every Year 10 student has an equal chance of being chosen.
Question 17
M1 980 / 30 (= 32.67), rounded to interval 33
M1 980 / 33 (= 29.7)
A1 cao: only 29 parts can be sampled, since 33 x 30 = 990 > 980
Answer: Interval = 33; only 29 parts can be sampled using this interval, since 33 x 30 = 990, which is greater than 980.
Question 18
M1 calculates at least one stratum using (region size / 730) x 50
A1 North 14, South 12, East 11, West 12 cao
B1 identifies that these four values sum to 49, not 50
B1 explains that rounding each stratum independently does not guarantee the rounded values sum to the required total, so one value (the one whose exact figure sits closest to the next rounding boundary, here North at 14.38) must be adjusted upward to restore a total of 50, oe
Answer: North 14, South 12, East 11, West 12 (total 49, not 50); since each value was rounded independently, the totals need not sum to 50 - the value closest to the next rounding boundary (North, 14.38) should be adjusted up to 15 to restore the total, giving final sample sizes North 15, South 12, East 11, West 12.
Question 19
B1 Method 2 only samples people who are already cycling to campus that morning (an opportunity/self-selected sample of cyclists), so non-cyclists have no chance of selection
B1 this would make the estimated proportion who cycle far too high, since (close to) everyone sampled by Method 2 cycles, unlike the true population of all 24000 students
B1 Method 1 is likely to suffer from non-response, since not every emailed student will reply
B1 if the students who choose to respond differ systematically from those who do not (e.g. cyclists feeling more strongly about the topic and being more likely to reply), this introduces non-response bias not present in Method 2's in-person approach, oe
Answer: Method 2 samples only people already cycling that morning, so it will greatly overestimate the true proportion of cyclists among all 24000 students. Method 1 instead relies on email responses, so it is vulnerable to non-response bias if the type of student who replies differs from those who do not - a problem Method 2, which interviews people in person, does not have.