A bakery makes 2000 loaves of bread each day. The manager wants to check that the loaves contain the right amount of salt. Each loaf tested is crushed and analysed in a laboratory.
State the population in this situation.
Suggest a suitable sampling frame.
Explain why the manager should take a sample rather than a census.
3 marks
Show answer
a) all 2000 loaves made that dayb) a numbered list of the 2000 loaves, e.g. by batch and position in the ovenc) testing destroys the loaf, so a census would leave nothing to sell
2
A leisure centre has 480 members. The manager wants a simple random sample of 30 members to ask about new opening hours.
Describe how the manager could take this sample.
3 marks
Show answer
Number the members 1 to 480, use a random number generator to pick 30 different numbers (ignoring repeats), and ask those members.
3
Name the sampling method used in each case.
A dentist picks a random number between 1 and 20, then contacts that patient on her list and every 20th patient after it.
A reporter stands outside a station at 8 am and interviews the first 25 people who agree to talk to her.
An interviewer is told to question 15 men and 15 women aged over 60, and stops each group when it is full.
3 marks
Show answer
a) systematic samplingb) opportunity (convenience) samplingc) quota sampling
4
The numbers of students in each year group at a secondary school are shown in the table.
Year group
7
8
9
10
11
Number of students
186
174
168
162
150
The head teacher wants a stratified sample of 60 students.
Find the number of students that should be chosen from each year group.
Describe how the students in Year 9 should then be chosen.
4 marks
Show answer
a) Year 7: 13, Year 8: 12, Year 9: 12, Year 10: 12, Year 11: 11b) number the 168 Year 9 students and pick 12 at random using a random number generator
5
An accountant has 600 invoices, numbered 1 to 600, and wants to check a sample of 40 of them.
Describe how she could take a systematic sample of 40 invoices.
Give one advantage of systematic sampling in this situation.
Explain why a systematic sample is not a simple random sample.
4 marks
Show answer
a) k = 600 ÷ 40 = 15; pick a random start from 1 to 15, then every 15th invoice after itb) quick and simple to carry out, and the sample is spread evenly through the invoicesc) once the start is chosen the rest are fixed, so not every set of 40 invoices is equally likely to be chosen
6
A market researcher in a shopping centre is told to interview 40 adults under 30 and 60 adults aged 30 or over. She approaches shoppers until each group is full.
State the name of this sampling method.
Give one advantage and one disadvantage of this method.
Explain what extra information she would need to take a stratified sample instead.
4 marks
Show answer
a) quota samplingb) advantage: quick and cheap with no sampling frame needed; disadvantage: not random, so it can be biased (e.g. only people in the centre at that time)c) a sampling frame: a list of the whole population split into the two age groups, so people can be chosen at random from each
7
Ravi is studying Daily Total Rainfall for Leeming in the large data set.
State the units of Daily Total Rainfall.
Some days are recorded as tr. State what this means.
Explain how Ravi should deal with the tr and n/a entries before he calculates the mean Daily Total Rainfall.
4 marks
Show answer
a) millimetres (mm)b) trace: less than 0.05 mm of rain fellc) replace tr with 0 (or a value such as 0.025 mm) and leave out the n/a days, since those data are not available
8
Holly downloads two columns from the large data set for Heathrow in 2015 but forgets to copy the headings.
Column P contains whole numbers, all between 0 and 8.
Column Q contains values between 996 and 1031.
Suggest which variable is in column P and state its units.
Suggest which variable is in column Q and state its units.
4 marks
Show answer
a) Daily Mean Total Cloud Cover, in oktasb) Daily Mean Pressure, in hectopascals (hPa)
9
Owen wants to show that summer days are warmer in Perth than in the UK. He compares the Daily Mean Temperature for Perth with the Daily Mean Temperature for Hurn, using the large data set for June and July 2015.
Using your knowledge of the large data set, explain why this comparison does not support Owen's aim.
Suggest how Owen could change his choice of data to make a fairer comparison of summer temperatures using only the large data set.
4 marks
Show answer
a) Perth is in the southern hemisphere, so June and July are winter there, not summerb) the large data set only covers May to October, so there is no Perth summer data; he should compare Hurn with a northern hemisphere station such as Beijing or Jacksonville in June and July
10
Zainab is studying Daily Total Sunshine, in hours, for Camborne in August 1987. She takes the first 8 days of the month as her sample.
Day
1
2
3
4
5
6
7
8
Sunshine (hours)
6.2
0.4
9.8
11.0
3.1
0.0
7.5
5.2
State the name of the sampling method Zainab used.
Calculate the mean of her sample.
The mean Daily Total Sunshine for all 31 days of August 1987 in Camborne was 5.8 hours. Comment on her sample mean.
Describe how Zainab could take a simple random sample of 8 days from August 1987 instead.
5 marks
Show answer
a) opportunity (convenience) samplingb) 5.4 hoursc) it is lower than 5.8, so the first 8 days underestimate the mean; days in a row often share the same weather, so the sample is not representatived) number the days 1 to 31 and use a random number generator to pick 8 different days
11
Ffion wants a sample of 32 days of Daily Mean Wind Speed from the five UK stations in the large data set for 2015. After removing days recorded as n/a, the number of days left at each station is shown below.
Station
Camborne
Heathrow
Hurn
Leeming
Leuchars
Days available
175
165
160
150
150
State the units of Daily Mean Wind Speed in the large data set.
Find how many days Ffion should take from each station for a stratified sample of 32 days.
Describe how she should choose the days for Hurn.
Give one reason why a stratified sample is better here than taking all 32 days from Camborne.
6 marks
Show answer
a) knotsb) Camborne 7, Heathrow 7, Hurn 6, Leeming 6, Leuchars 6c) number the 160 Hurn days and pick 6 at random with a random number generatord) wind speeds differ between stations, so the sample should represent every station in proportion
12
A town council wants to know what its 24 000 adult residents think about a new cycle lane. A councillor suggests two plans.
Plan 1: hand a questionnaire to every adult who walks past the town hall on a Monday morning until 200 have been completed.
Plan 2: use the electoral register, which lists the adults in each of the town's 4 wards, and take a stratified sample of 200.
Give two reasons why Plan 1 is likely to give a biased sample.
Ward A has 7200 adults. Find how many adults from Ward A would be in the Plan 2 sample.
Explain why Plan 2 is likely to give more reliable results than Plan 1.
Give one practical problem with Plan 2.
6 marks
Show answer
a) only people near the town hall are reached, and on a Monday morning people at work or school are missed (also people choose whether to take part)b) 60c) it uses random selection from a full sampling frame and represents each ward in proportion, so bias is reducedd) e.g. some chosen people may not reply, or the register may be out of date
Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.