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Either paper 2 marks
Zainab wants to find out how healthily people eat in she town.
She asks the first 30 people she sees in the queue at a burger van.
(a)Explain why Zainab's sample is likely to be biased.
(b)Describe one way she could improve the way the sample is chosen.
Show answer
(a) People queuing at a burger van are not typical of everyone in the town, and they were all asked in one place at one time.
(b) Choose people at random from a list of everyone in the town, and ask them at several different times and places.A sample is biased when it is not typical of the population, because some members of the population were more likely to be picked than others.
People queuing at a burger van are not typical of everyone in the town, and they were all asked in one place at one time.
Notice that the sample size is not the problem here. Asking \(30\) of the wrong people is no better than asking \(5\) of them.
The cure for bias is to give every member of the population the same chance of being chosen.
Choose people at random from a list of everyone in the town, and ask them at several different times and places.
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Either paper 3 marks
There are \(1200\) students in a large school. Two people each take a random sample to find out how many of them walk to school.
Rosa asks \(20\) students. \(13\) of them say they do.
Callum asks \(80\) students. \(26\) of them say they do.
(a)Use Callum's sample to estimate how many of the \(1200\) students walk to school.
(b)Whose estimate is likely to be the more reliable?
Explain your answer.
Show answer
(a) \(390\)
(b) Callum, because a larger sample is more likely to be typical of the population, so chance variation has less effect on the estimate.\(1200 \div 80 = 15\), so the population is \(15\) times the size of Callum's sample.
\(26 \times 15 = 390\)
About \(390\) students.
Both samples were chosen at random, so neither is biased. The difference is size.
Rosa asked \(20\) students. One unusual answer changes Rosa's estimate by \(1200 \div 20\) — a whole students count of \(1200 \div 20\) — while one unusual answer changes Callum's estimate by only \(15\).
So Callum's estimate is the more reliable: a larger sample is more likely to be typical of the population, and chance variation has less effect on it.
A larger sample never fixes bias, though. If the sample were chosen badly, asking more of the wrong people would only make the wrong answer more precise.
- (a) M1scale factor \(15\), or \(\frac{ 26 }{ 80 } \times 1200\)
- (a) A1estimate \(390\)
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Either paper 1 mark
Callum's school has Years 7 to 11.
Callum claims that most of the pupils in his school walk to school.
Callum asks 80 pupils from Years 7 to 9.
Give a reason why this sample may not be useful in testing Callum's claim.
Show answer
The sample leaves out Years 10 and 11. Pupils in those years may differ in how they travel to school, so the sample may not represent all the pupils.The school has Years 7 to 11, but Callum only asked pupils from Years 7 to 9.
So Years 10 and 11 are missing from the sample.
Pupils in those years may differ in how they travel to school, so a sample that leaves them out may not represent all the pupils in the school.
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Either paper 3 marks
A school has \(400\) pupils. A sample of \(50\) pupils is needed.
Three people each describe a different way of choosing the sample.
State the name of the sampling method each one is describing.
(a)Hugo numbers everybody in the population and then uses a random number generator to decide who is in the sample.
(b)Isla puts everybody in a list and then takes one at equal intervals all the way down it.
(c)Jonah splits everybody into age groups and then takes a number from each group in proportion to the size of that group.
Show answer
(a) Random sampling (a simple random sample)
(b) Systematic sampling
(c) Stratified samplingEverybody has exactly the same chance of being picked, and every possible group of that size is equally likely to come up. That is a simple random sample.
Picking at a fixed interval down a list is systematic sampling. It is quick and it does spread the sample out, but it is not random: once the first one has been chosen, everybody else in the sample is already decided.
The population is split into groups (strata) and each group keeps its share of the sample. That is stratified sampling, and it is used when the groups might answer differently, so leaving one of them short would bias the result.
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Either paper 3 marks
A toy factory makes \(1000\) model cars in a day.
Amara checks the quality of a systematic sample of \(50\) model cars, taken evenly through the day's production.
(a)Work out how often she should take a model car, so that the \(50\) model cars are spread evenly through the \(1000\).
(b)Amara says the 50 model cars are a random sample.
Explain why she is wrong.
Show answer
(a) every 20th model car
(b) Once the first model car has been chosen, every other model car in the sample is fixed, so not every group of 50 model cars is equally likely to be chosen.Spreading \(50\) checks evenly through \(1000\) model cars means one check in every group of
\(1000 \div 50 = 20\)
so she takes every \(20\)th model car.
Check it back: \(20 \times 50 = 1000\).
In a random sample every possible group of \(50\) model cars is equally likely.
Here that is not true. As soon as the first model car is picked, the other \(50 - 1\) are decided by the interval of \(20\) — so two model cars made one after the other can never both be in the sample.
Systematic sampling is quick and it does spread the checks out, but it is not the same thing as a random sample.
- (a) M1\(1000 \div 50\)
- (a) A1every \(20\)th model car
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Either paper 3 marks
There are \(800\) students at Tomas's school.
He asks a random sample of \(80\) of them whether they have a school lunch. \(23\) of them say they do.
(a)Use his sample to estimate how many students at the school have a school lunch.
(b)Explain why this is only an estimate.
Show answer
(a) \(230\)
(b) It is worked out from a sample, not from the whole school, so a different sample of 80 students would have given a different answer.The sample is one part of the school in \(800 \div 80 = 10\), so the whole school is \(10\) times the size of the sample.
Assume the sample is typical, and scale the result up by the same factor:
\(23 \times 10 = 230\)
About \(230\) students.
The same sum written as a proportion: \(\frac{ 23 }{ 80 } \times 800 = 230\).
Only \(80\) of the \(800\) students were actually asked. The other \(800 - 80 \) were not.
A second random sample of \(80\) students would almost certainly give a slightly different number, and so a slightly different estimate.
The estimate is only as good as how typical the sample is — which is why it was taken at random, and why a larger sample would be better still.
- (a) M1scale factor \(10\), or \(\frac{ 23 }{ 80 } \times 800\)
- (a) A1estimate \(230\)
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Either paper 4 marks
Ms Bhatt is taking a stratified sample of \(40\) students from the \(400\) students in her school.
The table shows how many students are in each year group.
Year group 9 10 11 Total Number of students 150 170 80 400 (a)Calculate how many Year \(10\) students should be in the sample.
(b)Calculate how many Year \(11\) students should be in the sample.
Show answer
(a) \(17\)
(b) \(8\)In a stratified sample every year group keeps its share, so first find what fraction of the school is being sampled:
\(\frac{ 40 }{ 400 } = \frac{1}{ 10 }\)
Now take that fraction of Year \(10\):
\(170 \times \frac{1}{ 10 } = 170 \div 10 = 17\)
\(17\) Year \(10\) students.
The same fraction again:
\(80 \div 10 = 8\)
Check the whole sample adds up: \(15 + 17 + 8 = 40\), which is the sample size asked for.
- (a) M1fraction sampled \(\frac{ 40 }{ 400 } = \frac{1}{ 10 }\), or \(170 \div 10\)
- (a) A1Year 10 sample \(17\)
- (b) M1same fraction applied, \(80 \div 10\)
- (b) A1Year 11 sample \(8\)
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Either paper 7 marksKim works at an airport in the UK. She records the number of planes landing between 11 am and 1 pm each day. The table shows her data for the first 10 days in January.
Day 1 2 3 4 5 6 7 8 9 10 Number of planes 78 88 90 34 77 87 91 88 83 81 (a)The airport was affected by fog on one of the days.
Which day do you think it was?
Give a reason for your answer.
(b)Kim uses the data to predict how many planes will land at the airport in a year.
In her method, she uses an estimate of \(83\) planes in each \(2\)-hour period throughout the day. The airport is closed on \(14\) days of the \(365\) days in the year. She assumes the same number of planes land on each day that it is open.
Work out Kim's prediction for the number of planes that will land in a year.
(c)Give one reason why Kim's prediction may not be reliable.
Show answer
(a) Day 4: only 34 planes landed, much fewer than on any other day (the other nine days were between 77 and 91).
(b) \(349596\ \text{planes}\)
(c) The data are for only 10 days in January, which may not be typical of the whole year; or the 2-hour period from 11 am to 1 pm may not be typical of the rest of the day.Fog stops planes landing, so the foggy day should have far fewer planes than a normal day.
Nine of the days have between \(77\) and \(91\) planes. Day \(4\) has only \(34\).
It was day \(4\), because \(34\) is much lower than all the others.
A day has \(24\) hours, so it splits into \(24 \div 2 = 12\) periods of \(2\) hours.
Planes in one day: \(83 \times 12 = 996\)
Days open: \(365 - 14 = 351\)
Planes in a year: \(996 \times 351 = 349596\)
The prediction treats every day of the year, and every period of the day, as if it were like the data.
The data cover only \(10\) days in January, and January may be quieter or busier than other months.
The period from 11 am to 1 pm is only \(2\) hours, so it may not be typical of the rest of the day.
Either of these is a valid reason.
- (a) A1Day 4
- (a) B1reason: 34 planes is much lower than every other day (77 to 91)
- (b) M1\(24 \div 2 = 12\) periods in a day
- (b) M1planes per day \(83 \times 12 = 996\)
- (b) M1days open \(365 - 14 = 351\)
- (b) A1\(996 \times 351 = 349596\)
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Either paper 4 marks
A survey was held at a railway station one day. A sample of the passengers was asked how important a quiet carriage is.
The pie chart shows the answers.
In total there were \(9900\) passengers.
(a)Estimate how many of the passengers think that a quiet carriage is very important.
Assume that the sample is representative of all the passengers.
(b)In fact, \(30\%\) of the sample were travelling with young children, but only \(15\%\) of the passengers were travelling with young children.
The people who were travelling with young children are less likely than the others to say that a quiet carriage is very important.
What does this suggest about the actual number of passengers who think a quiet carriage is very important?
Tick one box. Give a reason for your answer by comparing the percentage of the sample with the percentage of the passengers.
☐ higher than your estimate
☐ lower than your estimate
☐ the same as your estimate
Show answer
(a) \(1430\)
(b) Likely to be higher than the estimate: over-represented group, sample 30% against 15%.The "very important" sector is \(52^\circ\) out of \(360^\circ\), so it is \(\frac{ 52 }{360}\) of the sample.
Assuming the sample is representative, the same fraction of the passengers think a quiet carriage is very important:
\(\frac{ 52 }{360} \times 9900 = 1430\)
About \(1430\) passengers.
Compare the share of the sample with the share of all the passengers: \(30\%\) of the sample against \(15\%\) of the passengers, so this group is over-represented.
People who were travelling with young children are less likely than the others to say a quiet carriage is very important, so the share saying "very important" in the sample is too low.
The actual number is likely to be higher than the estimate.
- (a) M1fraction of the circle \(\frac{ 52 }{360}\) of \(9900\)
- (a) A1\(1430\)
- (b) B1the group is over-represented: \(30\%\) of the sample against \(15\%\) of the passengers, so the sample result is too low
- (b) A1tick higher than the estimate
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Either paper 2 marks
Leila wants to find out how people travel to work in she town.
She asks the first 20 people she sees at a bus stop.
(a)Explain why Leila's sample is likely to be biased.
(b)Describe one way she could improve the way the sample is chosen.
Show answer
(a) People standing at a bus stop are far more likely to travel by bus than people in general.
(b) Choose people at random from a list of everyone in the town rather than from one bus stop.A sample is biased when it is not typical of the population, because some members of the population were more likely to be picked than others.
People standing at a bus stop are far more likely to travel by bus than people in general.
Notice that the sample size is not the problem here. Asking \(20\) of the wrong people is no better than asking \(5\) of them.
The cure for bias is to give every member of the population the same chance of being chosen.
Choose people at random from a list of everyone in the town rather than from one bus stop.