Sampling & populations

A survey was carried out to find how many books children read during the holidays.

The following two questions were asked in a questionnaire.

Q1. Most children read lots of books.

Do you read lots of books?........

Q2. How many books do you read during the holidays?

None 1–10 books 10–20 books More than 20 books

Give one reason why Q1 was not a suitable question.

Give one reason why Q2 was not a suitable question.

The survey was carried out by asking children as they left the library on a Saturday morning.

Give one criticism of how the survey was carried out.

A survey was carried out to find how many books children read during the holidays.

The following two questions were asked in a questionnaire.

Q1. Most children read lots of books.

Do you read lots of books?........

Q2. How many books do you read during the holidays?

None 1–10 books 10–20 books More than 20 books

Give one reason why Q1 was not a suitable question.

Give one reason why Q2 was not a suitable question.

The survey was carried out by asking children as they left the library on a Saturday morning.

Give one criticism of how the survey was carried out.

NoomAir is a cargo plane company based in Nice.

Two of their planes have been hired to deliver cargo to the cities of Rome, Valencia and Tunis.

The second plane will fly from Nice to Valencia, and then on to Tunis. The pilot of this plane knows the following information. • Valencia is on a bearing of 234º from Nice. • The distance between Valencia and Tunis is 970 km.

Calculate the bearing of Tunis from Valencia.

NoomAir sells toy replicas of one of its cargo planes.

The volume of a large toy plane is 6 times the volume of a small toy plane. The length of a large toy plane is 30 cm.

Calculate the length of a small toy plane.

C-clips are used in motors and turbines as a type of fastener. Clip Zone makes metal C-clips. They have a uniform cross-section, as shown below.

O is the centre of the circular arcs AC and BD. OAB and OCD are straight lines. OA = OC = 50 mm. AB = CD = 4 mm. The C-clips have a uniform thickness of 2 mm.

C-clips are made by melting down metal bars and re-casting them. Calculate the number of C-clips that can be made from a metal bar with a volume of 1 500 000 mm³.

You must show all your working.

Clip Zone makes four types of fastener. The table below shows the number of each type produced each week.

Clip Zone takes a sample of these 149 000 fasteners to check the quality of its products. A stratified sample of 1500 fasteners is taken, based on the type of fastener. Calculate the number of each type of fastener that should be included in the sample.

You must show all your working.

Form-A-Part is a company that makes metal parts for use in engineering.

A metal part is made from a circular disc with a piece cut out. The part has uniform thickness, a diameter of 60 mm, and centre O. The diagram below shows the cross-section of the metal part.

Calculate the perimeter of the cross-section of this metal part.

Give your answer correct to the nearest millimetre.

Perimeter of the metal part is_{…........ ………………..……} mm, correct to the nearest millimetre.

The diagram below shows a metal part made by Form-A-Part. Each part consists of a cone sitting on top of a hemisphere. The diameter of the base of the cone and the diameter of the hemisphere are equal.

A customer has ordered 20 000 of these parts.

Form-A-Part has guaranteed that they will make all of the parts with the measurements shown in the diagram being correct to the nearest mm. All 20 000 parts will be given a protective coating that comes in tins. Each tin covers an area of 4 000 000 mm², correct to the nearest 100 000 mm². Form-A-Part must guarantee that they have enough tins of protective coating to coat all 20 000 parts. Calculate the minimum number of tins they will need.

You must show all your working.

Form-A-Part has decided to randomly sample these 20 000 metal parts to check their quality.

Use the following list of random numbers to select the first 5 parts for the sample. You must start with the first number in the list.

Explain clearly how you are using the numbers to select the sample.

The 5 parts selected will be

A survey was carried out to find how often primary school children play board games.

A questionnaire was designed for primary school children to answer. The following two questions were asked.

Q1. Do you live within 5 minutes’ walking distance of school?

Q2. How often do you play board games?

Never 1–5 times 5–10 times More than 10 times

For each question, give one reason why it is not suitable.

The survey was carried out by leaving copies of the questionnaire in the local supermarket.

Give one criticism of how the survey was carried out.

A survey was carried out in the U.S.A.

743 teenagers were interviewed. They were asked what type of information they posted on social media. The results were displayed on the internet, as shown below.

Posts on social media Percentage of American teenagers who say they have posted about the following:

Type of information:

Emotions and feelings 34%

How many times bigger is the percentage of the teenagers who posted about their

family than the percentage who posted about their religious beliefs?

What fraction of these teenagers posted about their emotions and feelings?

Give your answer in its simplest form.

Dewi looks at the type of information posted by these teenagers.

What is the modal type of information?

The modal type of information is_........

What information would have been needed in the original data so that the following

hypothesis could be tested?

In the U.S.A., teenage girls post about family more often than teenage boys.

How many of these 743 teenagers posted about their religious beliefs?

You must show all your working.

Lottie is confused by the data in the diagram.

She says,

The diagram is correct.

Explain why the bars do not add up to 100%.

A survey was carried out to find how often primary school children play board games.

A questionnaire was designed for primary school children to answer. The following two questions were asked.

Q1. Do you live within 5 minutes’ walking distance of school?

Q2. How often do you play board games?

Never 1–5 times 5–10 times More than 10 times

For each question, give one reason why it is not suitable.

The survey was carried out by leaving copies of the questionnaire in the local supermarket.

Give one criticism of how the survey was carried out.

Alan Frames is a company that employs 360 people.

6 of these people are to be selected to discuss changes to the company logo.

The manager has decided to use a systematic sampling method. He has a numbered list of all 360 people.

When using systematic sampling, where in the list should the manager start his selection of the 6 people?

Tick (√) one of the boxes.

The 60th name in the list

At a randomly chosen name

First person in the list

Last person in the list

A name by any multiple of 60 in the list

The manager actually starts by selecting the 4th name in his list. Complete the table below to give the position in the list of the 6 people who would be

selected using systematic sampling.

Evan will need to put strips of lead from B to C and from B to F. This is to ensure rainwater does not enter the room.

Calculate the total length of lead that Evan will need.

The roof of the structure is to be tiled. For the top edge, Evan has chosen to use half-round clay tiles.

The cross-section of a tile is shown below.

The length of each tile is 30 cm. XY is a semi-circular arc of radius 8 cm. Each tile has a uniform thickness of 1∙2 cm.

The tile manufacturer covers all 6 surfaces of each tile with a protective glaze.

Calculate the total surface area of one tile that is glazed.

Evan has bought roof tiles for the structure from Tuff Tiles.

Tuff Tiles makes 2000 of these roof tiles every hour. They randomly sample some of these 2000 tiles to check their quality.

Use the following list of random numbers to select the first 7 tiles for the sample. You must start with the first number in the list, explaining clearly how you are using the

numbers to select the sample.

3205 2002 1924 1521 2735 1205

0114 9533 9507 3969 4941 1521

8377 1098 3473 0769 1003 5421

5533 4788 1003 6432 5630 2483

The tiles selected will be

Delyth and Ronnie are both students at the local college.

Their houses and the college are all joined by straight roads, as shown in the diagram.

Delyth usually walks directly to college. Calculate how much further Delyth has to walk if she passes Ronnie’s house on her way

to college.

35 students were asked how far they travelled to the college. The results are recorded in the table below.

Distance, d (metres)	Frequency
100 d 200	9
200 d 1000	10
1000 d 3000	15
3000 d 7000	1

Calculate an estimate of the mean distance these 35 students travelled to the

college.

Estimate of the mean distance travelled by these 35 students is_........ m

There are 140 students who travel by bus to and from college.

Delyth wants to find out why these students do not walk to college. She has decided to use a systematic sampling method to select 7 of these students to form a discussion group.

The names of all the 140 students are in a list. Delyth has randomly selected the 2nd student in the list to join the discussion group.

Complete the table below to give the positions in the list of the 7 students selected to

join the discussion group.

Delyth and Ronnie are both students at the local college.

Their houses and the college are all joined by straight roads, as shown in the diagram.

Delyth usually walks directly to college. Calculate how much further Delyth has to walk if she passes Ronnie’s house on her way

to college.

35 students were asked how far they travelled to the college. The results are recorded in the table below.

Distance, d (metres)	Frequency
100 d 200	9
200 d 1000	10
1000 d 3000	15
3000 d 7000	1

Calculate an estimate of the mean distance these 35 students travelled to the

college.

Estimate of the mean distance travelled by these 35 students is_........ m

There are 140 students who travel by bus to and from college.

Delyth wants to find out why these students do not walk to college. She has decided to use a systematic sampling method to select 7 of these students to form a discussion group.

The names of all the 140 students are in a list. Delyth has randomly selected the 2nd student in the list to join the discussion group.

Complete the table below to give the positions in the list of the 7 students selected to

join the discussion group.

There are 600 pupils in a school.

8 of these pupils are to be selected to discuss changes to the school uniform.

The headteacher has a spreadsheet of the names of all 600 pupils. There are 600 rows of pupil names in the spreadsheet, starting at row 1. There is one pupil name on each row.

The headteacher uses a systematic sampling method.

The first pupil selected on the headteacher’s list is a boy whose name is in the 25th row.

Give the row numbers in the spreadsheet of the other 7 pupils who would be selected.

Complete the table below.

Explain how the headteacher selected the first pupil.

There are 600 pupils in a school.

8 of these pupils are to be selected to discuss changes to the school uniform.

The headteacher has a spreadsheet of the names of all 600 pupils. There are 600 rows of pupil names in the spreadsheet, starting at row 1. There is one pupil name on each row.

The headteacher uses a systematic sampling method.

The first pupil selected on the headteacher’s list is a boy whose name is in the 25th row.

Give the row numbers in the spreadsheet of the other 7 pupils who would be selected.

Complete the table below.

Explain how the headteacher selected the first pupil.

Môr-Dda is a company that makes buoys for use at sea.

Môr-Dda makes 4 types of buoy. Each week, they make the following numbers of each type of buoy.

This week, Môr-Dda plans to take a sample of the buoys it makes to check on the quality of their production. It plans to take a stratified sample of 12 buoys, based on the type of buoy.

Calculate the number of each type of buoy that should be included in the sample.

You must show all your working.

21 21

The design for a new buoy is shown below. It is made up of a cone attached to a hemisphere.

The base radius of the cone and the radius of the hemisphere are both 2 m.

The total volume of this new buoy is 10 m³.

Calculate the total height of the buoy.

Calculate an estimate of the percentage of passengers who were aged 60 or more.

On the graph paper opposite, draw a histogram to show the distribution of the ages of the

passengers.

Alun is a jeweller. He is designing a symmetrical pendant, as shown below.

The pendant will be made from solid silver, with a uniform thickness of 3 mm. In order to calculate the cost of making the pendant, Alun wants to calculate an estimate of the volume of the pendant. He has accurately drawn one of the symmetrical halves of the shape on graph paper.

Calculate an estimate of the volume of the whole pendant.

Use the graph opposite, with 6 strips of equal width.

Alun makes pendants that are mathematical shapes.

At the end of last month, Alun took a stratified sample of 30 of these 240 pendants to check their quality.

Calculate how many pendants of each shape were in Alun’s sample.

You must show all your working.

Mrs Butler asked all her friends how many pairs of shoes they own. Here are the results.

Mrs Butler says, ‘Most people in Wales own more than 12 pairs of shoes.’

Is Mrs Butler correct in using this data to come to this general conclusion?

You must give a reason for your answer.

Yes No

Write down different groups Mrs Butler could have chosen so that all of the data can be

analysed more accurately.

Use the table below to give the positions in the queue of the 8 people who would receive

vouchers.

Gerallt used his 20% off voucher to buy tickets. He paid £120 for tickets using the voucher.

How much would these tickets have cost Gerallt without a voucher?

Cost without a voucher £_........

Each of the 250 Year 10 pupils in Blaengwyn school study one foreign language at GCSE.

The table below shows how many pupils chose to study French, German and Spanish.

The Languages department is planning to take a group of 30 pupils on an educational trip to Europe.

Use a stratified sampling method to calculate the number of Year 10 pupils from each language group that should be taken on the trip.

You must show all your working.

Ffion thinks that the taller you are, the greater your waist measurement will be.

She recorded the height and waist measurements, in centimetres, for 6 people.

Draw a scatter diagram to display these measurements.

Use the graph paper below.

Give one reason why using this scatter graph to estimate the waist measurement of other

people is unlikely to give reliable results.

An engineering company employs 85 staff.

The company plans to carry out a survey on staff health. It will conduct the survey using a sample of 15 of its staff, stratified by job type.

50 engineers are employed by the company. Use the following extract from a table of random digits to choose 9 engineers for the sample. You must start with the first number in the list.

Describe clearly how you are using the numbers to select the sample.

29974 55479 07248 33999 17038 02475 49979 01218

The engineering company has a storage building, as shown below. The building is in the form of half a cylinder, with half a hemisphere attached at one end.

The company needs to paint all the exterior surfaces of the building, including the doors.

The measurements on the diagram are given correct to the nearest metre. The paint comes in tins that cover an area of 40 m², correct to the nearest m².

Calculate the smallest number of tins that would guarantee having enough paint to cover

these exterior surfaces.

Simon plans to make gloves.

One morning, Simon decided to carry out a survey to find the mean hand span of people in Wales.

He decided to sample systematically. He decided to sample from the first 240 people who pass him in the street during the morning.

He wanted to take 20 people’s hand span measurements.

Explain how Simon could use systematic sampling to obtain 20 measurements.

Yesterday morning, Simon only managed to sample 10 people. He calculated the mean hand span of these 10 people to be 22.8 cm. Yesterday afternoon, Simon recorded the hand spans of a further 20 people. The results for these 20 people are shown in the frequency table below.

Calculate an estimate of the mean of all 30 hand spans that Simon measured

yesterday.

What could Simon do to improve his estimate of the mean hand span of people in

Wales?

Simon plans to make gloves.

One morning, Simon decided to carry out a survey to find the mean hand span of people in Wales.

He decided to sample systematically. He decided to sample from the first 240 people who pass him in the street during the morning.

He wanted to take 20 people’s hand span measurements.

Explain how Simon could use systematic sampling to obtain 20 measurements.

Yesterday morning, Simon only managed to sample 10 people. He calculated the mean hand span of these 10 people to be 22.8 cm. Yesterday afternoon, Simon recorded the hand spans of a further 20 people. The results for these 20 people are shown in the frequency table below.

Calculate an estimate of the mean of all 30 hand spans that Simon measured

yesterday.

What could Simon do to improve his estimate of the mean hand span of people in

Wales?

The table shows the number of Year 11 pupils attending schools in Cwmifan.

In total there are 690 Year 11 pupils attending these three schools.

A new youth theatre has been set up in Cwmifan. On the opening night, a total of 80 Year 11 pupils from these three schools are going to be invited to attend.

Use a stratified sampling method to calculate the number of Year 11 pupils from each school who should be invited.

You must show all your working.

A survey was carried out to find how often teenagers buy DVDs.

The following two questions were asked in a questionnaire.

Q1. Where do you live?

Q2. How often do you buy DVDs?

Never 1-10 times 10-15 times More than 15 times

For each question give one reason why it is not suitable.

The survey was carried out by leaving copies of the questionnaire on the DVD shelves in a supermarket.

Give one criticism of how the survey was carried out.

For each question give one reason why it is not suitable.

The survey was carried out by leaving copies of the questionnaire on the DVD shelves in a supermarket.

Give one criticism of how the survey was carried out.

Students are taking tests in English and Welsh. The English test is marked out of 80. The Welsh test is marked out of 70.

Dyfed scores 35 in his English test. Estimate Dyfed’s score as a percentage.

Circle your answer.

4% 20% 23% 44% 51%

Liam scores 22 in his Welsh test. Estimate Liam’s score as a percentage.

Circle your answer.

Rowena states a hypothesis, ‘Boys do better than girls in their English tests.’

She displays the test marks for 5 girls and 5 boys in scatter diagrams.

Boys’ test results Mark in Welsh

Does the data support Rowena’s hypothesis?

You must give a reason for your answer.

How could Rowena improve the testing of her hypothesis?

Porth Ifan Hospital has made some changes to improve patient care.

A survey is to be used to find out the views of the hospital staff.

The table shows the total number of staff in each job type.

The survey is to be given to a sample of 75 staff.

Use a stratified sampling method to calculate the number of staff from each job type that should be asked to complete the survey.

You must show your working.

The hospital decides to take a random sample of its 120 doctors to select those needed for the survey. Use the following list of random numbers to select the first 5 doctors. You must start with the first number in the list, explaining clearly how you are using the

numbers to select the sample.