Sampling and populations

David collected data about 34 players on the 2022–2023 Manchester United Premier League football team.

Each player has a playing position identified as goalkeeper or defender or midfielder or forward.

The bar chart shows information about the playing positions of these players.

Which playing position has the greatest number of players?

Which playing position has twice as many players as the goalkeeper position?

David concludes that there are more defenders than midfielders who play for Premier League football teams.

Give one limitation of David’s conclusion.

Mandeep visited Marple Wood Farm and tallied the types of animals he saw.

Circle the word in the list below that best describes the type of data for the types of animals.

Quantitative Discrete Continuous Categorical

The incomplete tally chart and incomplete bar chart give some information about Mandeep’s results.

Complete the tally chart and the bar chart.

This is a sign at the entrance to the farm.

Mandeep tallied 55 animals during his visit. Using his results, Mandeep claims that the most common animal in the farm is a chicken.

Give two reasons why Mandeep’s claim may be incorrect.

The scatter diagram shows the marks each of ten students got for a Maths test and for a Music test.

One of these points is an outlier.

On the diagram, draw a circle around the outlier.

Excluding the outlier, describe the correlation shown in the diagram.

The outlier is removed from the data. The mean of the Maths marks is 52 The mean of the Music marks is 63

Excluding the outlier, draw a line of best fit that passes through the double mean point on the scatter diagram.

30 students did both tests. The scatter diagram shows the marks of 10 of these students.

Suggest one adjustment that could be made to improve the reliability of any conclusions made using the scatter diagram.

Elise wants to investigate the relationship between the size of mammals and the size of the fruits and seeds that these mammals eat. This is her plan.

Discuss the appropriateness of Elise’s hypothesis and her data collection strategy.

Elise wants to investigate the relationship between the size of mammals and the size of the fruits and seeds that these mammals eat. This is her plan.

Discuss the appropriateness of Elise’s hypothesis and her data collection strategy.

Chris is a manager at a theme park.

He wants to find out what food options visitors would like to be able to buy in the theme park.

State the population for this investigation.

Chris decides that he will take a convenience sample of visitors in the section of the park selling food.

Explain what is meant by a convenience sample.

Give one disadvantage of using a convenience sample.

Chris plans to use the data collection sheet below.

Discuss whether this data collection sheet is appropriate.

You should consider how Chris might use the data and describe any problems he might have when he uses the data collection sheet.

Chris suggests using a stem and leaf diagram to represent the data that he collects.

Discuss whether or not this would be a suitable diagram to represent his data.

Researchers used the Petersen capture recapture method to estimate the number of humpback whales off the coasts of California, Oregon and Washington.

They used aerial photographs to identify the whales, recording the whales’ unique markings in the first sampling rather than tagging the whales.

In 1995 they recorded the markings of 331 individual whales. In 1996 they identified 264 individual whales of which 104 had been identified in the previous year. (Source: digitalcommons.unl.edu)

Calculate an estimate for the number of humpback whales off the coasts of California, Oregon and Washington in 1996

Discuss the validity and reliability of this estimate.

A town council is proposing to build a new leisure centre. Michelle is going to carry out a survey to find out what all the people in the town think of the proposal.

Michelle thinks that she should take a sample rather than a census.

Give two reasons why Michelle might think this.

Michelle plans to use the electoral register as the sampling frame.

Explain what you understand by the term sampling frame.

Give one problem Michelle may have using the electoral register as the sampling frame.

Michelle intends to conduct a pilot study.

Give two reasons why it is a good idea to conduct a pilot study.

Michelle is writing a plan for her investigation into people’s views on the leisure centre proposal.

Write down what Michelle should include in her plan. You should include each of the following

• a sampling method

• a question she could ask in her questionnaire

• a statistical diagram she could use to show the results of the survey.

Explain why each of the things you have written down is appropriate.

The following is an extract from part of a row of a random number list.

68236 35335 71329

Use the random number list to complete the table for the first 5 random 2-digit numbers.

The most common blood type in the United Kingdom is O+

The percentage of people in the United Kingdom with O+ blood type is 38%

Asha uses a simulation method to estimate how many donors would be needed to find exactly 3 donors with O+ blood type.

Asha is going to use the following 2-digit numbers for her simulation.

Explain why this is an appropriate way to allocate the random numbers.

Asha runs trials using her simulation method. The result of each trial is the number of random numbers used until Asha gets exactly 3 donors with O+ blood type. The table below shows the results of her first 4 trials.

The set of random numbers used by Asha to complete the fifth trial are shown below.

60 13 12 86 73 10 98 95 43 46

Using this set of random numbers, find the result for the fifth trial. You must make it clear how you obtain your answer.

Asha finds the mean of her 5 results and decides that the results of her simulation are sufficient to predict the number of donors needed to find at least 3 with O+ blood type in the next blood donation session.

Explain whether the method that Asha uses to predict the number of donors required is appropriate.

Rose is investigating the number of brothers and sisters that students in her secondary school have.

To investigate this she asks 10 students in Year 8 and 10 students in Year 11 how many brothers and sisters they each have.

Assess Rose’s method for her data collection.

The vertical line graph shows the data that she collected.

How many students have 2 or more brothers and sisters?

Write down the mode.

Rose uses her vertical line graph to conclude that no student in her school has 5 or more brothers or sisters.

Assess whether or not Rose’s conclusion is appropriate.

A theme park in Staffordshire has around 30 000 visitors per day. (Source: www.statista.com)

Navine is a manager at the theme park. Navine is investigating what visitors think about the theme park. He is going to do a survey of visitors at the theme park.

Navine decides to question 30 people aged under 18 and 30 people aged 18 and over as they leave the theme park one day.

He plans to ask them face to face what their favourite ride was.

Name this sampling method.

Describe the population for this survey.

Assess Navine’s plan to get the opinions of the people who have visited the theme park.

A Science teacher wants to know the effects of revision on a student’s performance in an exam. She decides to carry out an experimental test on a group of 15 students to find out the effects of any revision.

Describe one way the teacher could carry out an experimental test.

Give one reason why the results of this experimental test could be unreliable.

Grace asked a sample of 60 people in her town if they had ever visited France or Spain.

17 people visited both France and Spain 23 people visited Spain only 33 people visited France

Draw a Venn diagram to represent this information.

Grace says

• more than half of the people in her sample have visited France

• therefore more than half of the people in her town have visited France

Discuss the validity of each of Grace’s comments.

Sheffield City Council has more than 8000 employees. These employees work at different locations in Sheffield. (Source: Wikipedia.org)

A researcher wants to find out about the different methods of transport that these employees use to get to work.

She plans to ask the 9 employees, who work in one library, how they got to work that morning.

Give two reasons why this is not a good plan.

A vet carried out a survey to see if there is a relationship between the weight of a dog and its lifespan. The vet found the mean weight, in kg, and the median lifespan, in years, of a sample of five dogs in each of ten breeds of dogs.

Here are her results.

(Source: adapted from www.instituteofcaninebiology.org)

The vet used statistical software to draw this scatter diagram for her results.

From the scatter diagram the vet concluded that there is a relationship between the weight of a dog and its lifespan.

Using the given statistical information,

assess whether or not the vet’s conclusion is appropriate,

assess the reliability of the vet’s conclusion.

David is investigating the prices of houses in Streetly, which is in the West Midlands.

David took a sample of 41 houses in Streetly from a property website and recorded the price of each of these houses.

The cumulative frequency table shows information about the prices of these houses.

Draw a cumulative frequency diagram for this information.

In David’s sample there were N houses with a price in the interval 300 < P ≤ 350

Find an estimate for the value of N.

David wants to use his investigation to predict the number of houses in Central London that have a price between £300 000 and £350 000

Assess whether or not it is sensible for David to make this prediction using the results of his investigation.

For a survey, Kayleigh counted the number of people in each of the 20 cars that arrived at her office car park one morning.

Here are her results.

Fill in the tally chart for this information and complete the frequency column.

Write down the mode.

At Kayleigh’s office there is a scheme to reduce the number of cars arriving at the office car park which have only 1 person in the car.

Kayleigh concludes from her survey that the scheme is not working.

Explain whether or not Kayleigh’s results support her conclusion.

John suggests that Kayleigh’s results are not suitable for her to reach a reliable conclusion.

Give one reason why John may be correct.

Bill investigated the heights of females competing in different sports.

Using data from the internet, he recorded the height, in cm, of each female in a sample of 15 female rugby players and the height, in cm, of each female in a sample of 15 female basketball players.

He drew the box plot below for the recorded heights of the 15 female basketball players.

(Source: englandrugby.com, gbbasketball.com)

The table gives information about the recorded heights of the 15 female rugby players.

Using the information in the table, draw on the grid above a box plot for the recorded heights of the 15 female rugby players.

Compare the two distributions of heights. Give three comparisons and interpret one of your comparisons.

Considering the method that Bill used to collect his data, comment on the reliability of your comparisons in part (b).

Bill investigated the heights of females competing in different sports.

Using data from the internet, he recorded the height, in cm, of each female in a sample of 15 female rugby players and the height, in cm, of each female in a sample of 15 female basketball players.

He drew the box plot below for the recorded heights of the 15 female basketball players.

(Source: englandrugby.com, gbbasketball.com)

The table gives information about the recorded heights of the 15 female rugby players.

Using the information in the table, draw on the grid above a box plot for the recorded heights of the 15 female rugby players.

Compare the two distributions of heights. Give three comparisons and interpret one of your comparisons.

Considering the method that Bill used to collect his data, comment on the reliability of your comparisons in part (b).

A hospital department has three specialist consultants, A, B and C.

The department manager uses 2-digit random numbers to carry out a simulation of the number of consultations per week for each specialist.

According to hospital records, the average number of consultations per week for each specialist is shown in the following table.

The department manager allocates the following 2-digit numbers to each specialist.

Explain why this is an appropriate way for the department manager to allocate the 2-digit numbers.

Also, according to hospital records, the expected total number of consultations in a week for all three specialists is 50

The table below shows the first 45 results from the simulation.

The last five random numbers used by the department manager are

06 74 59 14 29

Using these last five numbers, complete the table below to show the final result of

Each consultation is planned to last no longer than 20 minutes.

According to the simulation, how much more time a week should be allocated to specialist A than to specialist C for all of their consultations?

The department manager decides that the results of her simulation are sufficient for her to plan how much time is needed per week for consultations with each specialist.

Explain why the manager is not correct.

An adventure travel company runs expeditions for groups of 20 people.

The company provides all the food and equipment for these expeditions.

The manager of the company wants to trial a new sports drink to see if it is better than the usual sports drink.

He decides on the following plan.

• Before the expedition each of the 20 people will complete a questionnaire about their fitness and energy levels.

• The 20 people will be divided into two groups of 10 people, a test group and a control group.

• The test group will be given the new sports drink during the expedition.

• The control group will be given the usual sports drink during the expedition.

• After the expedition each of the 20 people will again complete a questionnaire about their fitness and energy levels.

• The energy levels of the two groups after the expedition will be compared.

Explain briefly why a control group should be used.

Explain how the manager could use matched pairs in order to select the people in the two groups.

Assess the reliability of any conclusions that the manager may reach as a result of using his plan.

Give two reasons for your answer.

Paul uses the Petersen capture recapture method in order to find an estimate for the number of trout living in a lake.

To do this, Paul catches an initial sample of 50 trout from the lake and he tags each trout. He then releases the trout back into the lake.

For his second sample, Paul catches a sample of 80 trout from the lake. He finds that 16 of these trout have tags.

Explain what needs to be true about his two samples for them to be valid for the capture recapture method.

Using his results, Paul is able to work out an estimate for the number of trout living in the lake.

Find Paul’s estimate for the number of trout and discuss the validity and the reliability of his estimate.

Andy is investigating the lateness of the trains that arrive at Manchester Piccadilly.

Explain why it is sensible for Andy to collect secondary data for his investigation.

Andy found out data on the lateness of each of 238 trains arriving at Manchester Piccadilly from London on Monday to Friday for one week.

The histogram below was drawn using Andy’s data.

A negative time for a train means that the train arrived early at Manchester Piccadilly.

In the week, 14 trains arrived early.

Find how many of the 238 trains did not arrive more than 10 minutes late at Manchester Piccadilly.

For a long distance journey, such as London to Manchester Piccadilly, a train is only considered to be late when it arrives more than 10 minutes late.

Andy says that more than 1 in 3 trains from London arrive late at Manchester Piccadilly.

Determine whether or not Andy is correct.

You should comment on the reliability of your conclusion.

Bethany is the union representative at the large company where she works. She is investigating the salaries of the employees at the company.

Bethany takes a simple random sample of 8 employees at the company and asks each of them to tell her their salary.

Describe what you understand by the term ‘simple random sample’.

The salaries of the 8 employees are listed below.

£16 000 £3 000 £23 000 £31 000

£21 000 £24 000 £17 000 £25 000

Work out the mean.

Show that the range is £28 000

Bethany believes that one of the salaries is an outlier.

Write down the salary that is most likely to be an outlier.

Explain why you think this value is an outlier.

Bethany removes the outlier and calculates the mean of the remaining seven salaries.

Without carrying out any further calculations, state whether the mean of the remaining seven salaries is greater than, is equal to or is less than the mean of all eight salaries. Give a reason for your answer.

Bethany uses the mean of the remaining seven salaries to draw conclusions about the salaries of all of the employees at her company.

Describe two things that could affect the reliability of her conclusions.

Naomi wants to find out what opinions people in her town have about holidays.

She designed a questionnaire and gave it to each person in a sample of 30 people who live in her town to complete.

Her questionnaire, her results and her conclusions are shown below.

Discuss whether or not the questions on Naomi’s questionnaire are appropriate and whether or not the results can be used to support Naomi’s conclusions.

John and Susan are investigating films.

John wants to find out the favourite type of film of the students at his school.

Susan wants to find out which type of film made the most money in UK cinemas last year.

State the population for

John’s investigation,

Susan’s investigation.

Susan plans to collect the information for her investigation from the internet.

Give advice to Susan so that she can ensure that the information she collects is reliable.

John is deciding between two different sampling methods for his investigation.

Method A: Quota sampling by sampling 20 students from each year group.

Method B: Opportunity sampling by selecting the first 50 students he sees one day.

For each method, give one reason why the method is not random.

Explain which of the two methods John should use in order to minimise bias. Give a reason for your answer.

Ben is investigating how far people travel to work each day. He asks the people in his office how far, in miles, they travel to work each day.

The grouped frequency table gives information about their answers.

Distance (d miles)	Frequency
0 d 5	7
5 d 10	10
10 d 15	9
15 d 20	6
20 d 25	3

Calculate an estimate of the mean distance. Give your answer correct to one decimal place.

Ben finds a newspaper report that says that the average distance travelled to work each day in England and Wales in 2001 was 8.3 miles. (Source: www.telegraph.co.uk)

From this Ben concludes that the distance travelled to work each day in England and Wales has increased since 2001

Give two reasons why Ben’s conclusion may not be reliable.

Ben is investigating how far people travel to work each day. He asks the people in his office how far, in miles, they travel to work each day.

The grouped frequency table gives information about their answers.

Distance (d miles)	Frequency
0 d 5	7
5 d 10	10
10 d 15	9
15 d 20	6
20 d 25	3

Calculate an estimate of the mean distance. Give your answer correct to one decimal place.

Ben finds a newspaper report that says that the average distance travelled to work each day in England and Wales in 2001 was 8.3 miles. (Source: www.telegraph.co.uk)

From this Ben concludes that the distance travelled to work each day in England and Wales has increased since 2001

Give two reasons why Ben’s conclusion may not be reliable.

Jon is investigating the colour of the eyes of all the students at his school. He recorded the colour of the eyes of a sample of 20 students in his class.

Work out how many of these students do not have brown eyes.

One of these students is selected at random.

Describe the likelihood that this student has grey eyes.

For Jon’s investigation, which statistical term best describes all of the students at Jon’s school?

Explain whether or not Jon could use his results to determine the most common colour of the eyes of teachers in his school.

Explain how the pie chart shows that fish made up the largest proportion of the total pet population of the UK in 2017

_........

The total pet population in the UK in 2017 was estimated to be 54 million.

Calculate an estimate for the number of fish in the total pet population of the UK in 2017

Melanie takes a random sample of 30 pet owners in her town and records the number of pets they each have.

She produced the following table for her results.

For Melanie’s sample, find the upper quartile of the number of pets owned.

Give an interpretation of the value of the upper quartile.

From this sample, Melanie concluded that everyone in her town owns at least 1 pet.

Comment on the validity of Melanie’s conclusion.

Cindy works in a library. She recorded the number of books borrowed by each person in a random sample of 20 people who borrowed books from the library one Monday. The table gives information about her results.

Cindy also recorded the number of books borrowed by each person in a random sample of 20 people who borrowed books from the library one Friday. The table gives two summary statistics for her results.

She claims that the mean number of books borrowed from the library on a Monday is greater than the mean number of books borrowed from the library on a Friday. Cindy also claims that there is a smaller spread in the number of books borrowed from the library on a Monday than on a Friday.

Comment on whether each of Cindy’s claims is supported by her results. You must give reasons for your answers.

Suzanne suggests repeating the investigation with a larger number of Mondays and Fridays.

Give a reason why Suzanne’s suggestion may be appropriate.

The table shows information from a survey about train passengers’ satisfaction with 10 categories of train facilities in spring 2018

In which of the 10 categories was

the ‘% satisfied or good’ the least?

the ‘% dissatisfied or poor’ the greatest?

How many people said ‘satisfied or good’ for punctuality?

Michelle claims that the percentages for ‘Cleanliness of the inside’ are the most reliable for these 10 categories.

Using the information in the table, suggest why she might think this.

Mehmet wants to conduct an investigation about school meals at his school. He wants to take a sample of students from his school.

Write down a sampling frame he could use.

Mehmet has two hypotheses:

● Younger students have school meals, rather than packed lunches, more often than older students do. ● More female students than male students are vegetarian.

Here are sections A, B and C of his plan for his investigation.

Name the sampling method that Mehmet plans to use.

Discuss whether or not Mehmet’s plans are appropriate. You must comment on each of the three sections A, B and C.

Mehmet also collects information about the amounts that the students pay for their school meals.

He sorts the costs of the meals into classes with equal widths of 5p.

The table gives some of his results.

Mehmet decides to use classes with equal widths of 10p instead of 5p.

Write down one advantage and one disadvantage of doing this.

According to an internet site, an estimate of the number of reindeer in a region of Ontario is 5000

(Source: www.ontario.ca

Giovani wants to verify this estimate.

He goes to the region of Ontario, captures a sample of 250 reindeer, attaches a tag to each reindeer and then releases the 250 reindeer back into the same region of Ontario.

Three days later, Giovani returns to the same region of Ontario and catches a sample of 98 reindeer.

He finds that 5 of these reindeer are tagged.

Giovani concludes that this information can be used to verify the estimate of 5000

Discuss the appropriateness of Giovani’s method and of his conclusion.

As part of your discussion you should show your calculations and state any assumptions made.