Statistical enquiry

Johan sells tickets for a walking tour, a boat tour and a bus tour.

The pictogram shows the number of people who chose each type of tour one Saturday morning in June.

Write down the modal type of tour.

How many more people chose the boat tour than the bus tour on this morning?

Johan thinks that the number of people choosing each type of tour will change in July compared to June.

He wants to investigate the number of people choosing each type of tour.

Write down a hypothesis that he could use for this investigation.

Suggest two things that Johan should do when collecting his data to ensure that he can compare his results for July with the results for June.

Louise wants to find out what local residents think about changes to car parking in the local town centre.

Louise is going to use a questionnaire.

Here is one of the questions she puts on the questionnaire.

Circle the word from the list that best describes the type of data that this question will collect.

quantitative discrete continuous qualitative

Here is a different question on her questionnaire.

Discuss whether or not this is a suitable question for the questionnaire.

Here is another question that Louise wants to put on the questionnaire.

This is not a suitable question. Explain why.

A company offers a number of activities for its employees. Yusra wants to investigate what employees at the company think about the activities.

Here is her plan for data collection, for calculations and for diagrams.

Discuss whether Yusra’s plans for data collection, for calculations and for diagrams are appropriate.

Benjamin collected data on the results for the male competitors in the Ironman Canada competition in 2022

The incomplete histogram and incomplete grouped frequency table give information about the times taken, in hours, for these competitors to complete the competition.

Time (t hours)	Frequency
8 t 9	2
9 t 10	23
10 t 11	58
11 t 12	134
12 t 13	188
13 t 14	183
14 t 15	..................
15 t 16	..................
16 t 17	102

Use the information in the histogram to complete the table.

Use the information in the table to complete the histogram.

Benjamin also collected data about the results of the male competitors in the Ironman UK competition in 2022

The histogram below gives information about the times taken, in hours, for these competitors to complete the Ironman UK competition.

Identify and interpret the type of skew shown in this histogram of the times for these competitors in the Ironman UK competition.

The time taken, in hours, for each competitor to complete the competition was recorded when the digital tag worn by the competitor passed the finish line.

Discuss the reliability of any conclusions made based on this method of data collection. Give a reason for your answer.

Benjamin wants to find out how competitors prepare for Ironman competitions. He plans to give a questionnaire to the finishers of his local competition. There are 1200 people in the competition. Benjamin intends to use systematic sampling and ask every 10th competitor who crosses the finish line.

Comment on the appropriateness of this plan for data collection.

Rebecca collected data from the internet about the litter size of wild boar in France and in Italy. The frequency polygons give information about the litter sizes.

(Source: www.academic.oup.com and www.researchgate.net) Rebecca concludes that the average wild boar litter size in France is bigger than in Italy.

Do the frequency polygons support Rebecca’s conclusion? Give a reason for your answer.

Give another comparison of the distributions of the wild boar litter sizes in France and in Italy.

Discuss how the data collection could reduce the reliability of these conclusions.

Agatha is investigating the times taken in the 2022 Olympic Women’s Skiathlon race.

There are two parts to the Skiathlon race. There is a 7.5 km classic stage and a 7.5 km free stage.

Agatha thinks there may be a relationship between the time taken by each skier to complete the classic stage and the time taken by each skier to complete the free stage. She suggests calculating a correlation coefficient to check.

Discuss the appropriateness of this suggestion. You should consider the type of data in your answer.

Agatha found the times taken for the event by all the skiers.

The incomplete histogram and incomplete grouped frequency table give information about the times taken, in minutes, to complete the 7.5 km classic stage of the race.

Use the information in the histogram to complete the table.

Use the information in the table to complete the histogram.

Agatha also drew this histogram for the overall times for the race.

Agatha concludes that the median is in the 50 t ≤ 52 minute time interval.

Does the histogram support Agatha’s conclusion? Show calculations to support your answer.

A ski manufacturer produces wax used on skis.

It has made a new type of wax and wants to find out if using this wax improves the skiers’ times.

Agatha thinks a matched pairs approach should be used to find out if the wax improves the time taken to complete the race.

Agatha suggests matching skiers based on the type of ski that they use and their times in the race. One skier in each pair would use their original wax and the other skier would use the new wax.

Comment on the appropriateness of this suggestion. Give a reason for your answer.

A council wants to encourage people to use the council leisure centres. The council has put advertisements in the town centres to tell people about the opening times and the available facilities in the leisure centres.

Nico has been asked to find out if the advertising has been successful. He plans to use cluster sampling where the clusters are customers at each leisure centre. For the selected clusters he will ask the customers to complete a questionnaire about the advertisements.

Comment on the suitability of Nico’s plan for sampling in this context.

Keti suggests that they should do a pilot study before they give out the questionnaire.

Give a reason why it is appropriate to do a pilot study.

The council also gave a voucher for one free admission to every child in the county.

The council thinks that some people have collected and used more than one of the free admission vouchers per child. Nico is going to include a question about this on the questionnaire. He thinks that he should use the random response technique.

Explain why it is appropriate to use the random response technique in this context.

Design a random response question that Nico could use on the questionnaire in order to collect this information.

Nico wants to find out if the number of people using the leisure centres increased after the advertisements and vouchers started a month ago.

He would also like to predict the number of people likely to use the leisure centres in the future.

Here is the plan for his investigation.

Discuss whether Nico’s plans for collecting, processing and presenting data are appropriate.

Freya wants to work out the life span of the light bulbs produced at her factory.

She selects a random sample of 50 light bulbs, and measures the time taken for each light bulb to fail.

Explain why conducting a random sample was more appropriate than conducting a census.

A teacher wants to investigate how many hours students spent revising for a test.

20 students have to complete a short questionnaire before they begin the test.

This is a copy of the questionnaire.

State one advantage of using a short questionnaire instead of interviewing each student.

State the type of question used as Question 2

The table below shows the data collected for Question 2

Give a reason why this data needs to be cleaned.

The teacher cleans the data. The teacher wants to display information about the test mark each student got compared to the number of hours this student revised.

The teacher plans to draw a diagram for this information. The diagram will be shown on a poster where these students can see it.

From the list below, circle the type of diagram it would be best to use for this target audience.

Box plot Pie chart Scatter diagram Cumulative frequency chart

A professor of sports science wanted to test if listening to music improved running performance.

She tested a sample of 20 adult males. All the males were similar in age and in fitness level.

The professor completed the study as a laboratory experiment. She asked all the males to run as far as they could in six minutes.

Each male ran twice. The first time he ran he listened to music. The second time he ran he did not listen to music.

Give two advantages of the professor completing this study as a laboratory experiment.

The professor believes the explanatory variable in this experiment is if the runners are listening to music or not.

State the response variable in this experiment.

One extraneous variable in this experiment is the type of music that the runners listen to.

How could the professor control this extraneous variable?

Explain why it is important to control extraneous variables in experiments.

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.

A group of homeowners are investigating the level of lead in their tap water.

Five homeowners each collected a 100 ml sample of their water at the same time each morning for 10 consecutive days.

All samples were sent to the same laboratory where the level of lead in each sample was measured.

Given these control measures, comment on the reliability of their results.

In previous tests in this area, the level of lead in the water was found to be normally distributed.

The mean level of lead was 7 µg/l and the upper action limit was 9.4 µg/l

Find the standard deviation used to set the upper action limit.

“ There are strict regulations concerning the amount of lead allowed in drinking water. The current drinking water standard is no more than 10 µg/ l ” (Source: www.stwater.co.uk)

One homeowner claims that the lead content in their tap water often exceeds 10 µg/l

Explain why this homeowner is likely to be wrong.

Control charts for the sample means and the sample ranges for the five homeowners over the ten days are shown below.

Decide what actions, if any, should be taken. Justify your answer by referring to both the sample means and the sample ranges.

A road safety officer records the speed of each vehicle travelling faster than the speed limit of 30 mph as they pass a site.

Here are the results.

Speed of vehicles (vmph)	Frequency
30 v 35	25
35 v 40	18
40 v 45	p
45 v 50	14
50 v 55	3

An estimate for the mean of these data was calculated to be 40.5 mph.

Find the missing value p in the table.

The government recommends that a speed camera should be installed if at least 20% of vehicles travel faster than the speed limit at a site. (Source: www.ageas.co.uk)

Explain why the data collected by the road safety officer cannot be used as evidence that a speed camera should be installed at this site.

The management of a factory is considering changing the working hours of their employees. Muhammad and Rose want to get the views of the employees in the factory.

Employees in the factory work on the production line or in the warehouse or in the office.

20 employees work on the production line. 15 employees work in the warehouse. 25 employees work in the office.

Muhammad plans to use a questionnaire. He plans to take a sample of the employees and ask them the questions on his questionnaire. For his sample, he decides to ask all of the employees who work on the production line.

Name this sampling technique.

Give two reasons why using this sampling technique may not be appropriate.

Muhammad wants to find out how many extra hours each employee would be willing to work each week.

Design a closed question that Muhammad could use in his questionnaire.

Muhammad collects the completed questionnaires. He finds that some of the employees on the production line have not responded.

Suggest two ways in which Muhammad could have reduced the number of non‑responses.

Rose decides to take a 10% systematic sample of all the 60 employees in the factory.

Describe in detail how this sample could be selected.

Rose plans to use a face-to-face interview.

How would using a face-to-face interview rather than a questionnaire improve the quality of the responses?

Roberta is investigating how the ages of brides getting married in the UK has changed from 2003 to 2013 She collects official data from the internet using the website ‘Office for National Statistics’.

Explain why this website will give reliable data.

Roberta wrote the following hypothesis before she collected her data,

Has the age of brides increased between 2003 and 2013?

Explain why it is not appropriate to use this as a hypothesis.

The table gives information about the data that Roberta collected.

Roberta wants to compare the proportion of brides in each age group and the total number of brides in each age group by presenting the results in charts for 2003 and 2013 She discusses how to present the results with Andria.

Andria thinks that they should use pie charts.

Roberta thinks that they should use comparative pie charts.

What advice would you give to Andria and to Roberta on their choice of charts?

Roberta used a circle with a radius of 4 cm for her comparative pie chart for 2003

Calculate the radius of the circle for her comparative pie chart for 2013 Give your answer correct to 2 decimal places.

Roberta is going to present her comparative pie charts to non-statisticians.

What extra information could she include on her pie charts to help the non-statisticians make comparisons?

Claire is planning an investigation into the length of time that a learner has to wait for a driving test. She wants to find out about how waiting time varies in different regions of the UK.

Here is her plan for data collection, for calculations and for diagrams.

Discuss whether Claire’s plans for data collection, for calculations and for diagrams are appropriate.

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.

Mobeen is investigating whether there is a difference in the amount of time spent reading by pupils in Green Park school and pupils at Golden Plains school.

He uses a census of all of the pupils at each school. Each pupil is asked to record the amount of time spent reading in a week.

Mobeen then collects this information from each student through an online database.

Part of the database is shown below.

Give two reasons why the data should be cleaned before processing.

Mobeen wants to compare the data for Green Park school with the data for Golden Plains school.

Once the data has been cleaned Mobeen plans to use all of the times to draw a single box plot.

Explain why this is not an appropriate thing to do.

Researchers investigated the effect of changing land usage on the amount of surface runoff water for areas of land around Brandywine Creek.

They recorded the change in forest cover area (km²) from 1992 to 2000 They also recorded the change in the amount of surface runoff (mm water) from 1992 to 2000

The table gives information about their data.

Calculate Spearman’s rank correlation coefficient for this data. Interpret this correlation in the context of the question.

Elizabeth wants to investigate possible drainage solutions that reduce surface runoff. She wants to establish whether a particular drainage solution reduces surface runoff.

Elizabeth plans to measure surface runoff in 10 areas for a year without the drainage solution. She will then install the drainage solution in these 10 areas and measure the surface runoff for a year.

Explain how Elizabeth’s plan to collect data controls some extraneous variables. You should include in your answer an example of an extraneous variable that is likely to be controlled in this investigation.

Elizabeth would like to reduce the time that she is collecting data to one year overall.

Describe how she could do this using a matched pairs approach.

Roxann wants to investigate the resting heart rates for members of her running club.

She believes that the resting heart rates will be normally distributed.

Here is the plan for her investigation.

Discuss whether Roxann’s plans for collecting and presenting data are appropriate.

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.

Claire is investigating sales of different types of vehicle over time. She plans to collect data on the numbers of motorcycles first registered in the UK over time.

Write down a suitable hypothesis for this investigation.

The time series graph shows some information about the numbers of motorcycles first registered in the UK from 2017 to 2019

Identify and interpret one example of seasonal trend shown by the time series graph.

Claire calculated 4-point moving averages for the information shown in the time series graph.

Explain why this is appropriate.

Claire also collected data on the numbers of cars first registered in the UK from 2017 to 2019

The time series graph shows some information about the numbers of cars first registered in the UK from 2017 to 2019 together with the first seven 4-point moving averages.

Compare the seasonal trend shown for the numbers of motorcycles first registered in the UK with the seasonal trend for the numbers of cars first registered in the UK.

The last three 4-point moving averages (thousands) for the number of cars registered in the UK from 2017 to 2019 are 576.0 575.3 573.9

Plot these three moving averages on the time series graph and draw a trend line.

Describe and interpret the trend in the numbers of cars first registered in the UK from 2017 to 2019

Connie is going to write a report on the difference in total rainfall between London and Aberdeen in 2019

She collects secondary data to investigate this.

What should Connie include in her report?

source of the data her telephone number her age
name of her school

Describe one way that she could obtain this secondary data.

The table shows the total rainfall, in cm, for each month in 2019 in London.

The mean monthly rainfall in Aberdeen in 2019 is 6.2 cm.

Connie considers the data in the table and concludes that the mean monthly rainfall for Aberdeen in 2019 is greater than the mean monthly rainfall in London in 2019

Is Connie correct? You must show how you get your answer.

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.

The choropleth map below represents a park that has been divided into 25 squares of equal area.

Arthur has collected data about litter in the park.

The number of pieces of litter collected in each square on one Saturday morning is shown.

Use the information in the choropleth map to calculate an estimate of the total number of pieces of litter that were collected that day.

Arthur works in this park. He has been asked to decide where a new bin should be placed in the park to help reduce the amount of litter. He concludes that the new bin should be placed in the corner of the park represented by the bottom right of the choropleth map.

Assess the validity of Arthur’s conclusion with reference to the choropleth map.

Ian suggests that the method Arthur used to collect his data is not suitable to reach a reliable conclusion.

Assess whether Ian’s suggestion is correct. Give a reason for your answer.

Mrs Singh is the Headteacher of a school with 1200 students.

She wants to find out about the ways the students at her school use the internet to complete their homework.

She is going to write a plan for her enquiry.

Write down one thing Mrs Singh should include in her plan for each of

• data collection

• choice of sampling method

• a diagram to represent the data collected.

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

Weronika works for a road traffic organisation. One day she is investigating the speeds of cars and the speeds of motorcycles along a motorway.

Here is part of the spreadsheet that Weronika used to record her results.

Speed (s miles per hour)	Percentage of cars	Percentage of motorcycles
30 s 40	3	0
40 s 50	18	two
50 s 60	42	7
60 s 70	27	124
70 s 80	6	56
80 s 90		11
Total	100	100

Give a reason why Weronika will need to clean the data.

Weronika concludes that the value of 124 in the spreadsheet must be wrong.

Explain why.

Here is the information about motorcycles from the spreadsheet with the data cleaned.

Speed (s miles per hour)	Percentage of motorcycles
30 s 40	0
40 s 50	2
50 s 60	7
60 s 70	24
70 s 80	56
80 s 90	11
Total	100

Use linear interpolation to work out an estimate of the median speed of the motorcycles.

The frequency polygon for the speeds of cars has been drawn on the grid.

On the same grid, draw the frequency polygon for the speeds of the motorcycles.

Using the two frequency polygons, compare the skew of the distribution of the speeds of the cars with the skew of the distribution of the speeds of the motorcycles. Explain what your comparison means in context.

Weronika works for a road traffic organisation. One day she is investigating the speeds of cars and the speeds of motorcycles along a motorway.

Here is part of the spreadsheet that Weronika used to record her results.

Speed (s miles per hour)	Percentage of cars	Percentage of motorcycles
30 s 40	3	0
40 s 50	18	two
50 s 60	42	7
60 s 70	27	124
70 s 80	6	56
80 s 90		11
Total	100	100

Give a reason why Weronika will need to clean the data.

Weronika concludes that the value of 124 in the spreadsheet must be wrong.

Explain why.

Here is the information about motorcycles from the spreadsheet with the data cleaned.

Speed (s miles per hour)	Percentage of motorcycles
30 s 40	0
40 s 50	2
50 s 60	7
60 s 70	24
70 s 80	56
80 s 90	11
Total	100

Use linear interpolation to work out an estimate of the median speed of the motorcycles.

The frequency polygon for the speeds of cars has been drawn on the grid.

On the same grid, draw the frequency polygon for the speeds of the motorcycles.

Using the two frequency polygons, compare the skew of the distribution of the speeds of the cars with the skew of the distribution of the speeds of the motorcycles. Explain what your comparison means in context.

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.

Explain what is meant by a simple random sample.

The manager of a health centre is reviewing the amount of time allowed for each patient to be seen by a doctor.

The manager wants to get the opinions of the people registered at the health centre.

She plans to give a questionnaire to each patient who has an appointment with a doctor next Monday. She wants each patient to complete the questionnaire at home and return the completed questionnaire to the health centre.

Assess the manager’s plan to get the opinions of the people registered at the health centre.

Here is an open question that the manager is considering for her questionnaire.

What do you think about the 5 minutes currently allowed for appointments?

This is not a good question.

Give one reason why.

Design a suitable closed question for the manager to use on her questionnaire so that she can decide how long should be allowed for each patient to be seen by a doctor.

When the manager has designed her questionnaire, she decides to pre-test it by using a pilot survey with a small sample of patients.

From the five options below, tick (✓) the three that are the best reasons for the manager to pre-test her questionnaire in this way.

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).

Irina is investigating whether the percentage of the population of a country living in urban areas has an effect on the life expectancy in that country.

Suggest a hypothesis Irina could use for her investigation.

Irina collected the following information about 10 countries.

The Urban population (%) refers to the percentage of the population of the country who live in urban areas.

Irina used statistical software to draw a scatter diagram for the information in the table.

Give a reason why a scatter diagram is an appropriate diagram to use.

For this investigation, which variable is the explanatory variable? Give a reason for your answer.

The scatter diagram from the statistical software is shown below.

Explain, giving a statistical reason, whether or not this scatter diagram supports your hypothesis in part (a).

For these 10 countries, the double mean point of the data is (63.9, 77.8).

Using this information, draw a line of best fit on the scatter diagram.

Using statistical software, Irina finds that the gradient of the line of best fit should be 0.19

Interpret the gradient of the line of best fit.

Irina now finds that South Africa has Urban population 65% and Life expectancy 63 years.

Determine how this information for South Africa fits with the relationship shown in the scatter diagram for the other countries.

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).

Irina is investigating whether the percentage of the population of a country living in urban areas has an effect on the life expectancy in that country.

Suggest a hypothesis Irina could use for her investigation.

Irina collected the following information about 10 countries.

The Urban population (%) refers to the percentage of the population of the country who live in urban areas.

Irina used statistical software to draw a scatter diagram for the information in the table.

Give a reason why a scatter diagram is an appropriate diagram to use.

For this investigation, which variable is the explanatory variable? Give a reason for your answer.

The scatter diagram from the statistical software is shown below.

Explain, giving a statistical reason, whether or not this scatter diagram supports your hypothesis in part (a).

For these 10 countries, the double mean point of the data is (63.9, 77.8).

Using this information, draw a line of best fit on the scatter diagram.

Using statistical software, Irina finds that the gradient of the line of best fit should be 0.19

Interpret the gradient of the line of best fit.

Irina now finds that South Africa has Urban population 65% and Life expectancy 63 years.

Determine how this information for South Africa fits with the relationship shown in the scatter diagram for the other countries.

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.

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.

The manager of an art shop wants to find out about the shopping habits of customers in his shop. He wants to find out how many customers leave the shop without buying anything.

For the next four weeks he plans to record how many customers leave the shop without buying anything each Monday.

Give one problem with the manager’s plan for data collection.

The manager asks his staff to record the number of people served at the till each hour.

‘Number of people served’ is an example of which type of data?

The staff record the number of people served per hour. Here is the completed table.

Identify two problems with this completed table.

The manager also collected data on the type of art that his customers produce.

The incomplete bar chart shows information about the type of art produced by some of his customers.

Photography is the type of art produced by 11 customers.

Complete the bar chart.

Which type of art is produced by half as many customers as those who produce paintings?

Diana is investigating the numbers of wolf pups in wolf dens and the numbers of fox pups in fox dens.

She collected data from the internet.

Data collected from the internet is an example of secondary data.

Explain why data collected from the internet is an example of secondary data.

Diana finds data on the number of wolf pups observed in a random sample of 16 wolf dens.

Here is the data.

3 3 4 4 5 5 5 5

5 6 6 7 7 8 8 9

(Source: college.cengage.com)

Diana also collects data on the number of fox pups in each den of a random sample of fox dens.

Here is information about the numbers of fox pups in the fox dens.

(Source: foxsurvey.ucdavis.edu)

Compare the distribution of the numbers of wolf pups in the wolf dens and the distribution of the numbers of fox pups in the fox dens. Justify your comparisons and interpret one of them in the context of the numbers of pups.

Explain how the use of secondary data affects the reliability of your conclusions in part (b)(i).

Anne is investigating the effect of caffeine on reaction times.

Here is what Anne has written down for her hypothesis and how she plans to collect, process and present her data.

Discuss whether Anne’s hypothesis and her plans for collecting, processing and presenting data are appropriate.

Anne is investigating the effect of caffeine on reaction times.

Here is what Anne has written down for her hypothesis and how she plans to collect, process and present her data.

Discuss whether Anne’s hypothesis and her plans for collecting, processing and presenting data are appropriate.

Matthew is investigating how the cost of computer data storage has changed over time.

Suggest a hypothesis Matthew could use.

Matthew finds the cost per terabyte of computer data storage in each first quarter of five successive years.

The table gives cost per terabyte, in US$, for computer data storage. It also gives two of the chain base index numbers, correct to one decimal place, for this information.

(Source: jcmit.net)

Find the chain base index numbers for 2018 and 2019 and write them in the table. Give each value correct to one decimal place.

Find the geometric mean of the four chain base index numbers. You must show your working. Give your answer correct to one decimal place.

Interpret your answer.

David is writing a project about iguanas.

He finds data on the internet posted by a biologist investigating iguanas on the Galapagos Islands.

The biologist captured and tagged a sample of iguanas in La Loberia. The tagged iguanas were then released back from where they were captured.

After a few days the biologist captured at random a second sample of iguanas in La Loberia and recorded the number of tagged iguanas in this sample.

Here is the biologist’s data.

(Source: www.discoveringgalapagos.org.uk)

Work out an estimate of the total number of iguanas in La Loberia.

Discuss the reliability of using the biologist’s data to work out an estimate of the total number of iguanas in La Loberia.

It is suggested that some of the tags might have fallen off the iguanas between the two samples.

If this had happened, describe the effect it would have on the estimate worked out in part (a).

A construction company employs three types of employee: builders, carpenters and electricians.

The table gives the number of each type of employee.

The owner of the construction company wants to find out what the employees feel about their jobs.

She plans to take a sample of 30 employees, stratified by type of employee.

Calculate the number of carpenters that should be in her sample.

The owner wants to know whether the employees in her sample believe that they work hard consistently. She knows that she will have to ask a sensitive question so she plans to use a random response question in order to find out this information.

Design a random response question that the owner could use on a questionnaire in order to collect this information.

A team of researchers investigated the time intervals between calls of tree frogs.

They investigated two species of tree frog: the Italian tree frog and the European tree frog.

They found that there was a correlation between air temperature, xC, and amount of time between calls, y milliseconds, for each of the two species of tree frog.

The table gives the equation of the regression line for the data for each of the two species of tree frog.

When x = 27, both equations give the same value of y, correct to the nearest whole number.

Explain what this means in context.

The equation of the regression line gives information about the relationship between temperature and time between calls for each of the two species of tree frog.

Using this information, compare the relationship for the Italian tree frog with the relationship for the European tree frog.

A second group of researchers investigated the relationship between temperature and the time between calls for a third species of tree frog. They also obtained the equation of the regression line for this species of tree frog.

The results of the different teams of researchers are to be compared.

Give one potential limitation of doing this.

People who completed higher education were asked what they were doing six months after they graduated.

The table shows information about the answers given by those who completed higher education in the UK for the years 2013 to 2017

(Source: Higher Education Statistics Agency

Write down the number of people in 2016 who answered Overseas work.

Write down all years in which the number of Non-response was greater than 120 000

Between which two consecutive years did the greatest increase in the number of people answering Further study take place?

For one of the categories in the table, the number of people decreased each year from 2013 to 2017

Write down this category.

Here are two methods that could have been used to collect this information.

Method 1: Conduct a face to face interview with each person who completed higher education.

Method 2: Send an email to each person who completed higher education inviting them to complete an online survey.

State which of Method 1 or Method 2 is the more appropriate method to use. Give a reason for your answer.

Sam works in a bank.

He wants to survey the people who work at the bank in order to investigate whether there is a relationship between a person’s height and the person’s salary.

Suggest a hypothesis for Sam’s investigation.

Sam plans to collect the data himself.

Complete correctly the sentence below.

Data collected by the person who is going to use the data is called_........ data.

Explain why Sam may have difficulty in collecting data about people’s salaries.

The list below contains some variables that Sam is planning to collect.

salary height age

The variables that Sam is planning to collect are all examples of which type of data?

Sam decides to take a sample of employees by putting the name of everyone who works at the bank in a list in alphabetical order and selecting every 5th name on the list.

Write down the name of this sampling method.

Part of a spreadsheet of data that has been collected by Sam is given below.

Sam tries to use his spreadsheet to calculate automatically the mean salary, the mean height and the mean age of the employees in his sample.

Sam’s computer gives him an error message for two of these calculations.

Write down the calculation for which there is not an error message. Give a reason why there is an error message for the other two variables.

Reeta read a report that said that these days more 18 to 24 year olds find out what is happening in the news from social media than find out by watching TV. She decided to investigate how the 12 000 students at her university find out what is happening in the news. Reeta wrote the following as a hypothesis: Do more students get their news from social media than from other sources?

Comment on whether it is appropriate to use this as a hypothesis.

Reeta planned to use a sample of the students at her university stratified by gender and by age.

Explain why this method of sampling would be appropriate.

Reeta found that she could not get a list of all the students at her university to use for her stratified sample. Instead she decided she would question students in the university cafeteria. Here is the data collection sheet Reeta plans to use.

Discuss whether Reeta’s data collection sheet is appropriate for her to use. You should consider how Reeta might use the responses and describe any problems she may have when she uses the data collection sheet.

Reeta’s friend George is at a different university with 18 000 students.

George carried out a similar investigation at his university so that he and Reeta could compare their results.

They each questioned the same percentage of students at the university they each attend.

They decided to compare their results by using comparative pie charts.

Explain why comparative pie charts would be appropriate.

Reeta used a circle with diameter 10 cm for her pie chart.

Calculate the diameter of the circle that George used for his pie chart.

Discuss two things that Reeta and George should have considered in planning their investigations to help improve the reliability and validity of their comparisons.

Sue is interested to know how much time children in the UK spend doing chores around the house each week.

She plans to ask a sample of 20 students at her school to record the number of hours they each spend doing chores in a week.

Explain why it is more appropriate for Sue to take a sample rather than a census in this case.

She will ask each of the 20 students to write down on a piece of paper the time they each spent doing chores around the house in one week.

Describe one problem Sue might encounter during the statistical enquiry process with regard to non‑response or unexpected outcomes.

You should explain how Sue could try to overcome the problem.

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.

Matthew is collecting information about road safety in his town.

He wants to collect information from drivers who live in his town.

Matthew plans to ask drivers at his local car park about their views on road safety. He plans to collect a quota sample of 20 drivers from each of three age groups.

Matthew’s three age groups of drivers are

39 years old or younger 40 years old to 59 years old 60 years old or older

Comment on whether Matthew’s plans are appropriate.

Matthew wants to know how many drivers, who live in his town, drive faster than the speed limit on the motorway. Matthew knows that he will have to ask a sensitive question so he plans to use the random response technique to find out this information.

Design a random response question that Matthew could use on a questionnaire in order to collect this information.

A supermarket manager is planning the design of a new petrol station.

He wants to use the results of spinning a coin to simulate on which side of a car the fuel filler cap is. He assumes that the fuel filler cap is equally likely to be on the left of the car as it is to be on the right of the car. He wants to use the simulation in order to predict how many cars will arrive at the petrol station before there are four consecutive cars with filler caps on the same side.

Explain how he could use a coin to carry out the simulation.

A new cafe is also opening at the supermarket. Three different types of hot drink – tea, coffee and hot chocolate – will be sold in the cafe.

The table gives, for each type of hot drink, the fraction of the total number of hot drinks sold in other cafes of this supermarket chain.

The manager wants to predict how many of each type of hot drink the cafe will sell in an hour the first Saturday the cafe is open.

The manager expects that there will be 40 customers in the cafe that hour.

He assigns the numbers shown in the table below to each type of hot drink.

He will then use a calculator to generate 40 random numbers between 0 and 9 to simulate 40 customers.

Comment on the suitability of the manager’s plan.

Jane collected data about the earnings of the 10 highest earning male tennis players and the 10 highest earning female tennis players.

Jane plans to use a scatter diagram to compare the earnings of the male tennis players with the earnings of the female tennis players.

Discuss whether or not a scatter diagram would be a suitable diagram to use.

Jane collected her data from the internet.

Suggest a possible reason why it might not be possible to collect primary data in this case.

Suresh is a secondary school student.

He wants to find out how many days off school each teacher has had in the last year.

Suresh plans to find out this information by using his school’s employee records.

Give one potential problem with Suresh’s data collection plan.

Ami and Tia work in a primary school.

Ami and Tia are investigating the numbers of days that people working for their school have been absent due to illness.

The table gives information about the number of days that each of 30 people working for the primary school have been absent due to illness in the last year.

Ami and Tia want to work out the average number of days absent due to illness.

Which of these two methods is correct, Ami’s or Tia’s? You must give a reason for your answer.

Find the median number of days absent due to illness.

Chris is carrying out an experiment to see if left handed people have a better memory than right handed people. He has two groups of people with the same number of people in each group.

Group A contains left handed people only and group B contains right handed people only.

Each person is given 15 objects to memorise and then they are tested to see how many objects they remember.

The variables for the experiment are shown in the table.

For each variable, put a tick () in the correct column of the table to show the type of variable.

Chris is not sure whether to work out the mean or the median number of objects each group remembered.

Describe a situation for which it would be more appropriate to work out the median than the mean.

Chris worked out these summary statistics for the number of objects people in group A remembered.

Lower quartile = 5 Median = 7 Upper quartile = 8

Sanjit is a member of group A.

He remembered 14 objects.

Determine whether or not the number of objects Sanjit remembered is an outlier.

A company wants to investigate the number of sick days its employees have off work.

The company uses a questionnaire.

Here is one of the questions on the questionnaire.

Assess the appropriateness of the method the company uses.

All the company’s employees completed the question.

615 ticked box A.

102 ticked box B.

Show that an estimate of the number of employees who ticked box A because they answered yes to the question is 137