Sampling & Data Collection (OCR AS Maths: Statistics)

The Office for National Statistics runs a compulsory census every ten years to gather information about all individuals and households in England and Wales.  The information gathered helps organisations make decisions on planning and funding for public services in each area, including transport, education and healthcare.

Suggest one advantage and one disadvantage of the Census only being carried out every ten years.

A local council wants to gather opinions from residents about opening a new care home.  They decide to conduct their own survey and need to pick between using a systematic or a stratified sample. 

Give two reasons why the council may be better off conducting their own survey rather than attempting to use census data to obtain the desired information.

Explain the main differences between a systematic sample and a stratified sample.

Every week an orangutan sanctuary measures the weight of each of its orangutans.

The weights, to the nearest kg, of ALL their 18 adult males are listed below:

                   52, 57, 63, 80, 56, 66, 101, 68, 55, 96, 70, 62, 66, 64, 99, 91, 55, 92

Using an opportunity sample of size six, calculate the mean weight of the male orangutans from the data set above.

Starting from the third data value, take a systematic sample of size six and re-calculate the mean weight of the male orangutans from the data set above.

Compare your results from parts (b) and (c) and state, with a reason, which sampling method is more reliable.

A supermarket wants to gather data from its shoppers on how far they have travelled to shop there. One lunchtime an employee is stationed at the door of the shop for half an hour and instructed to ask every customer how far they have travelled.

State and briefly describe an alternative method of non-random sampling that the employee could use to obtain the required data for a sample of 30 customers.

To check the quality of produce used in a restaurant kitchen, the head chef likes to taste one item from each box of produce as soon as it arrives at the restaurant.  Some produce is then kept in cold storage and used throughout the week.

A company wants to survey 15% of its staff to find out whether employees would like to continue working from home after the Covid-19 pandemic.  The company’s 580 members of staff are grouped by job as follows: 295 engineers, 11 managers, 154 office staff and 120 apprentices.

Suggest a suitable sampling method and explain how the company can use this method to obtain its sample.

The wingspans of a random sample of flamingos in a reserve were measured to the nearest centimetre.  The results are shown in the table below:

A toy shop, ‘Toys 4 U’, tests the battery life for a new toy by leaving a sample of the toys switched on until their batteries run out.

Give one reason why the shop decided to use a sample rather than a census.

Five toys out of a shipment of 5000 were tested and the battery life (in minutes) of each toy was recorded:

               172 minutes   252 minutes   248 minutes    155 minutes   161 minutes

The shop decides to display a sign beside the toys saying ‘Up to 4 hours battery life’.  Use the sample data above to comment on the sign.

Suggest one way in which the shop could improve its prediction.

Ramona wants to examine data about the number of people who work from home. As there are 10 regions in the large data set, she initially decides to look at a sample of 10 Local Authorities. She decides to compare data from the first Local Authority listed for each region, with the results shown in the table below:

The North East is made up of just 12 Local Authorities whereas the South East contains 67 Local Authorities. All other regions have between 21 and 47 Local Authorities each.

Give one reason for why the sample above is not a stratified sample.

Explain why a sample size of 10 would not be suitable if Ramona decided to use a stratified sample for her study.

Suggest a way Ramona could fix the problem outlined in part (b) in order to create a stratified sample. You must justify your suggestion with mathematical working.

An online magazine which offers both free and paid for content has a large number of readers.  Readers can view additional content by paying a monthly subscription fee.  Based on reviews on the magazine’s website, the editor of the magazine believes that an additional type of content could be introduced.  Before making any changes, the editor decides to carry out a sample survey to obtain the opinions of the readers. 

Define the population that would be associated with the magazine.

Give one advantage and one disadvantage that would have resulted from the editor using a census rather than a sample survey.

The editor decides to gather opinions from only those readers who subscribe to the additional content.  A random sample of 25 subscribers is selected for the sample survey.

Suggest a suitable sampling frame for the survey and identify the sampling units.

State two sources of uncertainty that may arise based on the chosen sampling method and sample size.

A group of 20 sloth sanctuaries across Central and South America record data on all the sloths in their care.  The sanctuaries have a central database where data for each sloth is collated.  A large variety of data on each individual sloth is collected and used to help decide when sloths are ready for releasing into the wild. 

Sanctuary owners want to look at a sample of sloths as part of a general health and well-being survey.  Sloths are said to have matured into adults when they reach the age of 5; before this they are classed as juveniles.  While juveniles may be treated as a single group for this survey, it is important that adult females and adult males be considered separately.

In the database there are currently 240 adult sloths, 60% of whom are male, and 64 juvenile sloths.  Explain how a stratified sample of size 35 could be taken.

Sanctuary owners want to look at a separate sample of sloths to get an idea of how many may be suitable for release.  Sloths can be released into the wild as long as they reach a heathy weight and are over the age of 3.  Male sloths when fully matured are generally heavier than females, although for juvenile sloths this weight difference is negligible.

Given that a quarter of the 64 juvenile sloths in the database are over the age of 3, explain how a stratified sample of size 35 could be taken to study suitability for release.

Stephan is researching the effects a new energy drink has on the glucose levels of students aged 13 to 18.  He decides to measure the blood glucose levels of 50 female students and 50 male students.

Stephan is provided with an alphabetical list of 350 male students aged 13 to 18, each of whom has agreed to supply a blood sample if asked.

Explain how Stephen could use a calculator or a random number generator to take a simple random sample from the male students aged 13 to 18.

Stephen has an equivalent list of 350 female students aged 13 to 18.

Explain how Stephen could take a systematic sample from the female students aged 13 to 18.

Freda needs to conduct a survey to investigate the type of ice cream people prefer.  She wants a random sample of people who eat ice cream.  She decides to stand in a busy high street on a Sunday afternoon and attempt to get shoppers to answer her questions.

Having been unsuccessful in obtaining enough data from her previous attempt, Freda decides to look at the electoral register for her town and select a new sample of people to contact.  She needs a sample of at least 50 people, so she decides to choose a person at random from the register and then use that person and every 5th person on the register after that (wrapping back around to the start of the register if necessary) to build her sample.

Suggest an alternative method for Freda to use and explain your reasons.

The CEO of Save My Exams, Jamie, wants to find out what users would like to see on the revision website in future.  He notices that around 15% of those who access the site have signed up to the mailing list to get content updates.

An employee suggests that they send out an email to all those who have signed up to the mailing list with a questionnaire for them to complete and return. 

Jamie decides to separate users by exam board to gather more detailed opinions.  A member of the Maths Content Team suggests the use a table of random numbers to select a random sample of 100 users from the 4581 IB mailing list subscribers.  The first five random numbers from the table are as follows.

                               02743      45290      19024      24337      90044

Explain how Jamie could use these random numbers to select the first few members in the sample.

A researcher measured heights of a random sample of giraffes.  The heights in metres are summarised in the table below.

All giraffes are allocated a unique four-digit ID number.  The researcher wants to randomly select five of the giraffes to test a new tracking device.  The researcher selects the first five giraffes with three zeros in their ID number.

A factory produces paper fruit baskets used for fruit pickers at ‘pick your own’ farms.  The breaking load of a paper fruit basket is the maximum load that it can carry before the basket handles break.  One ‘pick your own’ farm purchased 15 000 paper fruit baskets but wishes to test a sample of these to establish the breaking load of the baskets.

Suggest two reasons why a census would be unsuitable for this purpose.

The farm tests a random sample of six paper fruit baskets.  The loads required for the handles to break are shown below:

         2.035 kg       2.845 kg         2.528 kg        1.998 kg        2.212 kg        2.378 kg

The factory claims that the fruit baskets can carry 2 kg of fruit without breaking. Use the sample data to comment on this claim.

The large data set contains data on methods of travel to work for people in employment in 2001 and 2011. Data for Nottingham in the East Midlands in 2001 is summarised in the table below:

A stratified sample was taken for each method of travel in 2001, with the number of people travelling to work in a car or van given in the table below.

Given that the total sample size is 10200, use your knowledge of how stratified samples are constructed to complete the table with the appropriate numbers for the other methods of travel.

In a stratified sample of the methods of travel for Nottingham in 2011, the sample size is chosen so that, to the nearest integer, it represents the same proportion of the total number of people in employment as was used in the above stratified sample for 2001.
In the 2011 sample, the number of people who commuted to work on foot was calculated to be 1912.