Sampling & Data Collection (CIE A Level Maths: Probability & Statistics 2)

A census is when data is collected from every member of the population. In the UK, the Office for National Statistics runs a compulsory census every ten years, in years ending in 1 (2001, 2011, etc), to gather information about all individuals and households in England and Wales. This information 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 as opposed to using information from the 2011 census.

Give a reason why the council may be wise to conduct their own survey rather than using data from the 2011 census.

Briefly describe how the council could use a random number generator to select residents for their survey.

Clyde works at an orangutan sanctuary and part of his job involves measuring the weight of each of its orangutans once a week.

The weights, to the nearest kg, of ALL their 20 adult males are listed below:

52, 57, 63, 80, 56, 66, 101, 68, 55, 96, 70, 62, 66, 64, 99, 91, 55, 92, 73, 61

State a reason why, in this case, it is important that Clyde weighs every orangutan rather than taking a sample of them.

The sanctuary decides to trial a new medication by testing it on 5 of their 20 adult male orangutans. Clyde assigns each orangutan one unique 2-digit number (from 00 to 99), then uses a 2-digit random number generator to select which orangutans will be given the new medication.  Clyde ignores any numbers generated that are not assigned to an orangutan.

A supermarket wants to gather data from its shoppers on how far they have travelled to shop there.

Give a reason why sampling shoppers from the supermarket would be appropriate in this case.

One lunchtime an employee is stationed at the door of the shop and instructed to ask the first 10 customers how far they have travelled. Give two criticisms of using this method to sample shoppers.

Briefly explain why it is impossible to have a sampling frame for the population in this case.

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. The head chef opens each box in turn and takes the top item of produce to taste test.

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 has 580 members of staff.

Describe how the company can randomly sample its staff, stating clearly the sampling frame that could be used.

Work out the sampling fraction that the company will use.

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

For each scenario below, state, giving a reason, whether the scenario could lead to bias in the data collected.

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.

Briefly explain why, , the mean age of a sample of the magazine’s readers, is only an estimate of , the mean age of the parent population.

The editor decides to gather opinions from only the 3750 readers who subscribe to the additional content. A random sample of 25 subscribers is selected for the sample survey.

A charity running four sloth sanctuaries across a wild area of South America records data on all the sloths in its care. The numbers of sloths at the four sanctuaries are 24, 40, 16 and 56. To compare care across the 4 sanctuaries the charity decides to take a sample of sloths from each sanctuary and run some health tests on the sloths selected.

For the sanctuary with 40 sloths in its care, identify the sampling frame and briefly describe how the charity could efficiently use a three-digit random number generator to select the sloths from this sanctuary.

The charity decides to use a sampling fraction of one-eighth from each sanctuary. Determine the total number of sloths that will be selected for the health tests.

The results from the health tests showed that the sloths at the sanctuary with the smallest population weighed, on average, less than those from the other three sanctuaries, whereas the average weights of the sloths in those other three sanctuaries were similar.

Suggest a reason why this does not necessarily mean that the sloths at the sanctuary with the smallest population are in poorer health than those at the other three sanctuaries.

The charity wishes to determine which sloths are suitable for release into the wild. A sloth can be released once it has reached the age of 3 and is of a healthy weight. Explain why the information from the sample conducted by the charity is of little use to them in determining which sloths are suitable for release into the wild.

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

Stephan has a sampling frame which is an alphabetical list of 100 of the male students in his year group from the Sixth Form college he attends. All 100 have agreed to have a blood glucose test should they be selected in the sample.

Freda wants to conduct a survey to investigate the type of ice cream people prefer.

She decides to stand in a busy high street on a Sunday afternoon and attempt to get shoppers to answer her questions.

Explain why Freda’s sampling technique is not a random sample.

Having been unsuccessful in obtaining enough data from her previous attempt, Freda decides to look at the electoral register for her town and randomly select a new sample of people to contact.

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 using a table of random numbers to select a random sample of 100 users from the 4581 CIE 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 wishes to measure the heights of a random sample of giraffes from a large nature reserve where each of the 2400 giraffes is able to be tracked and identified.

When the current tracking system was first introduced, each giraffe was assigned a unique four‑digit ID number between 0001 and 2400.  The researcher wants to randomly select 100 of the giraffes to test a new, updated tracking system and decides to select the giraffes with the ID numbers 0001-0100.

Whilst attaching the new tracking devices to the 100 giraffes selected for the sample the researcher also measured the heights of the giraffes and worked out their mean height, .

Explain why  produces only an estimate of the population mean height, .

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.

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       1.975 kg       2.128 kg       1.898 kg            2.112 kg       1.999 kg

The farm decides to perform another 4 random sample tests, taking another sample of six paper baskets each time.  They calculate the mean breaking load for each sample.