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

A school has 40 staff and 490 students. Before this year's summer festival, the headteacher wants to survey a sample of the staff and students about their opinions of the festival.

The headteacher first suggests asking the staff and students who are helping to run the festival.

Give a reason why this method is unsatisfactory.

Instead, the headteacher numbers all 530 staff and students from 1 to 530 and uses a calculator to generate random digits. The first 20 random digits are as follows.

1 2 7 3 9 4 0 6 5 1 2 7 8 8 3 4 8 8 2 0

The numbers of the first two people in the sample are 127 and 394.

Use these digits to find the numbers of the next two people in the sample.

A local council wants to find out the opinions of residents about opening a new care home. The council numbers the 726 adults living in one village from 1 to 726 and uses a calculator to generate random digits. The first 20 random digits are as follows.

3 1 8 0 7 2 9 5 0 3 1 8 7 0 4 6 1 5 2 9

The numbers of the first two adults in the sample are 318 and 72.

Write down the numbers of the next two adults in the sample.

To choose more adults, the council starts again from the second random digit, 1, and obtains the numbers 180 and 503.

Explain why this method will not produce a random sample.

To check the quality of produce delivered to a restaurant, the head chef tastes one item from each box of produce as soon as it arrives. The chef opens each box in turn and tastes the item at the top of the box.

(i) Give a reason why it is necessary for the chef to test a sample rather than every item of produce.

(ii) Give a reason why the chef may not test every type of produce that is delivered.

(iii) Explain why the chef's method does not give a random sample of the produce.

The wingspans of a sample of 25 flamingos in a nature reserve were measured to the nearest centimetre.

(i) State one condition needed for the sample to be a random sample.

(ii) A researcher suggests that collecting the data on a weekday morning might cause the sample to be biased. State, with a reason, whether the researcher's suggestion is valid.

An online magazine offers both free content and paid-for content, and has a large number of readers. Before making any changes, the editor decides to carry out a sample survey to obtain the opinions of the readers.

(i) State one advantage of carrying out a sample survey rather than surveying all the readers.

(ii) The editor finds the mean age, $\overline{x}$ years, of the readers in a sample. Explain briefly why $\overline{x}$ is only an estimate of the mean age, $\mu$ years, of all the magazine's readers.

The editor decides to gather opinions from only the 3750 readers who pay for the additional content. A random sample of 25 of these readers is chosen.

Give a reason why this sample may not be representative of all the magazine's readers.

Stephan is researching the effects a new energy drink has on the glucose levels of students aged 13 to 18. He plans to measure the blood glucose levels of a random sample of 50 students.

Stephan uses an alphabetical list of 100 male students in his year group at the Sixth Form college he attends. All 100 have agreed to have a blood glucose test if they are selected.

Explain how Stephan could use a two-digit random number generator to choose a random sample of 50 students from his list.

Explain how the data could be biased.

A researcher wishes to measure the heights of a random sample of the 2400 giraffes in a large nature reserve. Each giraffe has a unique four-digit identification number from 0001 to 2400.

The researcher decides to select the giraffes with identification numbers 0001 to 0100.

(i) Explain why this method of selection will not produce a random sample.

(ii) Describe how the researcher could use random numbers to select a random sample of 100 giraffes.

The researcher measures the heights of the 100 giraffes in the sample and finds their mean height, $\overline{h}$ m.

Explain why $\overline{h}$ is only an estimate of the mean height, $\mu$ m, of all the giraffes in the reserve.

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 two reasons 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.

(i) Explain how Clyde's system of selecting orangutans will produce a random sample of the population.

(ii) Explain how Clyde could make more efficient use of the 2-digit random number generator.

A charity runs four sloth sanctuaries, which care for 24, 40, 16 and 56 sloths. The charity chooses a sample of sloths from each sanctuary for health tests.

At the sanctuary with 40 sloths, the sloths are numbered 1 to 40. The charity generates random three-digit numbers and allocates them to the sloths as follows.

000 to 024 to sloth 1, 025 to 049 to sloth 2, and so on, up to 975 to 999 to sloth 40

The first four random numbers generated are 137, 862, 140 and 503.

(i) Find the numbers of the first three sloths in the sample.

(ii) Explain why this method is more efficient than giving each sloth a single three-digit number.

The charity tests 3, 5, 2 and 7 sloths from the sanctuaries with 24, 40, 16 and 56 sloths respectively. The sloths tested at the sanctuary with 16 sloths weighed less, on average, than those tested at the other three sanctuaries.

Give a reason why this does not necessarily mean that the sloths at the sanctuary with 16 sloths are less healthy than those at the other sanctuaries.

A sloth can be released into the wild once it has reached the age of 3 and has a healthy weight.

Explain why the results of these tests are of little use for deciding which sloths are ready to be released into the wild.

Freda wishes to investigate the flavour of ice cream preferred by people living in her town. She stands in a busy high street on a Sunday afternoon and asks shoppers to answer her questions.

Give two reasons why this method of selecting a sample is unsatisfactory.

Freda instead decides to choose a random sample of people from the electoral register for her town, which is a list of the adults who are registered to vote.

Give a reason why this sample may still not be representative of the people in her town who eat ice cream.

Freda posts a questionnaire to each person in her sample and uses only the replies that she receives.

Explain why the people who reply may not form a random sample.

The owner of a revision website, Cram Central, wants to find out what users would like to see on the website in future. The website has over 650 000 users, and about 15% of them have signed up to its mailing list to get updates.

An employee suggests emailing a questionnaire to everyone on the mailing list and using the replies.

Give two reasons why the users who reply would not form a random sample of the website's users.

The owner decides instead to choose a random sample from the 4581 mailing-list subscribers who study one particular course. The subscribers are numbered from 1 to 4581 and the following random digits are used.

02743 45290 19027 44600 13880 00045 11279

The numbers of the first two subscribers in the sample are 274 and 3452.

Use the random digits to find the numbers of the next two subscribers in the sample.

A factory produces paper fruit baskets. The breaking load of a basket is the maximum load that it can carry before its handles break. A farm buys 15 000 of these baskets and wishes to test some of them to investigate their breaking load.

Give two reasons why it is necessary to test a sample rather than all 15 000 baskets.

The farm tests a random sample of 6 baskets. The breaking loads, in kg, are as follows.

2.035 1.975 2.128 1.898 2.112 1.999

(i) Without carrying out any calculations, comment on the factory's claim that the mean breaking load of the baskets is 2 kg.

(ii) Give one criticism of the sample used by the farm.