2.01d Select/critique sampling: in context

The small village of Tornep has a preservation society which is campaigning for a new by-pass to be built. The society needs to measure

1. the strength of opinion amongst the residents of Tornep for the scheme and
2. the flow of traffic through the village on weekdays. The society wants to know whether to use a census or a sample survey for each of these measures.
   1. In each case suggest which they should use and specify a suitable sampling frame. [4] For the measurement of traffic flow through Tornep,
   2. suggest a suitable statistic and a possible statistical model for this statistic. [2]

A large dental practice wishes to investigate the level of satisfaction of its patients.

1. Suggest a suitable sampling frame for the investigation. [1]
2. Identify the sampling units. [1]
3. State one advantage and one disadvantage of using a sample survey rather than a census. [2]
4. Suggest a problem that might arise with the sampling frame when selecting patients. [1]

A large dental practice wishes to investigate the level of satisfaction of its patients.

1. Suggest a suitable sampling frame for the investigation. [1]
2. Identify the sampling units. [1]
3. State one advantage and one disadvantage of using a sample survey rather than a census. [2]
4. Suggest a problem that might arise with the sampling frame when selecting patients. [1]

Before introducing a new rule the secretary of a golf club decided to find out how members might react to this rule.

1. Explain why the secretary decided to take a random sample of club members rather than ask all the members. [1]
2. Suggest a suitable sampling frame. [1]
3. Identify the sampling units. [1]

A school held a disco for years 9, 10 and 11 which was attended by 500 pupils. The pupils were registered as they entered the disco. The disco organisers were keen to assess the success of the event. They designed a questionnaire to obtain information from those who attended.

1. State one advantage and one disadvantage of using a sample survey rather than a census. [2]
2. Suggest a suitable sampling frame. [1]
3. Identify the sampling units. [1]

1. State two reasons why stratified sampling might be chosen as a method of sampling when carrying out a statistical survey. [2]
2. State one advantage and one disadvantage of quota sampling. [2]

(Total 4 marks)

Describe one advantage and one disadvantage of

1. quota sampling, [2]
2. simple random sampling. [2]

A telephone directory contains 50000 names. A researcher wishes to select a systematic sample of 100 names from the directory.

1. Explain in detail how the researcher should obtain such a sample. [2]
2. Give one advantage and one disadvantage of
   1. quota sampling,
   2. systematic sampling.
   [4]

A company that makes ropes for mountaineering wants to assess the breaking strain of its ropes.

1. Explain why a sample survey, and not a census, should be used. [2 marks]
2. Suggest an appropriate sampling frame. [1 mark]

Briefly explain what is meant by

1. a statistical model, [2 marks]
2. a sampling frame, [1 mark]
3. a sampling unit. [1 mark]

A college has 400 students. A journalist wants to carry out a survey about food preferences and she obtains a sample of 30 pupils from the college by the following method. • Obtain a list of all the students. • Number the students, with numbers running sequentially from 0 to 399. • Select 30 random integers in the range 000 to 999 inclusive. If a random integer is in the range 0 to 399, then the student with that number is selected. If the number is greater than 399, then 400 is subtracted from the number (if necessary more than once) until an answer in the range 0 to 399 is selected, and the student with that number is selected.

1. Explain why this method is unsatisfactory. [2]
2. Explain how it could be improved. [1]

A charity has 240 volunteers and wishes to consult a sample of them of size 20.

1. Explain briefly how a systematic sample can be taken using random numbers. [3]
2. Give one advantage and one disadvantage of using systematic sampling compared with simple random sampling. [2]

A retail company has 5200 employees in 100 stores throughout the United Kingdom. The company recently introduced a new reward scheme for its staff. The management team wanted to sample the staff to find out their opinions of the new scheme. Three possible sampling methods were suggested: Method A \quad Choose 100 people who work at the largest store Method B \quad Choose one person at random from each of the 100 stores Method C \quad List all employees in alphabetical order and assign each a number from 1 to 5200 Choose a random number between 1 and 52 Choose this person and every 52nd person on the list thereafter.

1. Give one disadvantage of using Method A compared with using Method B. [1 mark]
2. Give one advantage of using Method B compared with using Method C. [1 mark]
3. 1. Identify the method of sampling used in Method C. [1 mark]
   2. Give a reason why Method C does not provide a random sample. [1 mark]

The headteacher of a school wishes to collect the opinions of the students on a new timetable structure. To do this, a random sample of size 50, stratified by year group, will be selected. The school has a total of 720 students. The number of students in each of the year groups at this school is shown below.

Year group	10	11	12	13
Number of students	200	240	150	130

1. Find the number of students from each year group that should be selected in the stratified random sample. [3 marks]
2. State one advantage of using a stratified random sample. [1 mark]

The box plot below summarises the CO\(_2\) emissions, in g/km, for cars in the Large Data Set from the London and North West regions. \includegraphics{figure_12}

1. Using the box plot, give one comparison of central tendency and one comparison of spread for the two regions. [2 marks]
2. Jaspal, an environmental researcher, used all of the data in the Large Data Set to produce a statistical comparison of the CO\(_2\) and CO emissions in regions of England. Using your knowledge of the Large Data Set, give two reasons why his conclusions may be invalid. [2 marks]

Researchers are investigating the average time spent on social media by adults on the electoral register of a town. They select every 100th adult from the electoral register for their investigation.

1. Identify the population in their investigation. [1 mark]
2. 1. State the name of this method of sampling. [1 mark]
   2. Describe one advantage of this sampling method. [1 mark]

Jo is investigating the popularity of a certain band amongst students at her school. She decides to survey a sample of 100 students.

1. State an advantage of using a stratified sample rather than a simple random sample. [1]
2. Explain whether it would be reasonable for Jo to use her results to draw conclusions about all students in the UK. [1]

A researcher wants to find out how many adults in a large town use the internet at least once a week. The researcher has formulated a suitable question to ask. For each of the following methods of taking a sample of the adults in the town, give a reason why the method may be biased. Method A: Ask people walking along a particular street between 9 am and 5 pm on one Monday. Method B: Put the question through every letter box in the town and ask people to send back answers. Method C: Put the question on the local council website for people to answer online. [3]

An exercise gym opens at 6:00 a.m. every day. The manager decides to use a questionnaire to gather the opinions of the gym members. The first 30 members arriving at the gym on a particular morning are asked to complete the questionnaire.

1. What is the intended population in this context? [1]
2. What type of sampling is this? [1]
3. How could the sampling process be improved? [1]

A market gardener records the masses of a random sample of 100 of this year's crop of plums. The table shows his results.

1. Explain why the normal distribution might be a reasonable model for this distribution. [1]

The market gardener models the distribution of masses by \(N(47.5, 10^2)\).

1. Find the number of plums in the sample that this model would predict to have masses in the range:
   1. \(35 \leq m < 45\) [2]
   2. \(m < 25\) [2]
2. Use your answers to parts (b)(i) and (b)(ii) to comment on the suitability of this model. [1]

The market gardener plans to use this model to predict the distribution of the masses of next year's crop of plums.

1. Comment on this plan. [1]

The diagram below shows some "Cycle to work" data taken from the 2001 and 2011 UK censuses. The diagram shows the percentages, by age group, of male and female workers in England and Wales, excluding London, who cycled to work in 2001 and 2011. \includegraphics{figure_9} The following questions refer to the workers represented by the graphs in the diagram.

1. A researcher is going to take a sample of men and a sample of women and ask them whether or not they cycle to work. Why would it be more important to stratify the sample of men? [1]

A research project followed a randomly chosen large sample of the group of male workers who were aged 30-34 in 2001.

1. Does the diagram suggest that the proportion of this group who cycled to work has increased or decreased from 2001 to 2011? Justify your answer. [2]
2. Write down one assumption that you have to make about these workers in order to draw this conclusion. [1]