2.01d Select/critique sampling: in context

9 A teacher is investigating how pupils travel to and from school each day. Pupils can either travel by bus, train, car, bicycle or walk. The teacher decides to collect a sample of size 60 for the investigation.

1. The teacher lives in a village 10 miles away from the school. Explain how collecting a sample which just consists of pupils who live in the same village as the teacher might introduce bias. The table below shows how many students there are in each year.

   Year 7	Year 8	Year 9	Year 10	Year 11
   86	105	107	101	101

2. The teacher decides to use the method of proportional stratified sampling. Calculate the number of pupils in the sample who are in Year 9. The teacher generates a sample of 10 pupils from the 86 in Year 7 by listing them in alphabetical order and selecting the first name on the list and every ninth name thereafter.
3. Explain whether this method will generate a simple random sample of the pupils who travel in Year 7.

2. Bill owns a restaurant. Over the next four weeks Bill decides to carry out a sample survey to obtain the customers' opinions.

1. Suggest a suitable sampling frame for the sample survey.
2. Identify the sampling units.
3. Give one advantage and one disadvantage of taking a census rather than a sample survey. Bill believes that only \(30 \%\) of customers would like a greater choice on the menu. He takes a random sample of 50 customers and finds that 20 of them would like a greater choice on the menu.
4. Test, at the \(5 \%\) significance level, whether or not the percentage of customers who would like a greater choice on the menu is more than Bill believes. State your hypotheses clearly.

3 Jian owns a large group of shops. She decides to visit a random sample of the shops to check if the stocktaking system is being used incorrectly.

1. Suggest a suitable sampling frame for Jian to use.
2. Identify the sampling units.
3. Give one advantage and one disadvantage of taking a sample rather than a census. Jian believes that the stocktaking system is being used incorrectly in \(40 \%\) of the shops.
   To investigate her belief, a random sample of 30 of the shops is taken.
4. Using a 5\% level of significance, find the critical region for a two-tailed test of Jian's belief.
   You should state the probability in each tail, which should each be as close as possible to 2.5\% The total number of shops, in the sample of 30, where the stocktaking system is being used incorrectly is 20
5. Using the critical region from part (d), state what this suggests about Jian's belief. Give a reason for your answer. Jian introduces a new, simpler, stocktaking system to all the shops.
   She takes a random sample of 150 shops and finds that in 47 of these shops the new stocktaking system is being used incorrectly.
6. Using a suitable approximation, test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of shops where the stocktaking system is being used incorrectly is now less than 0.4 You should state your hypotheses and show your working clearly.

1. (a) Explain what you understand by a census.

Each cooker produced at GT Engineering is stamped with a unique serial number. GT Engineering produces cookers in batches of 2000. Before selling them, they test a random sample of 5 to see what electric current overload they will take before breaking down.
(b) Give one reason, other than to save time and cost, why a sample is taken rather than a census.
(c) Suggest a suitable sampling frame from which to obtain this sample.
(d) Identify the sampling units.

1. The Headteacher of a school is thinking about making changes to the school day. She wants to take a sample of 60 students so that she can find out what the students think about the proposed changes.

The names of the 1200 students of the school are listed alphabetically.

1. Explain how the Headteacher could take a systematic sample of 60 students.
2. 1. Explain why systematic sampling is likely to be quicker than simple random sampling in this situation.
   2. With reference to this situation,
      • explain why systematic sampling may introduce bias compared to simple random sampling
3. give an example of the bias that may occur when using this alphabetical list
When the Headteacher completes the systematic sample of size 60 she finds that 6 students were to be selected from Year 9. The Head of Mathematics suggests that a stratified sample of size 60 would be a more appropriate method. There were 200 students in Year 9.
4. Explain why this suggests that a stratified sample of size 60 may be better than the systematic sample taken by the Headteacher.

1. A company has 800 employees.

The manager of the company is going to take a sample of 80 employees.

1. Explain how this sample can be taken using systematic sampling. The company has offices in London, Edinburgh and Cardiff. The table shows the number of employees in each city.

   City	London	Edinburgh	Cardiff
   Number of employees	430	250	120

   The president of the company is going to take a sample of 100 employees to determine the average time employees spend in front of a computer each week.
2. Explain how this sample can be taken using stratified sampling.
3. Explain an advantage of using stratified sampling rather than simple random sampling.

1. A tennis club's committee wishes to select a sample of 50 members to fill in a questionnaire about the club's facilities. The 300 members, of whom 180 are males, are listed in alphabetical order and numbered \(1 - 300\) in the club's membership book.

The club's committee decides to use a random number table to obtain its sample.
The first three lines of the random number table used are given below. Starting with the top left-hand corner (319) and working across, the committee selects 50 random numbers. The first 2 suitable numbers are 241 and 278. Numbers greater than 300 are ignored.

1. Find the next two suitable numbers. When the club's committee looks at the members corresponding to their random numbers they find that only 1 female has been selected.
   The committee does not want to be accused of being biased towards males so considers using a systematic sample instead.
2. 1. Explain clearly how the committee could take a systematic sample.
   2. Explain why a systematic sample may not give a sample that represents the proportion of males and females in the club. The committee decides to use a stratified sample instead.
3. Describe how to choose members for the stratified sample.
4. Explain an advantage of using a stratified sample rather than a quota sample.

3. A company wants to survey its employees' attitudes to work. The company's workforce is located at three offices. The number of employees at each location is summarised in the table below. Each employee is located at only one office. A personnel assistant plans to survey the first 50 employees who arrive for work at the Bristol office on a Monday morning.

1. Give two reasons why this survey is likely to lead to a biased response. A personnel manager has access to the company's information system that holds details of each employee including their place of work. The manager decides to take a stratified sample of 150 employees.
2. Describe how to choose employees for this stratified sample.
3. Explain an advantage of using a stratified sample rather than a quota sample.

1. A local village radio station, LSB, decides to survey adults in its broadcasting area about the programmes it produces. \(L S B\) broadcasts to 4 villages \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D .
   The number of households in each of the villages is given below.

LSB decides to take a stratified sample of 200 households.

1. Explain how to select the households for this stratified sample.
   (3) One of the questions in the survey related to the age group of each member of the household and whether they listen to \(L S B\). The data received are shown below.

   \multirow{2}{*}{}	Age group
   	18-49	50-69	Older than 69
   Listen to LSB	130	162	65
   Do not listen to LSB	78	98	62

   The data are to be used to determine whether or not there is an association between the age group and whether they listen to \(L S B\).
2. Calculate the expected frequencies for the age group 50-69 that
   1. listen to \(L S B\)
   2. do not listen to \(L S B\) (2) Given that for the other 4 classes \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 4.657\) to 3 decimal places,
3. test at the \(5 \%\) level of significance, whether or not there is evidence of an association between age and listening to \(L S B\). Show your working clearly, stating the degrees of freedom and the critical value.

1. A researcher is hired by a cleaning company to survey the opinions of employees on a proposed pension scheme. The company employs 55 managers and 495 cleaners.

To collect data the researcher decides to give a questionnaire to the first 50 cleaners to leave at the end of the day.

1. Give 2 reasons why this method is likely to produce biased results.
2. Explain briefly how the researcher could select a sample of 50 employees using
   1. a systematic sample,
   2. a stratified sample. Using the random number tables in the formulae book, and starting with the top left hand corner (8) and working across, 50 random numbers between 1 and 550 inclusive were selected. The first two suitable numbers are 384 and 100 .
3. Find the next two suitable numbers.

3. A college manager wants to survey students' opinions of enrichment activities. She decides to survey the students on the courses summarised in the table below. Each student takes only one course.
The manager has access to the college's information system that holds full details of each of the enrolled students including name, address, telephone number and their course of study. She wants to compare the opinions of students on each course and has a generous budget to pay for the cost of the survey.

1. Give one advantage and one disadvantage of carrying out this survey using
   1. quota sampling,
   2. stratified sampling. The manager decides to take a stratified sample of 100 students.
2. Calculate the number of students to be sampled from each course.
3. Describe how to choose students for the stratified sample.

1. (a) Explain what you understand by a random sample from a finite population.
   (b) Give an example of a situation when it is not possible to take a random sample.

A college lecturer specialising in shoe design wants to change the way in which she organises practical work. She decides to gather ideas from her 75 students. She plans to give a questionnaire to a random sample of 8 of these students.
(c) (i) Describe the sampling frame that she should use.
(ii) Explain in detail how she should use a table of random numbers to obtain her sample.

3. A nursery has 16 staff and 40 children on its records. In preparation for an outing the manager needs an estimate of the mean weight of the people on its records and decides to take a stratified sample of size 14 .

1. Describe how this stratified sample should be taken. The weights, \(x \mathrm {~kg}\), of each of the 14 people selected are summarised as $$\sum x = 437 \text { and } \sum x ^ { 2 } = 26983$$
2. Find unbiased estimates of the mean and the variance of the weights of all the people on the nursery's records.
3. Estimate the standard error of the mean. The estimates of the standard error of the mean for the staff and for the children are 5.11 and 1.10 respectively.
4. Comment on these values with reference to your answer to part (c) and give a reason for any differences.

1. A company director decides to survey staff about changes to the company calendar. The company has staff in 4 different job roles

72 managers, 108 drivers, 180 administrators and 360 warehouse staff.
The director decides to take a stratified sample.

1. Write down one advantage of using a stratified sample rather than a simple random sample for this survey.
2. Find the number of staff in each job role that will be included in a stratified sample of 40 staff.
3. Describe how to choose managers for the stratified sample.

1. (a) Briefly describe the difference between a census and a sample survey.
   (b) Illustrate the difference by considering the case of a village council which has to decide whether or not to build a new village hall.

Given that the council decides to use a sample survey,
(c) suggest suitable sampling units.

2. A video rental shop needs to find out whether or not videos have been rewound when they are returned; it will do this by taking a sample of returned videos

1. State one advantage and one disadvantage of taking a sample.
2. Suggest a suitable sampling frame.
3. Describe the sampling units.
4. Criticise the sampling method of looking at just one particular shelf of videos.

3 A company has been commissioned to make 50 very expensive titanium components.
A sample of the components needs to be tested to ensure that they are sufficiently strong. However, this is a test to destruction, so the components which are tested can no longer be used.

1. Explain why it would not be appropriate to use a census in these circumstances. A manager suggests that the first 5 components to be manufactured should be tested.
2. Explain why this would not be a sensible method of selecting the sample. A statistician advises the manager that the sample selected should be a random sample.
3. Give two desirable features (other than randomness) that the sample should have.

2 A company manufactures batches of twenty thousand tins which are subsequently filled with fruit. The company tests tins from each batch to make sure that they are strong enough. The test is easy and cheap to carry out, but when a tin has been tested it is no longer suitable for filling with fruit.

1. 1. Explain why a sample size of 5 tins per batch may not be appropriate in this case.
   2. Explain why a sample size of 1000 tins per batch may not be appropriate in this case. The company tests a sample of 30 tins from each batch.
2. Explain why it would not be sensible for the sample to consist of the final 30 tins produced in a batch.
3. Give two features that the sample should have.

6 Answer part (iv) of this question on the insert provided. There are two types of customer who use the shop at a service station. \(70 \%\) buy fuel, the other \(30 \%\) do not. There is only one till in operation.

1. Give an efficient rule for using one-digit random numbers to simulate the type of customer arriving at the service station. Table 6.1 shows the distribution of time taken at the till by customers who are buying fuel.

   Time taken (mins)	1	1.5	2	2.5
   Probability	\(\frac { 3 } { 10 }\)	\(\frac { 2 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 10 }\)

   \section*{Table 6.1}
2. Specify an efficient rule for using one-digit random numbers to simulate the time taken at the till by customers purchasing fuel. Table 6.2 shows the distribution of time taken at the till by customers who are not buying fuel.

   Time taken (mins)	1	1.5	2	2.5	3
   Probability	\(\frac { 1 } { 7 }\)	\(\frac { 2 } { 7 }\)	\(\frac { 2 } { 7 }\)	\(\frac { 1 } { 7 }\)	\(\frac { 1 } { 7 }\)

   \section*{Table 6.2}
3. Specify an efficient rule for using two-digit random numbers to simulate the time taken at the till by customers not buying fuel. What is the advantage in using two-digit random numbers instead of one-digit random numbers in this part of the question? The table in the insert shows a partially completed simulation study of 10 customers arriving at the till.
4. Complete the table using the random numbers which are provided.
5. Calculate the mean total time spent queuing and paying.

8 A club secretary wishes to survey a sample of members of his club. He uses all members present at a particular meeting as his sample.

1. Explain why this sample is likely to be biased. Later the secretary decides to choose a random sample of members.
   The club has 253 members and the secretary numbers the members from 1 to 253 . He then generates random 3-digit numbers on his calculator. The first six random numbers generated are 156, 965, 248, 156, 073 and 181. The secretary uses each number, where possible, as the number of a member in the sample.
2. Find possible numbers for the first four members in the sample.

The manager of a clothing shop wishes to investigate how satisfied customers are with the quality of service they receive. A database of the shop's customers is used as a sampling frame for this investigation.

1. Identify one potential problem with this sampling frame. [1]

Customers are asked to complete a survey about the quality of service they receive. Past information shows that 35\% of customers complete the survey. A random sample of 20 customers is taken.

1. Write down a suitable distribution to model the number of customers in this sample that complete the survey. [2]
2. Find the probability that more than half of the customers in the sample complete the survey. [2]