2.01c Sampling techniques: simple random, opportunity, etc

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.

5. A dance studio has 800 dancers of which \begin{displayquote} 452 are beginners
251 are intermediates
97 are professionals

1. Explain in detail how a stratified sample of size 50 could be taken.
2. State an advantage of stratified sampling rather than simple random sampling in this situation. \end{displayquote} Independent random samples of 80 beginners and 60 intermediates are chosen. Each of these dancers is given an assessment score, \(x\), based on the quality of their dancing. The results are summarised in the table below.

   	\(\bar { x }\)	\(s ^ { 2 }\)	\(n\)
   Beginners	31.7	57.3	80
   Intermediates	36.9	38.1	60

   The studio manager believes that the mean score of intermediates is more than 3 points greater than the mean score of beginners.
3. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not these data support the studio manager's belief.

4. A company selects a random sample of five of its warehouses. The table below summarises the number of employees, in thousands, at each warehouse and the number of reported first aid incidents at each warehouse during 2017 The personnel manager claims that the mean number of reported first aid incidents per 1000 employees is the same at each of the company's warehouses.

1. Stating your hypotheses clearly, use a \(5 \%\) level of significance to test the manager's claim. Jean, the safety officer at warehouse \(C\), kept a record of each reported first aid incident at warehouse \(C\) in 2017. Jean wishes to select a systematic sample of 10 records from warehouse \(C\).
2. Explain, in detail, how Jean should obtain such a sample.

1. The names of the 400 employees of a company are listed alphabetically in a book.

The chairperson of the company wishes to select a sample of 8 employees.
The chairperson numbers the employees from 001 to 400

1. Describe how the list of numbers can be used to select a systematic sample of 8 employees.
2. State one disadvantage of systematic sampling in this case.
3. Write down the probability that the sample includes both the first name (employee 001) and the last name (employee 400) in the list.

4. Luka wants to carry out a survey of students at his school. He obtains a list of all 280 students.

1. Explain how he can use this list to select a systematic sample of 40 students. Luka is trying to make his own random number table. He generates 400 digits to put in his table. Figure 1 shows the frequency of each digit in his table. \begin{table}[h]

   Digit generated	0	1	2	3	4	5	6	7	8	9
   Frequency	36	42	33	41	44	43	48	38	32	43

   \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{table} A test is carried out at the \(10 \%\) level of significance to see if the digits Luka generates follow a uniform distribution. For this test \(\sum \frac { ( \mathrm { O } - \mathrm { E } ) ^ { 2 } } { \mathrm { E } } = 5.9\)
2. Determine the conclusion of this test.
   (3) The digits generated by Luka are taken two at a time to form two-digit numbers. Figure 2 shows the frequency of two-digit numbers in his table. \begin{table}[h]

   Two-digit numbers generated	\(00 - 19\)	\(20 - 39\)	\(40 - 59\)	\(60 - 79\)	\(80 - 99\)
   Frequency	31	49	30	42	48

   \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{table}
3. Test, at the \(10 \%\) level of significance, whether the two-digit numbers generated by Luka follow a uniform distribution. You should state the hypotheses, the degrees of freedom and the critical value used for this test. There are 70 students in Year 12 at his school.
4. State, giving a reason, the advice you would give to Luka regarding the use of his table of numbers for generating a simple random sample of 10 of the Year 12 students.

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 school has 15 classes and a sixth form. In each class there are 30 students. In the sixth form there are 150 students. There are equal numbers of boys and girls in each class. There are equal numbers of boys and girls in the sixth form. The head teacher wishes to obtain the opinions of the students about school uniforms.

Explain how the head teacher would take a stratified sample of size 40.
(7)

1. There are 64 girls and 56 boys in a school.

Explain briefly how you could take a random sample of 15 pupils using

1. a simple random sample,
2. a stratified sample.

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.

2. A lake contains 3 species of fish. There are estimated to be 1400 trout, 600 bass and 450 pike in the lake. A survey of the health of the fish in the lake is carried out and a sample of 30 fish is chosen.

1. Give a reason why stratified random sampling cannot be used.
2. State an appropriate sampling method for the survey.
3. Give one advantage and one disadvantage of this sampling method.
4. Explain how this sampling method could be used to select the sample of 30 fish. You must show your working.

1. A gym club has 400 members of which 300 are males.

Explain clearly how a stratified sample of size 60 could be taken.

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.

5. A student believes that there is a difference in the mean lengths of English and French films. He goes to the university video library and randomly selects a sample of 120 English films and a sample of 70 French films. He notes the length, \(x\) minutes, of each of the films in his samples. His data are summarised in the table below.

1. Verify that the unbiased estimate of the variance, \(s ^ { 2 }\), of the lengths of English films is 98.5 minutes \({ } ^ { 2 }\)
2. Stating your hypotheses clearly, test, at the 1\% level of significance, whether or not the mean lengths of English and French films are different.
3. Explain the significance of the Central Limit Theorem to the test in part (b).
4. The university video library contained 724 English films and 473 French films. Explain how the student could have taken a stratified sample of 190 of these films.

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.