2.01c Sampling techniques: simple random, opportunity, etc

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]

The names of the 720 members of a swimming club are listed alphabetically in the club's membership book. The chairman of the swimming club wishes to select a systematic sample of 40 names. The names are numbered from 001 to 720 and a number between 001 and \(w\) is selected at random. The corresponding name and every \(x\)th name thereafter are included in the sample.

1. Find the value of \(w\). [1]
2. Find the value of \(x\). [1]
3. Write down the probability that the sample includes both the first name and the second name in the club's membership book. [1]
4. State one advantage and one disadvantage of systematic sampling in this case. [2]

A hotel has 160 rooms of which 20 are classified as De-luxe, 40 Premier and 100 as Standard. The manager wants to obtain information about room usage in the hotel by taking a 10\% sample of the rooms.

1. Suggest a suitable sampling method. [1]
2. Explain in detail how the manager should obtain the sample. [4]

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]

As part of her statistics project, Deepa decided to estimate the amount of time A-level students at her school spend on private study each week. She took a random sample of students from those studying Arts subjects, Science subjects and a mixture of Arts and Science subjects. Each student kept a record of the time they spent on private study during the third week of term.

1. Write down the name of the sampling method used by Deepa. [1]
2. Give a reason for using this method and give one advantage this method has over simple random sampling. [2]

The results Deepa obtained are summarised in the table below.

Type of student	Sample size	Mean number of hours
Arts	12	12.6
Science	12	14.1
Mixture	8	10.2

1. Show that an estimate of the mean time spent on private study by A level students at Deepa's school, based on these 32 students is 12.56, to 2 decimal places. [3]

The standard deviation of the time spent on private study by students at the school was 2.48 hours.

1. Assuming that the number of hours spent on private study is normally distributed, find a 95% confidence interval for the mean time spent on private study by A level students at Deepa's school. [4]

A member of staff at the school suggested that A level students should spend on average 12 hours each week on private study.

1. Comment on this suggestion in the light of your interval. [2]

The 240 members of a bowling club are listed alphabetically in the club's membership book. The committee wishes to select a sample of 30 members to fill in a questionnaire about the facilities the club offers.

1. Explain how the committee could use a table of random numbers to take a systematic sample. [3]
2. Give one advantage of this method over taking a simple random sample. [1]

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 teacher wants to investigate the sports played by students at her school in their free time. She decides to ask a random sample of 120 pupils to complete a short questionnaire.

1. Give two reasons why the teacher might choose to use a sample survey rather than a census. [2 marks]
2. Suggest a suitable sampling frame that she could use. [1 mark]

The teacher believes that 1 in 20 of the students play tennis in their free time. She uses the data collected from her sample to test if the proportion is different from this.

1. Using a suitable approximation and stating the hypotheses that she should use, find the critical region for this test. The probability for each tail of the region should be as close as possible to 5\%. [6 marks]
2. State the significance level of this test. [1 mark]

1. The manager of a company that employs 250 travelling sales representatives wishes to carry out a detailed analysis of the expenses claimed by the representatives. He has an alphabetical (by surname) list of the representatives. He chooses a sample of representatives by selecting the 10th, 20th, 30th and so on. Name the type of sampling the manager is attempting to use. Describe a weakness in his method of using it, and explain how he might overcome this weakness. [3]

The representatives each use their own cars to drive to meetings with customers. The total distance, in miles, travelled by a representative in a month is Normally distributed with mean 2018 and standard deviation 96.

1. Find the probability that, in a randomly chosen month, a randomly chosen representative travels more than 2100 miles. [3]
2. Find the probability that, in a randomly chosen 3-month period, a randomly chosen representative travels less than 6000 miles. What assumption is needed here? Give a reason why it may not be realistic. [5]
3. Each month every representative submits a claim for travelling expenses plus commission. Travelling expenses are paid at the rate of 45 pence per mile. The commission is 10\% of the value of sales in that month. The value, in £, of the monthly sales has the distribution N(21200, 1100²). Find the probability that a randomly chosen claim lies between £3000 and £3300. [7]

A researcher wishes to take a sample of size 9, without replacement, from a list of 72 people involved in the trial of a new computer keyboard. She numbers the people from 01 to 72 and uses the table of random numbers given in the formula book. She starts with the left-hand side of the sixth row of the table and works across the row. The first two numbers she writes down are 56 and 32.

1. Find the other six numbers in the sample. [3 marks]
2. Give one advantage and one disadvantage of using random numbers when taking a sample. [2 marks]

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]

Kevin is the Principal of a college. He wishes to investigate types of transport used by students to travel to college. There are 3200 students in the college and Kevin decides to survey 60 of them. Describe how he could obtain a simple random sample of size 60 from the 3200 students. [4 marks]

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 manager of a factory wants to introduce a bonus scheme. The factory has 65 employees who work in production and 28 employees who work in the office. The manager decides to collect the opinions of a sample of these 93 employees.

1. Explain how the manager could collect a simple random sample of 20 employees. [3 marks]
2. The manager collected a simple random sample of 20 employees. The manager noticed that all 20 of the employees in the sample worked in production and therefore the sample was not representative. State a different method of sampling that would give a representative 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]