Linear programming and optimization

Describe briefly how to use a random number generator to obtain a sample of 10 students from a group of 50 students. [3]

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]

Describe one advantage and one disadvantage of

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

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]

Neesha wants to open an Indian restaurant in her town. Her cousin, Ranji, has an Indian restaurant in a neighbouring town. To help Neesha plan her menu, she wants to investigate the dishes chosen by a sample of Ranji's customers. Ranji has a list of the 750 customers who dined at his restaurant during the past month and the dish that each customer chose. Describe how Neesha could obtain a simple random sample of size 50 from Ranji's customers. [4 marks]

An electoral register contains 8000 names. A researcher decides to select a systematic sample of 100 names from the register. Explain how the researcher should select such a sample. [3 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]

Edna wishes to investigate the energy intake from eating out at restaurants for the households in her village. She wants a sample of 100 households. She has a list of all 2065 households in the village. Ralph suggests this selection method. "Number the households 0000 to 2064. Obtain 100 different four-digit random numbers between 0000 and 2064 and select the corresponding households for inclusion in the investigation."

1. What is the population for this investigation? Circle your answer. [1 mark]

   Edna and Ralph

   The 2065 households in the village

   The energy intake for the village from eating out

   The 100 households selected

2. What is the sampling method suggested by Ralph? Circle your answer. [1 mark]

   Opportunity

   Random number

   Continuous random variable

   Simple random

Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers. [5 marks]

Lenny is one of a team of people interviewing shoppers in a town centre. He is asked to survey 50 women between the ages of 18 and 29 Identify the name of this type of sampling. Circle your answer. [1 mark] simple random stratified quota systematic