Select random sample using random numbers

1 Jyothi wishes to choose a representative sample of 5 students from the 82 members of her school year.

1. She considers going into the canteen and choosing a table with five students from her year sitting at it, and using these five people as her sample. Give two reasons why this method is unsatisfactory.
2. Jyothi decides to use another method. She numbers all the students in her year from 1 to 82 . Then she uses her calculator and generates the following random numbers. $$231492 \quad 762305 \quad 346280$$ From these numbers, she obtains the student numbers \(23,14,76,5,34\) and 62 . Explain how Jyothi obtained these student numbers from the list of random numbers.

1 A residents' association has 654 members, numbered from 1 to 654 . The secretary wishes to send a questionnaire to a random sample of members. In order to choose the members for the sample she uses a table of random numbers. The first line in the table is as follows. $$\begin{array} { l l l l l l } 1096 & 4357 & 3765 & 0431 & 0928 & 9264 \end{array}$$ The numbers of the first two members in the sample are 109 and 643. Find the numbers of the next three members in the sample.

2 Andy and Jessica are doing a survey about musical preferences. They plan to choose a representative sample of six students from the 256 students at their college.

1. Andy suggests that they go to the music building during the lunch hour and choose six students at random from the students who are there. Give a reason why this method is unsatisfactory.
2. Jessica decides to use another method. She numbers all the students in the college from 1 to 256. Then she uses her calculator and generates the following random numbers. $$\begin{array} { l l l l l } 204393 & 162007 & 204028 & 587119 & 207395 \end{array}$$ From these numbers, she obtains six student numbers. The first three of her student numbers are 204, 162 and 7. Continue Jessica's method to obtain the next three student numbers.

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.