﻿ Unit 18 Section 1 : Random Samples

Unit 18 Section 1 : Random Samples

In this section we look at random samples and at the difference between a census and a sample.

In a census, information on every member of a population is considered. In the UK, a census is carried out every 10 years. The amount of work required to carry out a census means that it is an expensive process.

In a sample, a subset of the population is considered to try to obtain information about a particular problem or issue. Because a sample is normally much smaller than the whole population, it is quicker and easier to take and to analyse a sample than to carry out a census of the entire population. Sampling entails less effort and less expense. In some cases, it is essential to take a sample. For example, imagine a firm that uses quality control to test the light bulbs it manufactures, to see how long they last. (If every item was tested until it stopped working they would have no light bulbs left to sell!) They therefore take samples from the production and test these to see if the quality is up to standard.

In a random sample, every member of the population is equally likely to be included in the sample. One way of selecting a sample is to use random numbers, as demonstrated in the example below. You can find random numbers in books of statistical tables. You can also generate them using a calculator or a computer.

The diagram shows part of a table of random digits.

Random digits
98859 09884 45275 09467 93026 32912
26604 95099 93751 00590 93060 64776
82984 65780 94428 30160 86023 52284
70888 14063 96700 83008 17579 71321
77803 61872 86245 68220 66267 01379
11304 01658 82404 46728 35228 49673
53552 51215 45611 83927 00772 99295

Example 1

In a class there are 30 pupils. The teacher decides to take a random sample of 5 pupils to estimate the mean height of the pupils in the class. Select a random sample of 5 pupils from the list.

 1 Alan 10 Rachel 19 Sacha 28 Salif 2 Lucy 11 Ben 20 Halim 29 Annie 3 Tom 12 Emma 21 Daniella 30 Karen 4 Azar 13 Hannah 22 Joseph 5 Jayne 14 Grace 23 Anna 6 Nadima 15 Miles 24 Sophie 7 Matthew 16 James 25 Kathryn 8 Sushi 17 Joshua 26 Helen 9 Mohammed 18 Lisa 27 Fatoumata

To take a random sample you need to use a list of random digits, as follows:

The digits are taken in pairs to form 2-digit numbers, as shown above. All those numbers greater than 30 are discarded (as there are 30 pupils on the list). The process is continued until 5 different numbers between 1 and 30 have been obtained.

So the sample will be made up of the following pupils:

 7 Matthew 26 Helen 5 Jayne 27 Fatoumata 15 Miles

Example 2

Discuss whether or not the following situations produce random samples.

(a)

Mark is conducting a survey for a magazine. He stops people at random on a Saturday morning at his local shopping centre.

To produce a random sample, every member of the population must have an equal chance of being selected. In the case of Mark's sample, he is excluding people who are at work on that Saturday, as well as other people who haven't gone to that shopping centre. So although Mark is stopping people at random, he is not producing a random sample.

(b)

Granny Taylor's National Lottery numbers.

Most people use personal reasons when they select their National Lottery numbers. Granny Taylor may, for example, have used the number of grandchildren that she has, their birthdays, the number of her house, etc. If that is the case then she has not selected a random sample. However, if she bought a 'Lucky Dip' from her local shop then the computerised National Lottery till should have produced a random sample.

Note: For small populations it is relatively easy to produce a random sample. Simply number every member of the population, write those numbers on pieces of paper and put them into a hat or tombola. Mix them well and ask someone to pick out as many numbers as you need for your sample, then take the corresponding items from the population. This process clearly becomes unmanageable when we investigate large populations, which is why we tend then to use random number generators.

Exercises

Question 1

Use the random digits below to select a second sample of 5 from the class in Example 1.

7 1 9 5 4 3 5 9 1 6 8 4 5 3 2 1 7 6 6 0 1 2 3 3 7 0 2 2
6 3 7 1 3 5 3 3 2 3 6 5 2 4 6 5 1 1 3 0 8 5 7 3 9 6 5 5

Question 2

Use the table of random numbers shown at the start of this unit to select a sample of 10 pupils from the list in Example 1.

Question 3

There are 10 competitors in an athletics event. Their names are:

 Jimmy Jump Harold Hammer Tom Throw Dick Discus Harry Hop Paul Putt Sam Shot Liam Long Jake Javelin Victor Vault
(a)

Number the competitors from

 1 Jimmy Jump 2 Harold Hammer Tom Throw Dick Discus Harry Hop Paul Putt Sam Shot Liam Long Jake Javelin 10 Victor Vault
(b)

Use the following list of random digits to select a random sample of 3 of the competitors for drug testing.

 26 60 49 50 99 93 75 10 05 90 93 06 06 47 76 82 98 46 57 80 94 42 83 01 60 86 02 35 22 84 70 88

Question 4

A council wants to talk to the residents of a street to discuss a proposed traffic calming scheme. The houses in the street are numbered from 1 to 57.
Use the list of random digits in question 1 to identify a random sample of 10 of the houses for the council to visit.

The houses that should be visited are those numbered

, , , , , , , , ,
Question 5

In another road the houses are numbered from 1 to 539.
Use the list of random numbers in question 1 to identify a random sample of 6 houses for the council to visit.

The houses that should be visited are those numbered

, , , , ,
Question 6

The ages, in years, of the members of a computer club are listed below.

 Dee 12 Max 16 Ollie 18 Denise 14 Nazir 15 James 11 Tom 16 Jane 17 Hannah 14 Holly 11 Ferdi 11 Gemma 13 Richard 15 Kim 14 Nadia 16 Jai 13 Grant 12 Hugh 14 Victor 13 Juliette 13 Ben 13 Peter 14 Nigel 14 Ali 15
(a)

Number the club members from

 1 Dee 2 Max Ollie Denise Nazir James Tom Jane Hannah Holly Ferdi Gemma Richard Kim Nadia Jai Grant Hugh Victor Juliette Ben Peter Nigel 24 Ali
(b)

Use the list of random digits in question 1 to generate a random sample of 5 club members and calculate the mean age for your sample.

 Name Age
The mean age is years.
(c)

Use the list of random digits in question 1 in reverse order (i.e. 5 5 6 9, etc.) to generate a second random sample of 5 club members. Calculate the mean age for this new sample.

 Name Age
The mean age is years.

As can be seen, the two samples have different people, athough there is some overlap because the list of random digits is fairly short.

The sample means are not the same, although they are reasonably close. We would not normally expect the means of two random samples (from the same population) to be the same.

Question 7

Mr May wants to know the mean IQ of the pupils in his class. Would you recommend that he uses a sample or a census? State which you would use.

Mr. May's class is small enough for the data to be found quickly and easily for every pupil in the class. Using a smaller sample may produce a biased value that would not reflect the class as a whole.
Question 8

A large school has 1800 pupils. The headteacher wants to find out how far the pupils have to travel to school. Advise him whether to carry out a census or to use a sample.

The headteacher should use a sample because the population (1800) is very large. It would take too long to obtain the information from every pupil, which would not be cost effective as an accurate value is unlikely to be needed. A reasonable sample of, say, 100 pupils, chosen randomly from across the whole school, should produce a realistic estimate for the true mean travel time.