Unit 18 Section 3 : Stratified Random Samples

A stratified random sample is similar to a quota sample. The key difference is that the representatives from each category are chosen at random. In the previous section we looked at a quota sample that was made up of 4 boys and 6 girls.

In the quota sample the first 4 boys from the list were chosen and then the first 6 girls.

In a stratified random sample the 4 boys should be selected randomly from the 12 boys in the class. In the same way, the 6 girls should be selected randomly from the 18 girls in the class.

Example 1

Take a stratified random sample of size 5 from the class considered in the earlier sections.

The class consists of 18 girls and 12 boys, 30 pupils altogether. We want a sample of size 5 so we need to include = of the girls, and of the boys. This means that we need to choose 3 girls and 2 boys.

We choose a random sample of 3 girls. The girls in the class are:

1 Lucy 7 Hannah 13 Sophie
2 Jayne 8 Grace 14 Kathryn
3 Nadima 9 Lisa 15 Helen
4 Sushi 10 Sacha 16 Fatoumata
5 Rachel 11 Daniella 17 Annie
6 Emma 12 Anna 18 Karen

Using random digits gives:

So the sample will contain:

3 Nadima
9 Lisa
15 Helen

The boys are:

1 Alan 5 Mohammed 9 Joshua
2 Tom 6 Ben 10 Halim
3 Azar 7 Miles 11 Joseph
4 Matthew 8 James 12 Salif

Using random digits gives:

Note that the second 7 is ignored.

So the sample will also contain:

7 Miles
2 Tom

The complete stratified random sample will therefore consist of

Nadima, Lisa, Helen, Miles and Tom.

Example 2

A company has a total of 360 employees in four different categories:

Managers 36
Drivers 54
Administrative Staff 90
Production Staff 180

How many from each category should be included in a stratified random sample of size 20 ?

To create a sample of size 20 we need or of the workforce. So we take this fraction of the number of employees in each category.

Managers × 36   =   2
Drivers × 54   =   3
Administrative Staff × 90   =   5
Production Staff × 180   =   10
TOTAL   =   20

Exercises

Question 1

Create a stratified random sample of size 10 for the class considered in the Examples.

Use the following set of random numbers. Begin with the girls.

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










Question 2

A catering company employs the staff listed in the following table:

Delivery Drivers 12
Cooks 36
Cleaning Staff 4
Sales Staff 8

How many of each category of staff should be included in a stratified random sample of size 15 ?

Delivery Drivers
Cooks
Cleaning Staff
Sales Staff
Question 3

A garage services VW and Audi cars. They want to carry out a customer satisfaction survey.

(a)

What kind of sampling method should they use?

They might want to use a stratified random sample if they don't have equal numbers of VW and Audi owners.
(b)

They have 2000 regular customers of whom 650 have Audi cars.
How many VW owner and how many Audi owner will the sample of 40 contain?

VW:
Audi:
Question 4

A farmer owns 120 Jersey cows and 180 Friesians. How many of each breed of cow should he include in a stratified random sample of 50 for a survey of milk quality?

He should include Jersey cows and Friesians.
Question 5

A survey was carried out to determine the size people would prefer for a new coin. In the survey people were asked to select the size coin they preferred and indicate their sex by using the letter M or F.
Forty people took part and the results are shown below.

(a)

How many males and how many females would you include in a stratified random sample of size 10 ?

There are females and males altogether
so the sample should contain females and males.
(b)

Select a stratified random sample of size 10.

Use the random number table below. Begin with females.

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

Question 6

There are 1200 pupils in a school. Their methods of transport to school are given below.

Bus 720
Walk 310
Cycle 90
Car 80

How many pupils from each category should be included in a stratified random sample of size 50?

Bus
Walk
Cycle
Car
Bus: × 720 = 30
Walk: × 310 = 12
Cycle: × 90 = 3
Car: × 80 = 3

Rounding these to the nearest whole number, the sample would consist of
30 pupils who travel by bus, 13 pupils who walk, 4 pupils who cycle and 3 pupils who come by car.

Question 7

There are 1500 homes in one part of a city. These homes can be divided into 3 categories:

Privately owned homes 720
Homes rented from council 592
Homes rented from private landlords 188

The council is to carry out a survey into the quality of homes.

How many homes of each type should be included in a stratified random sample of size 200 ?

privately owned homes
homes rented from council
homes rented from private landlords