Question

The members of a population have been numbered 1-100. A sample of size 30 is to be taken from the population, using cluster sampling. The clusters are of equal size 10, where cluster #1 consists of the members of the population numbered 1-10, cluster #2 consists of the members of the population numbered 11-20, and so forth.

(a) Apply Procedure 1.2 on page 18 to determine the sample (i.e., the numbers corresponding to the members of the population that are included in the sample).

(b) Suppose that, in Step 2 of Procedure 1.2, clusters #2, #6, and #9 are selected. Determine the sample.

Short answer

Part (a) 1-10, 21-30, and 31-40 make up the sample.

Part (b) 11-20, 51-60, and 81-90 make up the sample.

Step-by-step solution

Part (a) Step 1. Given information.

The given statement is:

The members of a population have been numbered 1-100. A sample of size 30 is to be taken from the population, using cluster sampling. The clusters are of equal size 10, where cluster #1 consists of the members of the population numbered 1-10, cluster #2 consists of the members of the population numbered 11-20, and so forth.

Part (a) Step 2. Use cluster random sampling to determine the sample.

Clusters are used to categorize the population.

The population ranges from 1 to 100 people. It is also assumed that the clusters are of identical size.

As a result, there are ten clusters $\left(=\frac{100}{10}\right)$.

As a result, the cluster numbers and cluster members are tabulated below:

Clusters	Members
1	1-10
2	11-20
3	21-30
4	31-40
5	41-50
6	51-60
7	61-70
8	71-80
9	81-90
10	91-100

Now select a small sample of clusters at random.

A sample size of 30 is required because each cluster contains ten individuals. Thus, three $\left(=\frac{30}{10}\right)$clusters are chosen at random from a total of ten clusters.

Part (a) Step 3. To get 3 clusters from 10 clusters, use MINITAB.

Procedure for MINITAB:

Step 1: Select Calc > Random Data > Integer from the menu bar.

Step 2: Double your selected sample size in the Number of rows of data to generate a section, just in case there are repeats. Set the sample size to three.

Step 3: In the Store in the column, type Cluster in column C1.

Step 4: Enter 1 as the minimum value.

Step 5: Enter 10 as the maximum value.

Step 6: Click the OK button.

OUTPUT FROM MINITAB: 1 3 4

Therefore, 1, 3, and 4 are the three clusters chosen.

Part (a) Step 4. The sample is made up of members from the selected clusters 

A sample is made up of members from the selected clusters.

The clusters chosen here are 1, 3, and 4. Members 1 to 10, 21 to 30, and 31 to 40 are thus included in the sample.

Members 1-10, 21-30, and 31-40 make up the sample of 30 members utilizing cluster sampling.

Part (b) Step 1. When clusters 2, 6, and 9 are chosen, obtain a sample of size 30.

The clusters chosen here are 2, 6, and 9. Members of the sample range from 11 to 20, 51 to 60, and 81 to 90.

Members 11-20, 51-60, and 81-90 make up the sample of 30 members utilizing cluster sampling.