Question

The members of a population have been numbered 1-50. A sample of size 20 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 #1 and #3 are selected. Determine the sample.

Short answer

Part (a) Members 1-10 and 31--40 make up the sample of 20 members.

Part (b) Members 1-10 and 21- 30 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 1-50.

A sample of size 20 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.

To obtain a sample of size 20, use the cluster sampling approach.

• The population is separated into clusters in the first step. As we know that the population ranges from 1 to 50 people. It is also assumed that the clusters are 10 clusters in size.
• As a result, there are 5 clusters $\left(=\frac{50}{10}\right)$.
• Thus, cluster 1 has members ranging from 1 to 10,
• cluster 2 has members ranging from 11 to 20,
• cluster 3 has members ranging from 21 to 30,
• cluster 4 has members ranging from 31 to 40, and
• cluster 5 has members ranging from 41 to 50.

• Now Select a small sample of clusters at random.

Each cluster comprises ten members, requiring a sample size of 20.

Thus, 2 $\left(=\frac{20}{10}\right)$clusters are chosen at random from a total of 5 clusters.

Part (a) Step 3. To get 2 clusters from 5 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. Enter 2 as the sample size.

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

Step 4: Enter 1 for the minimum value and 5 for the maximum value.

Step 5: Select OK.

MINITAB output: Cluster 1 4

As a result, two clusters have been chosen: 1 and 4

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

Clusters 1 and 4 are chosen in this case. Members 1 to 10 and 31 to 40 are thus included in the sample.

Members 1-10 and 31--40 make up the sample of 20 members utilizing cluster sampling.

Part (b) Step 1. When clusters 1 and 3 are chosen, obtain a sample of size 20.

Clusters 1 and 3 are chosen in this case.

Members 1 to 10 and members 21 to 30 make up the sample.

Members 1-10 and 21- 30 make up the sample of 20 members utilizing cluster sampling.