Q 1.64. - Elementary Statistics

Question

The members of a population have been numbered 1-500. A sample of size 10 is to be taken from the population, using stratified random sampling with proportional allocation. The strata are of sizes 200, 150, and 150, where stratum #1 consists of the members of the population numbered 1-200, stratum #2 consists of the members of the population numbered 201-350, and so forth.

(a) Determine the sample sizes that will be taken from the strata.

(b) Apply Procedure 1.3 on page 21 to determine the sample (i.e., the numbers corresponding to the members of the population that are included in the sample).

Step-by-Step Solution

Answer

Part (a) 4, 3 and 3

Part (b) 

Stratum	Size	Numbered	Sample Size	Sample
1	200	1-200	4	108, 88, 159, 109
2	150	201-350	3	328, 339, 238
3	150	351-500	3	408, 384, 481

Part (a) Step 1. Given information.

The given statement is:

The members of a population have been numbered 1-500. A sample of size 10 is to be taken from the population, using stratified random sampling with proportional allocation. The strata are of sizes 200, 150, and 150, where stratum #1 consists of the members of the population numbered 1-200, stratum #2 consists of the members of the population numbered 201-350, and so forth. 

Part (a) Step 2. Find the size of each stratum.

The population in strata 1 ranges from 1 to 200 and the size is 200.

The population in strata 2 ranges from 201 to 350 and the size is 150.

The population in strata 3 ranges from 351 to 500 and the size is 150.

The sample size for the stratum 1 is:

$\frac{200}{500}\times 10=4$

The sample size for the stratum 2 is:

$\frac{150}{500}\times 10=3$

The sample size for the stratum 3 is:

$\frac{150}{500}\times 10=3$

Part (b) Step 1. Obtain a sample of size 10, and use stratified sampling.

For each stratum, take a simple random sampling to get the sample size.

Use the MINITAB technique to obtain four members from stratum 1:

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

Step 2. Enter double your intended sample size in the Number of rows of data to generate the section, just in case there are repeats. Set the sample size to four.

Step 3: In the Store in the column, type Stratum in column C1. 1

Step 4: Enter 1 as the minimum value.

Step 5: Type 200 for the maximum value.

Step 6: Click the OK button.

MINITAB output:

108, 88, 159, 109

Similarly, we have found the sample size for other strata following the MINITAB approach to finding the members for each stratum.

Change the sample size, minimum value, and maximum value for each region in the

Stratum 2:

328  339  238

Stratum 3:

408  384  481