Q 1.63 - Elementary Statistics

Question

The members of a population have been numbered 1-1000. A sample of size 20 is to be taken from the population, using stratified random sampling with proportional allocation. The strata are of sizes $300,200,400$, and 100, where stratum #1 consists of the members of the population numbered 1-300, stratum $\#2$ consists of the members of the population numbered 301-500, 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

a)

b)

Part a Step 1 Given Information

Procedure Stratified Random Sampling with Proportional Allocation is given information.

Part a Step 2 Explanation - Strata

Subdivide the population into groups (strata).

We know that there are 1000 people in the population, and that the population is divided into four strata with sizes of $300, 200, 400,$ and 100, with stratum #1 consisting of $1-300$ people. Similarly, stratum #2 has between 301 and 500 people in it, and so on.

Part a Step3 Explanation - Stratum to simple random

 For each stratum, create a basic random number with a size proportional to the stratum's size.

We can calculate the sample size for ith stratum using formula,

$n_i=n\times \frac{N_i}{N}$

Where $n_i$ is sample size of ithstratum

n is the total sample size

The population size of the ith stratum is $N_i$
N is the total population

n = 20 and N = 1000

Part b Step 1 Given Information

Procedure Stratified Random Sampling with Proportional Allocation is given information.

Part b Step 2 Explanation-Stratum #1 and stratum #2

To select the sample for each stratum, we will utilize a random number table.

For stratum #1, pick line 5 and column $7-8-9$ at random, then read down the three-digit number from top to bottom, then up the next column, and so on.

If a number appears in the column that is 0 or greater than 300, it should be discarded; otherwise, it should be kept.

The first number we chose was 82, followed by 8, 47,1, 16, and finally 163 in line 5 and columns 13-14-15.

As a result, the stratum #1 numbers chosen are $82,8,47,1,16$ and 163.

Similarly, for stratum #2, pick line 2 and column 1-2-3 at random, read down the three-digit number from top to bottom, and then up the next column.

If the number falls below 300 or exceeds 500 when passing across the column, it should be discarded; otherwise, it should be kept.

In line 16, the first number we choose is 358, followed by 332,490, and finally 441.

As a result, the numbers 358, 332, 490, and 441 were chosen from stratum #2.

Part b Step 3 Explanation-Stratum #3 and stratum #4

Similarly, for stratum #3, let choose randomly line number 8 and columns 5-6-7,

From top to bottom, read the three-digit number, then up the next column, and so on.

If a number falls below 500 or exceeds 900 while moving across the column, it should be discarded; otherwise, it should be kept.

We chose 813 as the first number, followed by $706,650,555,529,526,534$, and finally 864 in line 13 and column 8-9-10.

As a result, the stratum #3 numbers are $813,650,555,529,526,534$ and$864.$

For stratum #4, choose line 17 and column 6-7$8-9$ at random, then read down the four-digit number from top to bottom, up the next column, and so on.

If a number is less than 900 or greater than 1000, it should be discarded; otherwise, it should be kept.

In line 00 and column 1920-21-22, we chose 986 as the first number and 970 as the last.

As a result, the numbers 986 and 970 were chosen from stratum #4.

Part b Step 5 Final Answer