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).

Procedure Stratified Random Sampling with Proportional Allocation is given information.

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.

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

Procedure Stratified Random Sampling with Proportional Allocation is given information.

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.

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.