Sampling Without Replacement

Sampling is called without replacement when a unit is selected at random from the population and it is not returned to the main lot. First unit is selected out of a population of size N and the second unit is selected out of the remaining population of N - 1 units and so on. Thus the size of the population goes on decreasing as the sample size n increases. The sample size n cannot exceed the population size N. The unit once selected for a sample cannot be repeated in the same sample. Thus all the units of the sample are distinct from one another. A sample without replacement can be selected either by using the idea of permutations or combinations. Depending upon the situation, we write all possible permutations or combinations. If the different arrangements of the units are to be considered, then the permutations (arrangements) are written to get all possible samples. If the arrangement of units is of no interest, we write the combinations to get all possible samples.

Combination:

Let us again consider a lot (population) of 5 bulbs with 3 good ({G_1},{G_2}and{G_3}) and 2 defective ({D_1}and{D_2}) bulbs. Suppose we have to select two bulbs in any order there are ^5{C_2} = \frac{{5!}}{{2!3!}} = 10 possible combinations or samples. These combinations (samples) are listed as: {G_1}{G_2}, {G_1}{G_3}, {G_2}{G_3}, {G_1}{D_1}, {G_1}{D_2}, {G_2}{D_1}, {G_2}{D_2}, {G_3}{D_1}, {G_3}{D_2}, {D_1}{D_2}.

There are 10 possible samples and each of them has probability of selection equal to 1/10. The selected sample will be any one of these10 samples. The sample selected in this manner is also called simple random sample. In general, the number of samples by combinations is equal to

^N{C_n} =  \frac{{N!}}{{n!\left( {N - n} \right)!}}

.

Permutation:

Each combination generates a number of arrangements (permutations). Thus in general the number of permutations is greater than the number of combinations. In the previous example of bulbs, if the order of the selected bulbs is to be considered then the number of samples by permutations is given by

^5{P_2} = \frac{{5!}}{{\left( {5 - 2} \right)!}} =  20

.
These samples are:

{G_1}{G_2}

{G_2}{G_1}

{G_1}{G_3}

{G_3}{G_1}

{G_2}{G_3}

{G_3}{G_2}

{G_1}{D_1}

{D_1}{G_1}

{G_1}{D_2}

{D_2}{G_1}

{G_2}{D_1}

{D_1}{G_2}

{G_2}{D_2}

{D_2}{G_2}

{G_3}{D_1}

{D_1}{G_3}

{G_3}{D_2}

{D_2}{G_3}

{D_1}{D_2}

{D_2}{D_1}

Each sample has probability of selection equal to1/20. The selected sample keeping in view the order of the bulbs will be any one of these 20 samples. A sample selected in this manner is also called simple random sample because each sample has equal probability of being selected.

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