Sampling With Replacement

Sampling is called with replacement when a unit selected at random from the population is returned to the population and then a second element is selected at random. Whenever a unit is selected, the population contains all the same units, so a unit may be selected more than once. There is no change at all in the size of the population at any stage. We can assume that a sample of any size can be selected from the given population of any size.

 

This is only a theoretical concept, and in practical situations the sample is not selected by using this selection method. Suppose a population size N = 5 and sample size n = 2, and sampling is done with replacement. Out of 5 elements, the first element can be selected in 5 ways. The selected unit is returned to the main lot and now the second unit can also be selected in 5 ways.

 

Thus in total there are 5 \times 5 = 25 samples or pairs which are possible. Suppose a container contains 3 good bulbs denoted by {G_1},{G_2} and {G_3} and 2 defective bulbs denoted by {D_1} and {D_2}. If any two bulbs are selected with replacement, there are 25 possible samples, as listed in the table below:

 

 
{G_1}
{G_2}
{G_3}
{D_1}
{D_2}
{G_1}
{G_1}{G_1}
{G_1}{G_2}
{G_1}{G_3}
{G_1}{D_1}
{G_1}{D_2}
{G_2}
{G_2}{G_1}
{G_2}{G_2}
{G_2}{G_3}
{G_2}{D_1}
{G_2}{D_2}
{G_3}
{G_3}{G_1}
{G_3}{G_2}
{G_3}{G_3}
{G_3}{D_1}
{G_3}{D_2}
{D_1}
{D_1}{G_1}
{D_1}{G_2}
{D_1}{G_3}
{D_1}{D_1}
{D_1}{D_2}
{D_2}
{D_2}{G_1}
{D_2}{G_2}
{D_2}{G_3}
{D_2}{D_1}
{D_2}{D_2}

The number of samples is given by {N^n} = {5^2} = 25. The selected sample will be any one of the 25 possible samples. Each sample has an equal probability 1/25 of selection. A sample selected in this manner is called a simple random sample.