Simple Random Sampling with Replacement

                                                                                                                                    --Anasua Dutta

Simple Random Sampling

Simple random sampling (SRS) is a method of selection of a sample comprising of n number of sampling units out of the population having N number of sampling units such that every sampling unit has an equal chance of being chosen. The samples can be drawn in two possible ways.

· The sampling units are chosen without replacement in the sense that the units once are chosen are not placed back in the population.

 · The sampling units are chosen with replacement in the sense that the chosen units are placed back in the population.

Simple Random Sampling With Replacement (SRSWR):

SRSWR is a method of selection of n units out of the N units one by one such that at each stage of selection, each unit has an equal chance of being selected, i.e., 1/ N .

For Example:-

Consider a population of potato sacks, each of which has either 12, 13, 14, 15, 16, 17, or 18 potatoes, and all the values are equally likely. Suppose that, in this population, there is exactly one sack with each number. So the whole population has seven sacks. If I sample two with replacement, then I first pick one (say 14). I had a 1/7 probability of choosing that one. Then I replace it. Then I pick another. Every one of them still has 1/7 probability of being chosen. And there are exactly 49 different possibilities here (assuming we distinguish between the first and second.) They are: (12,12), (12,13), (12, 14), (12,15), (12,16), (12,17), (12,18), (13,12), (13,13), (13,14), etc.

Notations:-

These are some important notations:-

Probability of drawing a sample:

When n units are selected with SRSWR, the total number of possible samples are N^n. The Probability of drawing a sample is 1/N^n.

Alternatively, let ui be the ith unit selected in the sample. This unit can be selected in the sample either at first draw, second draw, …, or nth draw. At any stage, there are always N units in the population in the case of SRSWR, so the probability of selection of ui at any stage is 1/N for all i = 1,2,…,n. Then the probability of selection of n units u1, u2, ……un in the sample is:

Probability of drawing a unit:

P[selection of uj at kth draw] = 1/N .

Estimation of population mean and population variance:

One of the main objectives after selecting a sample is to know about the tendency of the data to cluster around the central value and the scatteredness of the data around the central value. Among various indicators of central tendency and dispersion, the popular choices are arithmetic mean and variance. So the population means and population variability is generally measured by the arithmetic mean (or weighted arithmetic mean) and variance, respectively. There are various popular estimators for estimating the population mean and population variance. Among them, sample arithmetic mean and sample variance is more popular than other estimators. One of the reasons to use these estimators is that they possess nice statistical properties. Moreover, they are also obtained through well-established statistical estimation procedures like maximum likelihood estimation, least squares estimation, method of moments, etc., under several standard statistical distributions. One may also consider other indicators like median, mode, geometric mean, harmonic mean for measuring the central tendency and mean deviation, absolute deviation, Pitman nearness, etc., for measuring the dispersion. The properties of such estimators can be studied by numerical procedures like bootstrapping.

1.    Estimation of population mean

 

1.    Variance of the estimate

Application in R

Here we have taken an inbuilt dataset (Plant Growth) in R to show the commands used for Simple Random Sampling with replacement.



CONCLUSION:-

Lastly, we can conclude that, when we sample with replacement, the two-sample values are independent. Practically, this means that what we get on the first one doesn't affect what we get on the second. Also, sampling without replacement is more efficient than sampling with replacement as the variance of SRSWOR is less than that of SRSWR shown in the R code. 











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