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. 











Comments

Post a Comment

Popular posts from this blog

Selection of samples:SRSWR vs SRSWOR(2048114)

Image
  Selection of samples(SRSWR VS SRSWOR) Anindita Sarkar(2048114) INTRODUCTION- SIMPLE RANDOM SAMPLING (with replacement Vs without replacement) :- The simplest of the methods of probability sampling which is usually called the method of random sampling. In this method, an equal probability of selection is assigned to each available units of the population at the first and each subsequent draw. Thus, if the number of units in the population is N, then the probability of selection of any unit at first draw is 1/N and at the second draw is 1/N-1 etc, which are ultimately equal to 1/N. The sample obtained using the above method is called “ Simple Random Sampling”. Since this result is independent of the specified unit it follows that every one of the units in the population has the same chance of being included in the sample under the procedure of simple random sampling. Difference Between SRSWOR and SRSWR: (i)                   If the selected units are not being replaced back
Read more

Probability Proportion to size ordered Sampling – Desraj Estimator

Image
Karan Singhal  Introduction: - Sampling is a process used in statistical analysis in which a predetermined number of observations are taken from a larger population . The methodology used to sample from a larger population depends on the type of analysis being performed. One such sampling method is Probability Proportion to size without replacement ordered Estimate. As we know, sampling without replacement is more efficient than with replacement, and this rule also applies to PPS sampling. Since the probability of inclusion changes by draws or selected units' order, then ppswor is divided into ordered estimator and unordered estimator for better clarity. Furthermore, t o overcome the difficulty of changing expectations with each draw, associate a new variate with each draw such that its expectation is equal to the population value of the variate under study. Such estimators take into account the order of the draw. They are called ordered estimates. The order of the value obtain
Read more

Overview of Systematic Sampling

Image
  SYSTEMATIC SAMPLING By: UJJAINI DALAL CHRIST UNIVERSITY, BANGALORE S ystematic sampling is a commonly employed technique if the complete and up-to-date list of the sampling units in available. This consists in selecting only the first unit at random, the rest being automatically selected according to some predetermined pattern involving regular spacing of units. Systematic sampling is a statistical method that researchers use to zero down on the desired population they want to research.   Systematic sampling is an extended implementation of probability sampling in which each member of the group is selected at regular periods to form a sample. How Systematic Sampling Works When we  are sampling, ensure you represent the population fairly. Systematic sampling is a symmetrical process where the researcher chooses the samples after a specifically defined interval. Sampling like this leaves the researcher no room for bias regarding choosing the sample. Let us suppose that N samp
Read more