2Objective of this lecture After completion of this lecture students will be able to:Understand the concept of Sampling distirubtionUnderstand sampling with or without replacementUnderstand sampling distibution of different sample statistics
3Samples drawing procedures There are two ways to draw samplesSampling with replacementIn this scheme after selecting a sample, sample is put back into the population after observation. In this technique, each time each sample is having equal probability of selection.Sampling without replacementSamples are not replaced in the population once it is drawn and observed. So each time probability of remaining samples decreased by one unit
4WHY SAMPLE WITH REPLACEMENT? When selecting a relatively small sample from a large population, it makes no significant difference whether we sample with replacement or without replacement.Sampling with replacement results in independent events that are unaffected by previous outcomes, and independent events are easier to analyze and they result in simpler formulas.
5SAMPLING DISTRIBUTION OF THE MEAN The sampling distribution of the mean is the probability distribution of the sample means, with all samples having the same size n.In general the sampling distribution of any statistic is the probability distribution of that statistic.
6SAMPLING VARIABILITY The value of a statistic, such as the sample mean , depends on the particular values included in the sample, and it generally varies from sample to sample. This variability of a statistic is called sampling variability.
7How many samples?When sampling is done without replacement then total number of samples are (Ncn), where N=size of population (hypothetical) whereas n= size of sampleWhen sampling is done with replacement then total number of samples are (Nn).
8ExampleIf size of hypothetical population is 1,4,5,6, and sample size is 2, then how many samples will be there if we useSampling with replacementSampling without replacement
10Example-1: Drawing samples from population Assume that a population consists of 5 similar containers having the following weights (in Kilograms):2, 4, 6, 8, 10.Find the mean µ and the standard deviation δ of the given population.Draw random samples of size 2 containers with replacement and calculate the mean weight of each sample.Form a frequency distribution of sample mean and sampling distribution of sample mean.Find the mean and standard deviation of the sampling distribution of sample mean.Verify the results.
15Example-2 Sampling distribution when sample size is 3 (with replacement) With the same data in above example try it with sample size = 3.We will draw few samples for your understanding.Rest of the portion will your home exercise.
17Example-3 Sampling distribution (without replacement case) Assume N=0,3,6, n=3Find the sampling distribution of the sample mean .Calculate the mean and the standard deviation of sample mean and verify thatδ2 = [δ2/n]*[N-n/N-1]Where N-n/N-1 is fpc (fintie population correction factor and only introduced when n≥5%of N