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**Virtual COMSATS Inferential Statistics Lecture-3**

Ossam Chohan Assistant Professor CIIT Abbottabad

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**Objective of this lecture**

After completion of this lecture students will be able to: Understand the concept of Sampling distirubtion Understand sampling with or without replacement Understand sampling distibution of different sample statistics

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**Samples drawing procedures**

There are two ways to draw samples Sampling with replacement In 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 replacement Samples are not replaced in the population once it is drawn and observed. So each time probability of remaining samples decreased by one unit

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**WHY 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.

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**SAMPLING 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.

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**SAMPLING 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.

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How 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 sample When sampling is done with replacement then total number of samples are (Nn).

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Example If size of hypothetical population is 1,4,5,6, and sample size is 2, then how many samples will be there if we use Sampling with replacement Sampling without replacement

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Continued…..

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**Example-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.

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**Mean and standard deviation of population**

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**Random samples (with replacement)**

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**Sampling distribution of mean**

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**Example-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.

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**Example-3 Sampling distribution (without replacement case)**

Assume N=0,3,6, n=3 Find the sampling distribution of the sample mean . Calculate the mean and the standard deviation of sample mean and verify that δ2 = [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

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