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Asian School of Business PG Programme in Management (2005-06) Course: Quantitative Methods in Management I Instructor: Chandan Mukherjee Session 18: Sampling Distribution (a demo) Definitions Statistic = A function of the sample values Example: Sample Mean, Sample Variance are Statistics Estimator: When a Statistics is used to estimate a parameter of the distribution from which the sample has been drawn, it is called an Estimator of the parameter. Value of a Statistic varies from sample to sample. Distribution of the all possible values of an Estimator is called the Sampling Distribution of the Estimator. An Estimator is called Unbiased if the mean of its Sampling Distribution is equal to the parameter. Standard Deviation of the Sampling Distribution of an Estimator is called its Standard Error. Symmetric Population Distribution Population: 1 2 2 3 3 3 3 4 5 X 1 2 3 4 5 All Mean F 1 2 4 2 1 10 = 3.0 Variance = 1.2 80% Symmetric Population Distribution: Random Sample Without Replacement, Size 2 List of All Possible Sample Means Sample 1 Sample 2 1 2 2 3 3 3 3 4 4 5 1 1.5 1.5 2.0 2.0 2.0 2.0 2.5 2.5 3.0 X2 2 2 3 3 3 3 4 4 5 1.5 1.5 2.0 2.0 2.5 2.5 2.0 2.5 2.5 3.0 2.0 2.5 2.5 3.0 3.0 2.0 2.5 2.5 3.0 3.0 3.0 2.5 3.0 3.0 3.5 3.5 3.5 3.5 2.5 3.0 3.0 3.5 3.5 3.5 3.5 4.0 3.0 3.5 3.5 4.0 4.0 4.0 4.0 4.5 4.5 2.0 2.5 2.5 2.5 2.5 3.0 3.0 3.5 2.5 2.5 2.5 2.5 3.0 3.0 3.5 3.0 3.0 3.0 3.5 3.5 4.0 3.0 3.0 3.5 3.5 4.0 3.0 3.5 3.5 4.0 3.5 3.5 4.0 4.0 4.5 4.5 Distribution of Sample Mean: The Sampling Distribution X2 F RF 1.5 2.0 2.5 3.0 3.5 4.0 4.5 All 4 4.4 10 11.1 20 22.2 22 24.4 20 22.2 10 11.1 4 4.4 90 100.0 Mean = 3.0 Variance = 0.53 91% Asymmetric Population Distribution Population: 1 1 1 1 2 2 2 3 3 4 X 1 2 3 4 All Mean F 4 3 2 1 10 = 2.0 Variance = 1.0 90% Asymmetric Population Distribution: Random Sample Without Replacement, Size 2 List of All Possible Sample Means Sample 1 Sample 2 1 1 1 1 2 2 2 3 3 4 1 1.0 1.0 1.0 1.5 1.5 1.5 2.0 2.0 2.5 X2 1 1 1 2 2 2 3 3 4 1.0 1.0 1.0 1.0 1.0 1.0 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2.0 1.5 1.5 1.5 1.5 2.0 2.0 2.0 2.0 2.0 2.0 2.5 2.5 2.5 2.0 2.0 2.0 2.0 2.5 2.5 2.5 3.0 2.5 2.5 2.5 2.5 3.0 3.0 3.0 3.5 3.5 1.0 1.0 1.5 1.5 1.5 2.0 2.0 2.5 1.0 1.5 1.5 1.5 2.0 2.0 2.5 1.5 1.5 1.5 2.0 2.0 2.5 2.0 2.0 2.5 2.5 3.0 2.0 2.5 2.5 3.0 2.5 2.5 3.0 3.0 3.5 3.5 Distribution of Sample Mean: The Sampling Distribution X2 1.0 1.5 2.0 2.5 3.0 3.5 ALL Mean F RF 12 13.3 24 26.7 22 24.4 20 22.2 8 8.9 4 4.4 90 100.0 = 2.0 Variance = 0.44 96% Lessons from the Demo •Consider a random sample and the sample mean. •If the distribution of the universe (population) is symmetric then the sampling distribution of the sample mean is also symmetric around the population mean. •If the distribution of the universe (population) is asymmetric then the sampling distribution of the sample mean is asymmetric around the population mean. •In both cases the sample mean is an unbiased estimator of the population mean. Theoretical Results X ~ N(μ, σ2) Random sample of size n. Xn = Sample Mean Then, Xn ~ N(μ, σ2/n) •If X has an arbitrary distribution, then also the above result about the sample mean holds if n is sufficiently large. •How large is sufficiently large for the result to be valid? •Well, that depends on the distribution of X. Theoretical Results X ~ N(μ, σ2) Random sample of size n. xn ~ N (0,1) / n xn ~ t( n 1) s/ n Where n 1 2 s2 ( x x ) i n n 1 1 Theoretical Results In Binomial Distribution in n.p ≥ 5 and n.(1-p) ≥ 5 then x n. p ~ N (0,1) n. p.(1 p) Which implies pˆ p ~ N (0,1) p.(1 p) / n Where x is the number of successes in n trials

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