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07+Sampling+distribution - A Zhensykbaev Sampling...

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A. Zhensykbaev Sampling distributions Sampling distributions. Sampling theory is a study of relationships existing between a population and samples drawn from the population. It is of great value in many connec- tions. It is useful, for example, in estimating unknown population quantities (such as population mean and variance), often called population parameters or briefly parameters, from a knowledge of corresponding sample quantities (the sample mean X , sample variance s 2 , sample standard deviation s ), often called sample statistics or briefly statistics. Parameter is a numerical measure that describes a characteristic of the population. Sample Statistic is a numerical measure computed from a sample to describe a characteristic of the population. Population parameters are usually unknown. We use sample statistics to estimate the parameters of a population. Sampling theory is also useful in determination whether the observed differences between two samples are due to chance variation or whether they are really significant. The answers involve the use of so-called test of significance and hypotheses that are important in the theory of decisions. In general, a study of the inferences made concerning a population by using samples drawn from it, together with indications of the accuracy of such inferences by using probability theory, is called statistical inference. Sample statistics are themselves random variables. The probability distri- bution of a sample statistic (the mean, the standard deviation) is called the sampling distribution for the statistic. Example of a sampling distribution. The population consists of the measurements { 0 , 2 , 10 } with equal probabilities: P (0) = P (2) = P (10) = 1 3 . A random sample of 2 measurements is selected from the population. Find sampling distribution of the sample mean x . Solution . 1. List possible samples and its mean (it is presented in the table below). For example if x = (0; 2) x = 0 + 2 2 = 1 1
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A. Zhensykbaev Sampling distributions Sample 0; 0 0; 2 0; 10 2; 0 2; 2 2; 10 10; 0 10; 2 10; 10 Mean x 0 1 5 1 2 6 5 6 10 2. Determine probability each sample: sample space consists of 9 samples. Hence the probability of each of them equals to 1/9. 3. Determine the number of simple events containing in the event: becau- se the mean x = 0 occurs in one sample, P ( x = 0) = 1 / 9 ; similarly x = 1 occurs in two samples and P ( x = 1) = 2 / 9 etc. So, we have the sampling distribution of the sample mean x Mean x 0 1 2 5 6 10 P ( x ) 1/9 2/9 1/9 2/9 2/9 1/9 If a sample statistic has a sampling distribution with the mean equal to the population parameter the statistic is intended to estimate, the statistic is called an unbiased estimator of the parameter.
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