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sampling distribution

# sampling distribution - Sampling Distributions Populations...

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Sampling Distributions Populations and Samples The Sampling Distribution of the mean ( σ known) The Sampling Distribution of the mean ( σ unknown) The Sampling Distribution of the Variance

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Populations and Samples Population: A set or collection of all the objects, actual or conceptual and mainly the set of numbers, measurements or observations which are under investigation. Finite Population : All students in BITS Pilani Goa Campus Infinite Population : Total water in the sea or all the sand particle in sea shore. Populations are often described by the distributions of their values, and it is common practice to refer to a population in terms of its distribution.
Finite populations are described by the actual distribution of its values and infinite populations are described by corresponding probability distribution or probability density. For example, we may refer to a number of flips of a coin as a sample from a “binomial population” or to certain measurements as a sample from a “normal population”. “Population f ( x )” means a population is described by a frequency distribution, a probability distribution or a density f ( x ).

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If a population is infinite it is impossible to observe all its values, and even if it is finite it may be impractical or uneconomical to observe it in its entirety. Thus it is necessary to use a sample . Sample: A part of population collected for investigation which needed to be representative of population and to be large enough to contain all information about population. Random Sample (finite population): A set of observations X 1 , X 2 , …, X n constitutes a random sample of size n from a finite population of size N , if its values are chosen so that each subset of n of the N elements of the population has the same probability of being selected.
Random Sample (infinite Population): A set of observations X 1 , X 2 , …, X n constitutes a random sample of size n from the infinite population ƒ( x ) if: 1. Each X i is a random variable whose distribution is given by ƒ( x ) 2. These n random variables are independent. We consider two types of random sample: those drawn with replacement and those drawn without replacement . Sampling with replacement: In sampling with replacement, each object chosen is returned to the population before the next object is drawn. We define a random sample of size n drawn with replacement , as an ordered n- tuple of objects from the population, repetitions allowed.

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Example 1: A population consists of the set S = {4, 7, 10}. The space of all random samples of size 2, drawn with replacement, consists of all ordered pair ( a , b ) including repetitions of numbers in S .There are 9 such pairs, which are (4, 4), (4, 7), (4, 10), (7, 4), (7, 7), (7, 10), (10, 4), (10, 7), (10, 10) The Space of random samples drawn with replacement: If samples of size n are drawn with replacement from a population of size N , then there are N n such samples. In any survey involving sample of size n , each of these should have same probability of being chosen. This is equivalent to making a collection of all N n
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sampling distribution - Sampling Distributions Populations...

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