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Unformatted text preview: Chapter 6 Sampling and Sampling Distributions A random sample is a set of random variables ,, . . . , (upper case notation) that are: 1 X 2 X n X x identically distributed. That is, each of these random variables has mean P and variance; and 2 V x independently distributed. That is, 0)X,X( Cov j i for any ji . Typically, the population parameters (such as and) are unknown. 2 V Econ 325 – Chapter 6 1 A sample of data are the observed numerical outcomes ,, . . . , (lower case notation). The sample mean can be calculated as: 1 x 2 x n x ¦ n 1i i x n 1 x Clearly, x will not be identical to the population mean P . For a second sample of n observations denote the numerical outcomes as: ,, . . . , * 1 x * 2 x * n x From this sample the sample mean is: ¦ n 1i * i * x n 1 x The two calculated sample means x and * x will be different numbers and neither will be the same as the population mean P . That is, different samples of n observations have different numerical observations and therefore, the calculated sample means are different. Econ 325 – Chapter 6 2 The sample mean of the random variables,, . . . , is defined as: 1 X 2 X n X ¦ n 1i i X n 1 X X is a linear combination of random variables and, therefore, is also a random variable. X has a probability distribution known as the sampling distribution . The sampling distribution of a sample statistic is the probability distribution of the values it could take over all possible samples of size n drawn from the population. What are the properties of the sampling distribution of X ? First, state the mean: P P » ¼ º « ¬ ª ¦ ¦ )n( n 1 )X(E n 1 X n 1 E)X(E n 1i i n 1i i That is, P )X(E . This says – for a large number of samples (say 1000 samples), each with n observations, the average of the calculated sample means will approach the population mean . Econ 325 – Chapter 6 3 Now state the variance: n ) n( n 1 )X( Var n 1 X n 1 Var )X( Var 2 2 2 n 1i i 2 n 1i i V V » ¼ º « ¬ ª ¦ ¦ ce independen use Problem: How did the assumption of independence...
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 Spring '10
 WHISTLER
 Economics, Normal Distribution, Standard Deviation, Variance, Probability theory, probability density function

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