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Unformatted text preview: Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Lecture 11: Random Samples, Weak Law of Large Numbers and Central Limit Theorem Devore: Section 5.35.5 October, 2011 Page 1 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Definition of a Statistic • A statistic is any quantity whose value can be calculated from sample data. Prior to obtaining data, there is uncertainty as to what value of any particular statistic will result. • A statistic is a random variable denoted by an uppercase letter; a lowercase letter is used to represent the calculated or observed value of the statistic. October, 2011 Page 2 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 • Example Consider a sample of n = 3 cars of a particular type; their fuel efficiencies may be x 1 = 30 . 7 mpg, x 2 = 29 . 4 mpg, x 3 = 31 . 1 mpg. • It may also be x 1 = 28 . 8 mpg, x 2 = 30 . mpg and x 3 = 31 . 1 mpg • This implies that the value of the mean ¯ X is different in these cases. Clearly, ¯ X is a statistic. The first sample has the mean ¯ X 1 = 30 . 4 mpg and the second one has ¯ X 2 ≈ 30 mpg October, 2011 Page 3 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Statistic Examples • A sample mean ¯ X of the sample X 1 ,...,X n is a statistic; ¯ x is one of its possible values • The value of the sample mean from any particular sample can be regarded as a point estimate of the population μ . • Another example is the sample standard deviation S , while s is its computed value • Yet another example is the difference between the sample means for two different populations ¯ X ¯ Y October, 2011 Page 4 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Sampling distribution • Each statistic is a random variable and, as such, has its own distribution • Consider two samples of size n = 2 ; if X 1 = X 2 = 0 , ¯ X = 0 with probability P ( X 1 = 0 ∩ X 2 = 0) • On the other hand, if X 1 = 1 but X 2 = 0 or X 1 = 0 and X 2 = 1 , we have ¯ X = 0 . 5 with probability P ( X 1 = 1 ∩ X 2 = 0) + P ( X 1 = 0 ∩ X 2 = 1) • This distribution is called the sampling distribution to emphasize its description of how the statistic varies in value across all possible sample October, 2011 Page 5 Statistics 511: Statistical Methods...
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 Fall '08
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 Statistics, Central Limit Theorem, Law Of Large Numbers, Standard Deviation, Purdue University, Dr. Levine

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