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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 5.3-5.5 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 . 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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