Lecture+7+su10+_standard+error_ - Sociology 210 Lecture 7...

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02/23/11 Soc 210 summer 2010 1 Sociology 210 Lecture 7: Standard Error
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02/23/11 Soc 210 summer 2010 2 Review: Sampling Distributions A statistic is any function computed from the data in a sample You can think of any statistic as a variable, because its value will vary from one sample to another in the same population This set of statistics from different samples has a probability distribution which we call a sampling distribution A sampling distribution is a general term for a distribution of statistics obtained by selecting all possible samples of a specific size from a population The sampling distribution of the sample mean is the collection of sample means for all possible random samples of a particular size ( n ) that can be obtained from a population
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02/23/11 Soc 210 summer 2010 3 Properties of the Sampling Mean or the Mean of the Means of All Possible Samples Suppose you repeatedly draw random samples of size n , with replacement, from a population of size N and calculate the sample mean from each sample. You can consider the sample mean to be a variable, , with a mean, , and a standard deviation If a population variable X has mean μ and standard deviation σ, then the sample mean for random sample of size n drawn with replacement has mean and standard deviation given by Also, if X is approximately normally distributed, so is X X μ X σ X μ n X X σ σ μ μ = = and X σ X
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02/23/11 Soc 210 summer 2010 4 Example A population consists of three individuals with values on the variable X = {4, 7, 10}. Random samples of size 2 are drawn with replacement. Compute the population mean, μ , and standard deviation, σ There are only three cases in the population, so the mean and standard deviation of the population are μ = X/N = (4 + 7 + 10)/3 = 21/3 = 7 and Find the sampling distribution (probability distribution) for the sample mean. First, you need to calculate the mean value for every sample, as given in the following table ( a , b ) (4, 4) (4, 7) (4, 10) (7, 4) (7, 7) (7, 10) (10, 4) (10, 7) (10, 10) 4 5.5 7 5.5 7 8.5 7 8.5 10 [ ] 6 3 18 3 / ) 7 10 ( ) 7 7 ( ) 7 4 ( / ) ( 2 2 2 2 = = - + - + - = - = N x μ σ
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