Unformatted text preview: oblem: How did the assumption of independence simplify the variance formula ? That is, Var ( X ) = σ n 2 This gives the result that as the sample size n increases the variance of the sample mean decreases. The standard deviation of the sampling distribution of X is called the standard error of X . This is: 4 se( X ) = σ n Chapter 7 Now introduce the assumption of normality. Let the random sample X 1 , X 2 , . . . , X n be a set of normally distributed and independent random variables with mean μ and variance σ . 2 It follows that X is also normally distributed (recall that an earlier result stated that a linear combination of normally distributed random variables is also normally distributed). That is, 2
X ~ N⎛ μ , σ n ⎞ ⎜
⎠ PDF of X for n=25 and n=100 n = 100 n = 25 Note: the total area under each curve is equal to one. 5 Chapter 7 The graph on the previous page shows the probability density function of the sampling distribution of the sample mean. This is centered at μ . The graph demonstrates that as the...
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- Spring '10