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Lecture1

# xn n 2 we are interested in the parameter 1

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Unformatted text preview: X 1 , . . . , X n , such that 1.  The Xi’s are independent RV’s; and 2.  Every X i has the same probability distribu\$on. We also say that the Xi’s are independently and iden3cally distributed (IID). •  Then the sta\$s\$c, i.e., a func\$on of the X i’s, follows a distribu\$on (called sampling distribu3on) that also depends on the unknown parameter θ . We make inference on θ based on the sampling distribu\$on. We typically write X 1 , . . . n for hypothe\$cal random sample, and the lower , X case x 1 , . . . , x for their observed values. n •  Stat 431 8 Example: Revenue Neutral Tax Bill (Cont’d) •  •  Parameter of interest: average change in tax paid over all tax return forms Sta\$s\$c: average change in tax paid over the sampled tax return forms •  Sampling distribu\$on of the sta\$s\$c: iid –  Assuming a normal popula\$on X1 , . . . , Xn ∼ N (µ, σ 2 ) –  We are interested in the parameter µ ￿ 1 ¯ –  The sta\$s\$c is X = n Xi –  By STAT 430 knowledge, we know 1 ¯ X ∼ N (µ, n σ 2 ) •  From the observed data: –  Observed average change is  ­\$219, with sample SD = \$725 ≈ σ •  What do...
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