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# Central limit theorem o ˆ p sample proportion

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Central Limit Theorem o ˆ p : Sample Proportion Variance of Sample Proportion Application of Central Limit Theorem o s 2 : Sample Variance Degrees of Freedom χ 2 distribution of Sample Variance Estimation (25%) o Estimators Unbiasedness Precision o Confidence Intervals (Mean) Finding Critical Values for confidence intervals Known Population Variance Unknown Population Variance t-distribution For Population Proportions

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o Useful Formulas The following formulas may be helpful for certain problems. Some formulas that are easily derived from others are not included. Continuous: 0 max max max 2 2 0 min min min min ( ) 1; ( ) ( ) ; [ ] ( ) ; [ ] ( ) ( ) x x x x x x x x P x dx F x P x dx E X xP x dx Var X x P x dx μ σ μ = = = = = = - 2 2 2 ( ) ( ) ( ); , ; Y X Y X X P a X b F b F a If Y a bX a b and b Z μ μ μ σ σ σ - = - = + = + = = Uniform over [a,b]: 2 ( ) ( ) ; [ ] ; [ ] 2 12 a b b a P x c E X Var X + - = = = Normal: 2 2 ( ) 2 2 2 1 ( ) ; [ ] ; [ ] 2 x P x e E X Var X μ σ μ σ πσ - - = = = Binomial Approximation: ( , (1 )) S N np np p - : Proportional Approximation: (1 ) ( , ) S p p P N p n n - = : ; Joint Probability ( ) ( , ) ( , ) ( ) ( , ) y P X x Y y P X x Y y P x y P X x P x y = = = = = = = = 2 2 2 2 2 2 2 2 [ ] ( , ) ( ) [ ] [( ) ] [ ] [ ] ; [ ] +2abCov[X,Y] Additionally, if X,Y are normal, the sum aX+bY is also normal.
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