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Lecture9

# Lecture9 - March 5 th Large Sample Confidence interval for...

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Unformatted text preview: March 5 th ( Large Sample ) Confidence interval for a population proportion p Let . . . ~ ( ), 1, , i i d i X Bernoulli p i n = L , please find the 100(1-α)% CI for p. 1) Point estimator : 1 ˆ n i i X p X n = = = ∑ (ex. 1000 n = , ˆ 0.6 p = ) 2) 100(1-α)% C.I. for p <Thm> Central Limit Theorem ( ) (0,1) ( ) n X E X N Var X → ∞- → When n is large enough, we have ( ) ~ (0,1) ( ) X E X Z N Var X- = & Application #1. Inference on μ when the population distribution is unknown but the sample size is large ~ (0,1) / X Z N n μ σ- = & ~ (0,1) / X Z N S n μ- = & (Slutsky’s Theorem) => 100(1 )% α- C.I. for μ /2 S X Z n α ± Application #2. Inference on one population proportion p when the population is Bernoulli(p) *** ( ) ~ (0,1) ( ) X E X Z N Var X- = & 1 1 ( ) ( ) ( ) i i X E X E E X np p n n n = = = ⋅ = ∑ ∑ , ( ~ ( , )) i X B n p ∑ Q 2 2 1 1 (1 ) ( ) ( ) ( ) (1 ) i i X p p Var X Var Var X np p n n n n- = = =- = ∑ ∑ 1 ~ (0,1) (1 ) X p Z N p p n- =- & ˆ ~ (0,1)...
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Lecture9 - March 5 th Large Sample Confidence interval for...

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