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Lecture 21-2007 - Confidence Intervals Lecture XXI Interval...

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Unformatted text preview: Confidence Intervals Lecture XXI Interval Estimation • As we discussed when we talked about continuous distribution functions, the probability of a specific number under a continuous distribution is zero. • Thus, if we conceptualize any estimator, either a nonparametric estimate of the mean or a parametric estimate of a function, the probability of the true value equal to the estimated value is obviously zero. • Thus, usually talk about estimated values in terms of confidence intervals. Specifically, as in the case when we discussed the probability of a continuous variable, we define some range of outcomes. Confidence Intervals • Amemiya notes a difference between confidence and probability. Most troubling is our classic definition of probability as “a probabilistic statement involving parameters.” This is troublesome due to our inability without some additional Bayesian structure to state anything concrete about probabilities. • Example 8.2.1. Let X i be distributed as a Bernoulli distribution, i=1,2,…,n. Then, Therefore, we have ( 29 n p p p N X T A / ) 1 ( , ~- = ( 29 ( 29 1 , ~ 1 N n p p p T Z A-- = • Why? By the Central Limit Theory • Given this distribution, we can ask questions about the probability. Specifically, we know that if Z is distributed N (0,1), then we can define ( 29 k Z P k < = γ Table of Normal Probabilities k γ/2 γ 1.0000 0.1587 0.3173 1.5000 0.0668 0.1336 1.6449 0.0500 0.1000 1.7500 0.0401 0.0801 1.9600 0.0250 0.0500 2.0000 0.0228 0.0455 2.3263 0.0100 0.0200 • The values of γ k can be derived from the standard normal table as ( 29 k k n p p p T P γ = <-- 1 •...
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Lecture 21-2007 - Confidence Intervals Lecture XXI Interval...

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