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Lecture21-2010

# Lecture21-2010 - Condence Intervals Lecture XXI Charles B...

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Confdence Intervals: Lecture XXI Charles B. Moss October 19, 2010 I. Interval Estimation A. As we discussed when we talked about continuous distribution functions, the probability of a speciFc number under a continuous distribution is zero. B. 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 obvi- ously zero. C. Thus, usually talk about estimated values in terms of conFdence intervals. SpeciFcally, as in the case when we discussed the proba- bility of a continuous variable, we deFne some range of outcomes. However, this time we usually work the other way around deFning a certain conFdence level and then stating the values that contain this conFdence interval. II. ConFdence Intervals A. Amemiya notes a di±erence between conFdence and probability. Most troubling is our classic deFnition of probability as ”a proba- bilistic statement involving parameters.” This is troublesome due to our inability without some additional Bayesian structure to state anything concrete about probabilities. B. Example 8.2.1. Let X i be distributed as a Bernoulli distribution, i =1 , 2 , ··· n . Then, T = ¯ X A N p, p (1 p ) n ! (1) 1

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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XXI Fall 2010 Table 1: Con±dence Levels / 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 Therefore, we have Z = T p s p (1 p ) n A N (0 , 1) (2) 1. Why? By the Central Limit Theory. 2. Given this distribution, we can ask questions about the prob- ability. Speci±cally, we know that if Z is distributed N (0 , 1) , then we can de±ne γ k = P ( | Z | <k )( 3 ) Building on the normal probability, the one tailed probabili- ties for the normal distribution are presented in Table 1.
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Lecture21-2010 - Condence Intervals Lecture XXI Charles B...

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