Slides21-2010 - Condence Intervals Lecture XXI Charles B Moss Charles B Moss Condence Interval 1 20 1 Interval Estimation 2 Condence Intervals

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Confdence Intervals: Lecture XXI Charles B. Moss October 21, 2010 Charles B. Moss () Confdence Interval October 21, 2010 1 / 20
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1 Interval Estimation 2 Confdence Intervals Charles B. Moss () Confdence Interval October 21, 2010 2 / 20
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Interval Estimation As we discussed when we talked about continuous distribution functions, the probability of a speciFc 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 conFdence intervals. SpeciFcally, as in the case when we discussed the probability 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. Charles B. Moss () Confdence Interval October 21, 2010 3 / 20
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Confdence Intervals Amemiya notes a diference between conFdence and probability. Most troubling is our classic deFnition o± 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, T = ¯ X A N ± p , p (1 p ) n ² (1) There±ore, we have Z = T p r p (1 p ) n A N (0 , 1) (2) Charles B. Moss () Confdence Interval October 21, 2010 4 / 20
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Continued I Why? By the Central Limit Theory. I Given this distribution, we can ask questions about the probability. Specifcally, we know that iF Z is distributed N (0 , 1) , then we can defne γ k = P ( | Z | < k )( 3 ) Building on the normal probability, the one tailed probabilities For the normal distribution are presented in Table 1.
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.

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Slides21-2010 - Condence Intervals Lecture XXI Charles B Moss Charles B Moss Condence Interval 1 20 1 Interval Estimation 2 Condence Intervals

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