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Slides21-2010 - Condence Intervals Lecture XXI Charles B...

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Confidence Intervals: Lecture XXI Charles B. Moss October 21, 2010 Charles B. Moss () Confidence Interval October 21, 2010 1 / 20
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1 Interval Estimation 2 Confidence Intervals Charles B. Moss () Confidence 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 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. However, this time we usually work the other way around defining a certain confidence level and then stating the values that contain this confidence interval. Charles B. Moss () Confidence Interval October 21, 2010 3 / 20
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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, T = ¯ X A N p , p (1 p ) n (1) Therefore, we have Z = T p p (1 p ) n A N (0 , 1) (2) Charles B. Moss () Confidence Interval October 21, 2010 4 / 20
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Continued 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 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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