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Unformatted text preview: Introduction to Confidence Intervals 1 Introduction When analyzing data, we view our observations x 1 ,...,x n as a realization of a random sample X 1 ,...,X n from a distribution with unknown parameters. We then use x 1 ,...,x n to compute estimates of these unknown parameters. How good are these estimates? • In the last chapter, we studied the probability distributions of estimators (random variables). An estimate is a realization of an estimator. • The quality of the estimate depends on the distribution of the estimator. • When we report an estimate of a parameter, it is important to include some information about its quality. • In some situations, we can report: estimate ± margin of error where the margin of error is determined to ensure a level of confidence that the parame ter is contained in this range of values. The margin of error depends on the distribution of the estimator. Definition: confidence interval for a parameter an interval in which we are confident the parameter of the distribution we are estimating is contained. It is a realization of a random confidence interval, which has an associated confidence level , defined as the probability that the random interval contains the parameter. Usually we use confidence levels of 90%, 95%, or 99%. Example 1.1: Suppose that (0 . 55 , . 71) is a confidence interval for θ , the population proportion of workers in the Twin Cities that drive to work alone, based on a 95% confidence level and a realization of a random sample from Bern( θ ). • The probability that θ is in this interval (0 . 55 , . 71) is either 0 or 1, we don’t know since θ is unknown. • The confidence interval with an associated confidence level of 95% is simply a realiza tion of a random 95% confidence interval, which has probability of 0.95 (approximately in this example) of containing the population proportion θ . • If we repeated the experiment that generates realizations of the random 95% confidence interval, many times, we would expect roughly 95% of the new 95% confidence intervals to contain the population proportion θ . 1 2 Approximate confidence interval (CI) for θ , the suc cess probability (population proportion) of a Bernoulli distribution. 2.1 How to derive a 95% (approximate) confidence interval for θ In this setting, X 1 ,...,X n are a random sample from Bern( θ ). From Corollary A and B, we know that if nθ ≥ 15 and n (1 θ ) ≥ 15, then the sample proportion ˆ P approximately has the N ( μ ˆ P ,σ ˆ P ) distribution, where: μ ˆ P = θ σ ˆ P = r θ (1 θ ) n The 2.5th and 97.5th percentile of the Standard Normal distribution are z . 025 = 1 . 96 and z . 975 = 1 . 96, which imply that P ( 1 . 96 < Z < 1 . 96) = 95% where Z is Standard Normal. Now, using the Normal–Standard Normal rule, the random variable: ˆ P θ q θ (1 θ ) n approximately has the Standard Normal distribution. Let’s replace Z with the expression above to get: 95% ≈ P 1 . 96 < ˆ P μ ˆ P σ...
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 Spring '08
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 Normal Distribution, Standard Deviation

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