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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 , 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( ). The probability that is in this interval (0 . 55 , . 71) is either 0 or 1, we dont know since 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 . 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 . 1 2 Approximate confidence interval (CI) for , the suc- cess probability (population proportion) of a Bernoulli distribution. 2.1 How to derive a 95% (approximate) confidence interval for In this setting, X 1 ,...,X n are a random sample from Bern( ). From Corollary A and B, we know that if n 15 and n (1- ) 15, then the sample proportion P approximately has the N ( P , P ) distribution, where: P = P = r (1- ) 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 NormalStandard Normal rule, the random variable: P- q (1- ) n approximately has the Standard Normal distribution. Lets replace Z with the expression above to get: 95% P- 1 . 96 < P- P...
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This note was uploaded on 05/06/2011 for the course STAT 5021 taught by Professor Staff during the Spring '08 term at Minnesota.
- Spring '08