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Unformatted text preview: 7.1 Basic Properties of Confidence intervals Point estimate does not say how close it might be to parameter . An alternative is to report an interval of possible values of . This is called interval estimate or confidence interval. Thus, Confidence level: degree of the reliability of an interval. Confidence interval: point estimate margin of error Suppose we have a population with mean and standard deviation . The population mean is unknown and our task is to estimate its value. We draw a random sample and calculate its mean x . Since x is normally distributed (at least for large n), the variable / x z n  = has standard normal distribution (at least approximately). In one of the previous problems we were looking for the zscores that cut given percentage of central area around the mean in standard normal distribution; that is, we solved a problem like, for example, P(?<z<?)=0.95. Corresponding zvalues are: 1.96 and 1.96. This can be written: 0.95 ( 1.96 1.96) ( 1.96 1.96) / ( 1.96 1.96 ) ( 1.96 1.96 ) ( 1.96 1.96 ) 1 0.05 x P z P n P x x n n P x x n n P x x n n  = < < = < < = <  <  + = + = < < + =  1 Where =0.05 is the probability that x doesnt fall in between the boundaries above. The interval 1.96 1.96 x x n n + is called the 95% confidence interval. Such interval can be found for ANY chosen error level Problem: Find an % confidence interval for a true mean of a population. That is, compute the formula for confidence intervals for the mean : Finally, 95% confidence means that on average in 95 out of 100 estimations the interval will contain an estimated parameter. The examples of other confidence levels: C in % z* 90% 95% 99% 2 For instance, for x =46 confidence level 95% a conclusion can be that "we are 95% confident that is between 41 and 51" (if we take many samples of size n, then in the long run 95% of them will capture the real mean ) Remember: It is possible that your confidence interval doesnt contain the mean of the population: that its one of the 5% who does not contain it! HW assignment: Section 7.1: #17 odd, 4* 3 7.2 Large Sample Confidence Intervals for Population Mean and Proportion The formula Onesided and twosided confidence intervals A large sample lower and upper confidence bound Choosing the sample size Confidence interval: Point estimate...
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This note was uploaded on 07/25/2008 for the course STT 430 taught by Professor Nane during the Spring '08 term at Michigan State University.
 Spring '08
 NANE

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