7.1-7.3 - 7.1 Basic Properties of Confidence intervals...

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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 z-scores 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 z-values 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: #1-7 odd, 4* 3 7.2 Large Sample Confidence Intervals for Population Mean and Proportion The formula One-sided and two-sided 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.

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7.1-7.3 - 7.1 Basic Properties of Confidence intervals...

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