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Unformatted text preview: Recap for Confidence Interval for the Mean ( Known)A confidence interval is a range of values that has a prespecified probability of including the true value of the parameter. Therefore, a confidence interval for the mean is a range of values that has a prespecified probability of including the mean, . The formula for a confidence interval for the mean when is known is:xz/2/n .The value of /2z(sometimes written as just z) depends on the level of confidence. The higher the level of confidence, the larger the zscore. For example, an 80% confidence interval for the mean is:x1.28n, whereas a 95% confidence interval for is: x1.96n. You should know how to determine the value of /2zfor the desired confidence level.Example: Genron Corp. would like to determine the mean length of life for the highperformance model of tires that it supplies to an American automobile manufacturer. Forty tires are randomly selected from the manufacturing process and subjected to accelerated lifetesting. The mean from the sample is 33,127 miles. Assume that the standard deviation for the population of tires (that is, ) is 7,500 miles. a. Construct a 95% confidence interval for the true mean life of this model of tire.132b. Carefully interpret the interval that you constructed in part a. in the context of the problem.133Confidence Intervals for the Mean, , When isUnknownThe tdistributionThe tdistribution, rather than the zdistribution (the standard normal) is used when we would like to estimate , but the population standard deviation, , is also unknown. In this case, when we take a sample to estimate with x, we also estimate with s. The tdistribution looks a lot like thestandard normal. The t and the z distributions become more similar as the sample size, n, gets larger. The formula for a confidence interval for the mean when and are both unknown is:/2,n1x tsnWhere:/2,n1x t=snTwo assumptions should be met when constructing a confidence interval for with a tscore. They are:1. Random sampling 2. Sampling from a normal population (xs are normal).As noted above, a tdistribution looks a lot like a normal distribution. However, when n is small, the tails of the tdistribution are much heavier than a normal distribution. As 134n gets larger and larger, the tdistribution looks more and more like the standard normal distribution....
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This note was uploaded on 06/06/2011 for the course MGSC 291 taught by Professor Rollins during the Fall '09 term at South Carolina.
 Fall '09
 Rollins

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