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Unformatted text preview: Week 5&6: Introduction to Inference Week 5&6: Introduction to Inference Confidence Intervals In statistics, when we cannot get information from the entire population, we take a sample. However, as we have seen before statistics calculated from samples vary from sample to sample. When we obtain a statistic from a sample, we do not expect it to be the same as the corresponding parameter. It would be desirable to have a range of plausible values which take into account the sampling distribution of the statistic. A range of values which will capture the value of the parameter of interest with some level of confidence. This is known as a confidence interval . 2 / 46 Week 5&6: Introduction to Inference Confidence Intervals A confidence interval gives possible values for a parameter, not a statistic. For example, we use the sample mean to form a confidence interval for the population mean, . We use the sample proportion to form a confidence interval for the population proportion, . We NEVER say, The confidence interval of the sample mean is . We say, A confidence interval for the true population mean, , is . 3 / 46 Week 5&6: Introduction to Inference Confidence Intervals If a value is not covered by a confidence interval (it is not included in the range), then it is not a plausible value for the parameter in question and should be rejected as a plausible value for the population parameter. In general, a confidence interval is centered on our best guess for the parameter, the appropriate statistic. We can find confidence intervals for any parameter of interest, however we will be primarily concerned with the CIs for the following two parameters: Population mean, Population proportion, , in Week 10 4 / 46 Week 5&6: Introduction to Inference Confidence Intervals Confidence Interval for Population Mean Here we make use of the sampling distribution of the sample mean to develop a confidence interval for the population mean, , from the sample mean: 5 / 46 Week 5&6: Introduction to Inference Confidence Intervals Confidence Interval for Population Mean The unknown true mean of the sampling distribution is . We know from the central limit theorem that the sampling distribution can be normal if the sample size is large. We know, from our study of normal distributions, the proportion of the values between two values (for example, two standard deviations). We can thus say that we are 95% confident that a sample mean we find is within this interval. This is the same as saying that if we took many, many samples and found their means, 95% of them would fall within two standard deviations of the true mean. If we took a hundred samples, we would expect that about 95 sample means would be within this interval....
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 Spring '08
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 Statistics

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