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Unformatted text preview: 1 Chapter 6  Confidence Intervals and Hypothesis Testing Why do we even bother analyzing data? We want to draw conclusions from the data. Why cant we just accept our sample mean or sample proportion as the official mean or proportion for the population? Every time we estimate the statistics , x p (sample mean and sample proportion), we get a different answer due to sampling variability. Two most common types of formal statistical inference: Confidence Intervals : when we want to estimate a population parameter Significance Tests : when we want to assess the evidence provided by the data in favor of some claim about the population (yes/no question about the population) Confidence Intervals allow us to estimate a range of values for the population mean or proportion. The true mean or proportion for the population exists and is a fixed number, but we just dont know what it is. Using our sample statistic, we can create a net to give us an estimate of where to expect the population parameter to be. Confidence interval = net Population parameter = invisible, stationary butterfly We dont know exactly where the butterfly is, but from our sample, we have a pretty good estimate of the location. If we take a single sample, our single confidence interval net may or may not include the population parameter. However if we take many samples of the same size and create a confidence interval from each sample statistic, over the long run 95% of our confidence intervals will contain the true population parameter (if we are using a 95% confidence level). 2 If you increase the sample size ( n ) , you decrease the size of your net (or your margin of error). If you increase your confidence level ( C ) , then you increase the size of your net (or your margin of error). A smaller net is good because it gives you more information. It is a smaller range for where to expect your true population parameter. Freeman applet: Go to course website, Freeman link, statistical applets, confidence interval. Confidence interval formulas look like estimate margin of error. We write the intervals as (lower bound, upper bound). Confidence Interval for a Population Mean, : * x x z n , where z* is the value on the standard normal curve with area C between z* and z* . z* 1.645 1.960 2.576 C 90% 95% 99% (Table D at the back of the book also contains more z* values on the bottom row.) What if your margin of error is too large? Here are ways to reduce it: Increase the sample size (bigger n ) Use a lower level of confidence (smaller C ) Reduce x 3 Sample Size, n, for Desired Margin of Error, m: 2 * x z n m Note that it is the sample size, n , that influences the margin of error. The population size has nothing to do with it....
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 Spring '08
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