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CHAPTER 11: Data Analysis and Interpretation: Part I. Describing Data, Confidence Intervals, Correlation 363BOX 11.3INTERPRETING CONFIDENCE INTERVALS FOR A SINGLE MEAN:RINGS AND STAKESHaving calculated the .95 confidence interval for apopulation mean we may state thatthe odds are 95/100 that the obtained confidencein ter val contains the true population mean.The confidence interval either does or doesnot contain the true mean (e.g., Mulaik, Raju, &Harshman, 1997). A .95 probability associatedwith the confidence interval for a mean refers tothe probability of capturing the true populationmean if we were to construct many confidence intervalsbased on different random samples of thesame size. That is, confidence intervals aroundthe sample mean tell us what happens if we wereto repeat this study under the same conditions(e.g., Estes, 1997). In 95 of 100 replications wewould expect to capture the true mean with ourconfidence intervals.Having calculated the 95% confidence intervalfor a population mean we should NOT state thatthe odds are 95/100 that the true mean falls in thisinterval.This statement may seem to be identical to thestatement above. It isn’t. Keep in mind that thevalue in which we are interested is fixed, a constant;it is a population characteristic or parameter.Intervals are not fixed; they are characteristicsof sample data. Intervals are constructed fromsample means and measures of dispersion thatare going to vary from study to study and, consequently,so do confidence intervals.Howell (2002) provides a nice analogy to helpunderstand how these facts relate to our interpretationof confidence intervals. He suggests wethink of the parameter (e.g., the population mean)as a stake and confidence intervals as rings. Fromthe sample data the researcher constructs ringsof a specified width that are tossed at the stake.When the 95% confidence interval is used, therings will encircle the stake 95% of the time andwill miss it 5% of the time. “The confidence statementis a statement of the probability that the ringhas been on target; it is not a statement of theprobability that the target (parameter) landed inthe ring” (Howell, 2002, p. 208).The narrower the interval, the better is our interval estimate of the populationmean. You can see by examining the formulas for the upper and lower limitsthat the width of the interval depends on both the t statistic and the standarderror of the mean. Both of these values are related to sample size such that eachdecreases as sample size increases; however, increases in sample size have themost effect on the standard error. Consider that doubling the sample ___ size in theabove example would produce a standard error of 1.81 (14/ 60 ) and consequentlya much narrower confidence interval. The bottom line: Increasing samplesize will improve the interval estimate of the mean.Confidence Intervals for a Comparison Between Two Independent Group MeansThe procedure and logic for constructing confidence intervals for a differencebet ween means is similar to that for setting confidence intervals for a singlemean. Because our interest is now in the difference between the populationmeans (i.e., “the effect” of our independent variable) we substitute __ X 1 __ X 2