Contents

360 PART V: Analyzing and Reporting ResearchKey Concept(c) the effect size for each of the major independent variables; and(d) how to best present pictorial summaries of the data (e.g., figure showingmean performance across conditions).Note: Although a graph showing mean performance in the two groups of thevocabulary study could be drawn, a figure usually is not needed when only twogroup means are involved. Pictorial summaries become more important whensummarizing the results of studies with more than two groups.Stage 3: Using Confidence Intervals to ConfirmWhat the Data Reveal• An important approach to confirming what the data are telling us is to constructconfidence intervals for the population parameter, such as a mean ordifference between two means.In the third stage of data analysis we seek to confirm impressions of theevidence obtained as we familiarized ourselves with the data and obtainedsummary measures. A major approach in this third stage is the calculation of aconfidence interval for a population parameter. A confidence interval (CI) maybe calculated for a single population mean or population mean difference. Wefirst review the use of confidence intervals for one population mean. Then weintroduce confidence intervals for the difference between two population meansand discuss the interpretation of intervals when there are three or more means.Confidence intervals may already be familiar to you under a different name.Have you not heard reports in the media of survey results based on a sample ofrespondents? And with these reports have you sometimes heard a “margin oferror” presented? In Box 11.2 we review the concept of margin of error and itsrelation to a confidence interval.Confidence Intervals for a Single Mean The mean of a random sample from apopulation is a point estimate of the population mean. However, we can expectvariability among sample means from one situation to another due to randomvariation. The estimated standard error of the mean ( s X __ ) provides informationabout the “normal” range of sampling error. In computing a confidence intervalwe specify a range of values that we state with a certain degree of confidenceincludes the population mean. As you may suspect, the larger the interval wespecify, the greater our confidence that the mean will be included; but largerintervals give us less specific information about the exact value of the populationmean. As a compromise, researchers have agreed that the 95% confidenceinterval and the 99% confidence interval are the best intervals to use when aninterval estimate of the population mean is desired.The confidence interval is centered about our point estimate ( __ X ) of the populationmean, and the boundaries of the 95% confidence interval can be calculatedusing the following formulas:Upper limit of 95% confidence interval: __ X [ t .05 ][ s __ X ]Lower limit of 95% confidence interval: __ X [ t .05 ][ s __ X ]