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CHAPTER 11: Data Analysis and Interpretation: Part I. Describing Data, Confidence Intervals, Correlation 369When sample sizes are equal, the estimated standard error is then defined asX _____s pooled ns__where n sample size for each groupDegrees of freedom are then calculated as k(n 1), where k is equal to the numberof independent groups.Looking again at Figure 11.3, we can see that for 9-month-olds, the mean numberof investigative behaviors for the sample was 4.75. The expression [ t .05 ][s __ X ] inthe equation for the 95% confidence interval in this analysis is 1.14. We can be95% confident that the interval between 3.61 and 5.89 (4.75 1.14) contains thepopulation mean for 9-month-olds. Thus, the sample of 16 nine-month-old infantsin this study is used to estimate the average number of investigative behaviorsthat would be demonstrated if the larger population of 9-month-olds were testedin this situation. For 15-month-olds, the mean number of investigative behaviorswas 1.63, and we can be 95% confident that the interval between .49 and 2.77(1.63 1.14) contains the population mean. The sample mean for 19-month-oldswas .69, and the 95% confidence interval has a lower bound of 0.0 (restricted bythe range of permissible values) and an upper bound of 1.83 (.69 1.14).Box 11.5 provides information about how to interpret confidence intervalswhen there are three or more means.BOX 11.5INTERPRETING CONFIDENCE INTERVALS WHEN THERE ARE THREEOR MORE MEANS: DO INTERVALS OVERLAP?In many research situations, we are not reallyinterested in estimating the specific value of thepopulation mean. For example, we aren’t reallyinterested in knowing the average number oftimes 9-month-olds can be expected to grasp atpictures. Instead, we are interested in the patternof population means and comparing the relationshipsamong population means (Loftus & Masson,1994). That is, we wish to be able to compare thebehavior of different groups. This, too, can be accomplishedusing confidence intervals. Consideronce again the data from DeLoache et al.’s study.We can use our estimates of the populationmeans to ask: Do infants in the different agegroups demonstrate different amounts of investigativebehaviors? To answer this question wecan examine the overlap of the 95% confidenceintervals in Figure 11.3. Remember, the confidenceinterval is associated with a probability(e.g., .95) that the interval contains the populationmean; the width of the interval tells us how preciseis our estimate. We want to keep in mindthat confidence intervals are intended to provideinformation about how well we have estimateda population value, usually a mean. Confidenceintervals are not statistical tests like the t-testor F-test, where the emphasis is on comparingdirectly two or more means to see if the differencesare “statistically significant.” Nevertheless,as we stated previously, researchers often are interestedin the pattern of population means, andwe can use confidence intervals to help us detectthese patterns.When the intervals do not overlap, we canbe confident that the population means differ.Nonoverlapping intervals tell us that the populationmeans estimated by the sample means areprobably not the same. For example, the 95% confidenceinterval for 9-month-olds does not overlapwith the interval for 15-month-olds. From this(continued)