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368 PART V: Analyzing and Reporting ResearchFIGURE 11.3Mean number of investigative behaviors with 95% confidence intervals for 9-month-olds,15-month-olds, and 19-month-olds. (From DeLoache et al., 1998; used with permission.)6Mean Number of Investigative Behaviors95% Confidence IntervalsMean Number of Behaviors5432109 Months 15 Months 19 MonthsAgeestimate of the population value. Each of the bars in Figure 11.3 represents a 95%confidence interval. However, the calculation of this interval in a multigroupstudy differs slightly from that when only one mean is present. Specifically, whencalculating the estimated standard error of the mean, we may make use of thepooled variance from all the groups in the study. Let us illustrate.The formula for the 95% confidence interval is the same as it was when therewas only one mean:Upper limit of 95% confidence interval: __ X [ t 0.5 ][ s __ X ]Lower limit of 95% confidence interval: __ X [ t 0.5 ][ s __ X ]However, the calculation of s __ X differs from that with one mean; so, too, does thecalculation of the degrees of freedom for the critical value of t. To estimate thestandard error of the mean, we may pool the variances from the various groupsto obtain one measure of variability. In this case we pool the information fromas many groups as we have in the study. When the comparison involves two ormore means from independent groups, the estimated standard error of the meanis calculated as follows. First, we find the standard deviation based on the pooledvariance: 12 2 2( ns pooled ___________________________________1 1) s 1 ( n 2 1) s 2 ( n 3 1) s 3 . . .____________________________________(n 1 1) ( n 2 1) ( n 3 1) + . . .1The pooled estimate of the population standard deviation is equivalent to the square root ________ of themean square error in a between-groups analysis of variance (ANOVA). That is, s pooled MSerror .See Chapter 12 for discussion of ANOVA.