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computing confidence interval

You compute that confidence interval similarly to the confidence interval for the difference of two means, but using the q distribution which avoids the problem of inflating α:

xbar sub i minus xbar sub j plus or minus q of alpha, r, df sub w times square root of 0.5 times MS sub w times quantity 1 over n sub i plus 1 over n sub j

where x̅i and x̅j are the two sample means, ni and nj are the two sample sizes, MSW is the within-groups mean square from the ANOVA table, and q is the critical value of the studentized range for α, the number of treatments or samples r, and the within-groups degrees of freedom dfW. The square-root term is called the standardized error (as opposed to standard error).

Using the studentized range, developed by Tukey, overcomes the problem of inflated significance level that I talked about earlier. If sample sizes are equal, the risk of a Type I error is exactly α, and if sample sizes are unequal it’s less than α: the procedure is conservative. In terms of confidence intervals, if the sample sizes are equal then the confidence level is the stated 1−α, but if the sample size are unequal then the actual confidence level is greater than 1−α (NIST 2012 section