Tukey's method is used in ANOVA to create confidence intervals for all pairwise differences between factor level means while controlling the family error rate to a level you specify. It is important to consider the family error rate when making multiple comparisons because your chances of making a type I error for a series of comparisons is greater than the error rate for any one comparison alone. To counter this higher error rate, Tukey's method adjusts the confidence level for each individual interval so that the resulting simultaneous confidence level is equal to the value you specify.
You are measuring the response times for memory chips. You sampled 25 chips from five different manufacturers.
You decide to examine all 10 comparisons between the five plants to determine specifically which means are different. Using Tukey's method, you specify that the entire set of comparisons should have a family error rate of 0.05 (equivalent to a 95% simultaneous confidence level). Minitab calculates that the 10 individual confidence levels need to be 99.35% in order to obtain the 95% joint confidence level. These wider Tukey confidence intervals provide less precise estimates of the population parameter but limit the probability that one or more of the confidence intervals does not contain the true difference to a maximum of 5%. Understanding this context, you can then examine the confidence intervals to determine whether any do not include zero, indicating a significant difference.
Comparison of 95% confidence intervals to the wider 99.35% confidence intervals used by Tukey's in the previous example. The reference line at 0 shows how the wider Tukey confidence intervals can change your conclusions. Confidence intervals that contain zero indicate no difference. (Only 5 of the 10 comparisons are shown due to space considerations.)