Suppose you have four groups of data, and you want to do...

Suppose you have four groups of data, and you want to do hypothesis tests ( $t$ -tests) to compare all possible pairs of means. a. How many pairwise comparisons can be done with four groups called A, B, C, and D? Show all possible pairs, starting with $\mathrm{AB}$. b. Using the Bonferroni Correction, which significance level should you use for each comparisons if you want an overall significance level of $0.05 ?$

Suppose you have four groups of data, and you want to do hypothesis tests ( $t$ -tests) to compare all possible pairs of means.
a. How many pairwise comparisons can be done with four groups called A, B, C, and D? Show all possible pairs, starting with $\mathrm{AB}$.
b. Using the Bonferroni Correction, which significance level should you use for each comparisons if you want an overall significance level of $0.05 ?$

Key Concepts

Pairwise Comparisons

In statistical analysis, pairwise comparisons involve comparing every possible pair of group means to determine if there are significant differences between them. This method is useful when researchers want to understand not just whether there is an overall difference among groups, but which specific groups differ from each other.

Combination Formula

The number of unique pairs that can be formed from a given number of groups is calculated using the combination formula n choose 2. This formula determines how many ways one can select 2 items from n without considering the order, which directly gives the number of pairwise comparisons.

Bonferroni Correction

The Bonferroni Correction is a technique used in multiple hypothesis testing to control the family-wise error rate. When multiple tests are performed, the chance of encountering at least one type I error increases, so the Bonferroni method divides the desired overall significance level by the number of tests to obtain a more stringent significance threshold for each individual comparison.

Family-wise Error Rate

The family-wise error rate is the probability of making one or more false rejections (type I errors) among a set of statistical tests. By controlling this error rate, as done by the Bonferroni Correction, researchers ensure that the chance of incorrectly declaring a significant result when there is none remains within acceptable bounds.

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