Finding Critical Values Assume that we have two treatments (A and B) that produce quantitative results, and we have only two observations for treatment A and two observations for treatment B. We cannot use the Wilcoxon signed-ranks test given in this section because both sample sizes do not exceed 10.





a. Complete the accompanying table by listing the five rows corresponding to the other five possible outcomes, and enter the corresponding rank sums for treatment A.

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with completing the table by listing all possible outcomes for the ranks assigned to treatments A and B, and calculating the rank sums for treatment A. Each rank (1, 2, 3, 4) must be assigned to either treatment A or B, and we need to consider all possible combinations.
2. Step 2: Determine the total number of combinations. Since there are two observations for treatment A and two for treatment B, we are essentially dividing the four ranks into two groups. The number of ways to choose 2 ranks for treatment A out of 4 is given by the combination formula: \( \binom{4}{2} = 6 \). This means there are 6 possible outcomes.
3. Step 3: List all possible combinations of ranks for treatment A. For example, one combination could be ranks 1 and 2 assigned to treatment A, and ranks 3 and 4 assigned to treatment B. Another combination could be ranks 1 and 3 assigned to treatment A, and ranks 2 and 4 assigned to treatment B. Continue listing all combinations.
4. Step 4: Calculate the rank sum for treatment A for each combination. For each combination, add the ranks assigned to treatment A. For example, if ranks 1 and 2 are assigned to treatment A, the rank sum is \( 1 + 2 = 3 \). Repeat this calculation for all combinations.
5. Step 5: Fill in the table with the combinations and their corresponding rank sums for treatment A. Ensure that all 6 combinations are represented, and verify that the rank sums are correctly calculated.

Key Concepts

Rank Sums

Rank sums are calculated by assigning ranks to observations from two or more groups and summing the ranks for each group. In this context, treatment A has a rank sum of 3, which indicates the total rank assigned to its observations. This method is often used in non-parametric tests to compare groups when the assumptions of parametric tests are not met.

Non-parametric Tests

Non-parametric tests are statistical methods that do not assume a specific distribution for the data. They are particularly useful when sample sizes are small or when data do not meet the assumptions required for parametric tests, such as normality. The Wilcoxon signed-ranks test is a common non-parametric test, but in this case, it cannot be used due to the small sample size.

Combinatorial Outcomes

Combinatorial outcomes refer to the different possible arrangements or combinations of observations from two treatments. In this scenario, with two observations each from treatments A and B, there are a limited number of ways to rank these observations. Understanding these combinations is essential for completing the table and calculating the rank sums for treatment A across all possible outcomes.