1. **Problem 1: Wilcoxon Signed-Rank Test for Twin Attention Span** We want to test if the vitamin supplement (Twin B) leads to a longer attention span than the normal drink (Twin A) at significance level $\alpha=0.05$. 2. **Calculate differences (Twin B - Twin A):** $$\begin{array}{c|ccccccccc} \text{Pair} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \text{Twin A} & 22 & 17 & 30 & 21 & 22 & 23 & 13 & 19 & 25 \\ \text{Twin B} & 26 & 22 & 27 & 26 & 22 & 29 & 18 & 20 & 26 \\ \text{Difference} & 4 & 5 & -3 & 5 & 0 & 6 & 5 & 1 & 1 \end{array}$$ Note: Differences of zero (Pair 5) are excluded. 3. **Rank the absolute differences (ignoring zeros):** Absolute differences: 4,5,3,5,6,5,1,1 Sorted absolute differences: 1,1,3,4,5,5,5,6 Assign ranks (average ranks for ties): - 1 and 1: ranks 1 and 2, average rank = 1.5 each - 3: rank 3 - 4: rank 4 - 5,5,5: ranks 5,6,7 average rank = 6 each - 6: rank 8 4. **Assign ranks to differences with signs:** | Difference | Absolute | Rank | Signed Rank | |------------|----------|------|-------------| | 4 | 4 | 4 | +4 | | 5 | 5 | 6 | +6 | | -3 | 3 | 3 | -3 | | 5 | 5 | 6 | +6 | | 6 | 6 | 8 | +8 | | 5 | 5 | 6 | +6 | | 1 | 1 | 1.5 | +1.5 | | 1 | 1 | 1.5 | +1.5 | 5. **Calculate sums of positive and negative ranks:** - Positive ranks sum $W_+ = 4 + 6 + 6 + 8 + 6 + 1.5 + 1.5 = 33$ - Negative ranks sum $W_- = 3$ 6. **Test statistic $W$ is the smaller of $W_+$ and $W_-$:** $$W = 3$$ 7. **Critical value for $n=8$ (non-zero differences) at $\alpha=0.05$ (one-tailed) is 5.** Since $W=3 < 5$, we reject the null hypothesis. **Conclusion:** There is significant evidence at the 0.05 level that the vitamin supplement increases attention span. --- 8. **Problem 2: Wilcoxon Signed-Rank Test for Shampoo Ratings** Test if the darker-colour shampoo ratings are generally higher than original ratings at $\alpha=0.05$. 9. **Calculate differences (Darker - Original):** $$\begin{array}{c|cccccccc} \text{Customer} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \text{Original} & 5 & 7 & 5 & 5 & 2 & 4 & 8 & 8 \\ \text{Darker} & 5 & 4 & 3 & 8 & 4 & 6 & 9 & 10 \\ \text{Difference} & 0 & -3 & -2 & 3 & 2 & 2 & 1 & 2 \end{array}$$ Exclude zero difference (Customer 1). 10. **Rank absolute differences:** Absolute differences: 3,2,3,2,2,1,2 Sorted: 1,2,2,2,2,3,3 Ranks: - 1: rank 1 - 2,2,2,2: ranks 2,3,4,5 average rank = 3.5 each - 3,3: ranks 6,7 average rank = 6.5 each 11. **Assign signed ranks:** | Difference | Absolute | Rank | Signed Rank | |------------|----------|------|-------------| | -3 | 3 | 6.5 | -6.5 | | -2 | 2 | 3.5 | -3.5 | | 3 | 3 | 6.5 | +6.5 | | 2 | 2 | 3.5 | +3.5 | | 2 | 2 | 3.5 | +3.5 | | 1 | 1 | 1 | +1 | | 2 | 2 | 3.5 | +3.5 | 12. **Sum positive and negative ranks:** - $W_+ = 6.5 + 3.5 + 3.5 + 1 + 3.5 = 18$ - $W_- = 6.5 + 3.5 = 10$ 13. **Test statistic $W$ is the smaller sum:** $$W = 10$$ 14. **Critical value for $n=7$ at $\alpha=0.05$ (one-tailed) is 2.** Since $W=10 > 2$, we fail to reject the null hypothesis. **Conclusion:** There is insufficient evidence at the 0.05 level to conclude that the darker shampoo is rated higher.