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The Wilcoxon matched-pairs signed-ranks test is a non-parametric statistical test. It is used to compare two related samples, matched samples, or repeated measurements on a single sample. This test is often referred to simply as the Wilcoxon test. It evaluates whether the median difference between pairs of observations is zero.
It considers the magnitude and direction of differences between matched pairs. In our example, we have dependent samples and we are testing at a significance level of \( \alpha = 0.05 \).
Remember, the Wilcoxon signed-ranks test is similar to the paired t-test, but it does not assume a normal distribution of differences. Therefore, it is a good alternative when the normality assumption is in question or when the sample size is small.

Non-parametric tests are a set of statistical tests that do not assume a specific distribution for the data. They are particularly useful when we cannot make assumptions about the population from which our sample is drawn.
Some key points about non-parametric tests:

• They do not assume the data follow a normal distribution.
• They are less powerful than parametric tests when the assumptions of the parametric tests are met.
• Useful for small sample sizes and ordinal data.
• Examples include the Wilcoxon signed-rank test, Mann-Whitney U test, and Kruskal-Wallis test.

In our context, the Wilcoxon signed-ranks test is a non-parametric test that helps us to compare the medians of two related samples without making assumptions about their distribution.

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves making a claim or hypothesis and then using sample data to test whether this claim is likely to be true.
Steps in Hypothesis Testing include:
1. **Formulate Hypotheses**: The null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_1\)). In our exercise, \( H_{0}: M_{D}=0 \) and \( H_{1}: M_{D} eq 0 \)
2. **Select Significance Level**: Commonly used levels are 0.05 or 0.01. Our exercise uses \( \alpha = 0.05 \). This represents a 5% risk of concluding that a difference exists when there is no actual difference.
3. **Calculate Test Statistic**: Derived from the sample data. For the Wilcoxon test, the test statistic is the smaller of the sums of the signed ranks.
4. **Determine Critical Value**: From statistical tables or distributions corresponding to the test used.
5. **Compare Statistic and Critical Value**: Decide whether to reject the null hypothesis based on this comparison.
If the test statistic is greater than the critical value, as in our example, we fail to reject the null hypothesis, meaning there is not enough evidence to support a difference. This methodical approach helps ensure our conclusions are based on data and not random chance.