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When performing a sign test, one of the primary steps is to define the null hypothesis, which provides a starting point for statistical analysis. The null hypothesis (\( H_0 \) ) is a statement of no effect or no difference. It is essentially a skeptical position, which assumes that any observed differences in the data are due to chance rather than a real effect.

For instance, in the given exercise, the null hypothesis is that there is no significant difference between the medians of the two populations being compared, mathematically stated as \( H_0: M_1 - M_2 = 0 \). This hypothesis is what we presuppose to be true before collecting any data, and our test aims to determine whether the observed data provide enough evidence to reject this assumption.

The alternative hypothesis (\( H_1 \) or \( H_a \)) provides contrast to the null hypothesis. It suggests that there is an effect or a difference, and in the context of the sign test, it posits that the medians of the two populations are not equal. This hypothesis reflects the research question or the suspicion that led to the investigation.

In our example, the alternative hypothesis is defined as \( H_1: M_1 - M_2 eq 0 \), which means we suspect that there is a significant difference between the population medians. The alternative hypothesis is what we seek to support by showing that the null hypothesis is unlikely given the data.

A test statistic is a calculated value used to determine whether to reject the null hypothesis. It is derived from sample data and measures the degree of agreement between the null hypothesis and the observed data. For the sign test, the test statistic (\( T \) ) is determined by the smaller number of positive or negative differences in paired observations.

In the exercise provided, it was necessary to calculate the differences for each pair and then count the number of positive and negative differences. The test statistic was the minimum of these counts (\( T = \min(6, 1) = 1 \)). This value is then compared to a critical value to help decide if there is enough evidence to reject the null hypothesis.

The rejection region defines the values of the test statistic that would lead to rejection of the null hypothesis. It is determined based on the significance level of the test, denoted by \( \alpha \). A common significance level used in hypothesis testing is 0.05, but in the exercise, the significance level was set as \( \alpha \leq 0.10 \).

By consulting statistical tables or software with a given \( \alpha \) value and the number of pairs (\( n \) ), we find the critical value (\( T_\alpha \) ) that delineates the rejection region. In this case, the rejection region is \( T \leq T_{0.95} \). Since the observed test statistic (\( T = 1 \) ) is greater than the critical value (\( T_{0.95} = 0 \)), it falls outside the rejection region, meaning we do not reject the null hypothesis.