Question

Give the rejection region for a test to detect rank correlation if the number of pairs of ranks is 25 and you have these \(\alpha\) -values: a. \(\alpha=.05\) b. \(\alpha=.01\)

Short answer

The rejection regions are as follows: a. For \(\alpha = 0.05\), the rejection region is when the absolute value of the test statistic (Spearman Rank Correlation Coefficient) is greater than 0.396: \(|r_s| > 0.396\) b. For \(\alpha = 0.01\), the rejection region is when the absolute value of the test statistic is greater than 0.519: \(|r_s| > 0.519\) These rejection regions apply to a two-tailed test, where we are looking for either a strong positive or strong negative correlation, and reject the null hypothesis if the correlation coefficient is too large, either positively or negatively, compared to the critical values.

Step-by-step solution

Identify the number of pairs of ranks and the corresponding degree of freedom

The problem states that the number of pairs of ranks is 25. The degree of freedom (df) can be calculated as the number of pairs minus 2. So, for our problem, we have: df = 25 - 2 = 23

Find the critical values for the given \(\alpha\)-values

We need to find the critical values for the given \(\alpha\)-values using the Spearman Rank Correlation Coefficient table. Let's denote the critical values as \(r_{\alpha}\). a. For \(\alpha = 0.05\), the table gives us a critical value at the degree of freedom of 23: \(r_{0.05} = 0.396\) b. For \(\alpha = 0.01\), the table gives us a critical value at the degree of freedom of 23: \(r_{0.01} = 0.519\)

Determine the rejection region

The rejection region is the set of values for which we would reject the null hypothesis (that there is no correlation), and we find this by comparing the actual test statistic (Spearman Rank Correlation Coefficient) to the critical values. a. For \(\alpha = 0.05\), the rejection region is when the absolute value of the test statistic is greater than \(r_{0.05}\): \(|r_s| > 0.396\) b. For \(\alpha = 0.01\), the rejection region is when the absolute value of the test statistic is greater than \(r_{0.01}\): \(|r_s| > 0.519\) These rejection regions are for the two-tailed test, where we are looking for either a strong positive or strong negative correlation, and reject the null hypothesis if the correlation coefficient is too large, either positively or negatively, compared to the critical values.