Question

Give the rejection region for a test to detect negative rank correlation if the number of pairs of ranks is 12 and you have these \(\alpha\) -values: a. \(\alpha=.05\) b. \(\alpha=.01\)

Short answer

Answer: For a significance level of 0.05, the rejection region for negative rank correlation is when the calculated Spearman's rank correlation coefficient (rs) is less than -0.576. And for a significance level of 0.01, the rejection region for negative rank correlation is when the calculated rs is less than -0.760.

Step-by-step solution

Find the critical values

To find the critical values for the given significance levels (\(\alpha=0.05\) and \(\alpha=0.01\)) and sample size (number of pairs of ranks, n = 12), you should consult a table of critical values for Spearman's rank correlation coefficient (\(r_s\)). This table usually lists the critical values for a two-tailed test, so you will need to find the one-tailed critical values by halving the significance levels: a. For \(\alpha=0.05\), the one-tailed significance level is \(\frac{0.05}{2} = 0.025\). b. For \(\alpha=0.01\), the one-tailed significance level is \(\frac{0.01}{2} = 0.005\). Now look up the critical values in the \(r_s\) table using the one-tailed significance levels and the number of pairs of ranks (12). You will find the following critical values: a. For \(\alpha=0.05\) (0.025 in one-tailed test), the critical value is approximately \(\pm0.576\) b. For \(\alpha=0.01\) (0.005 in one-tailed test), the critical value is approximately \(\pm0.760\)

Determine the rejection region for negative rank correlation

Now that you have the critical values, you can determine the rejection regions for negative rank correlation: a. For \(\alpha=0.05\), the rejection region for negative rank correlation (given by \(r_s<0\)) is \(r_s<-0.576\). If the calculated \(r_s\) is less than -0.576, you can reject the null hypothesis in favor of the alternative hypothesis (negative rank correlation) at the 0.05 significance level. b. For \(\alpha=0.01\), the rejection region for negative rank correlation (given by \(r_s<0\)) is \(r_s<-0.760\). If the calculated \(r_s\) is less than -0.760, you can reject the null hypothesis in favor of the alternative hypothesis (negative rank correlation) at the 0.01 significance level.