In a study to investigate the effects of regular exercise on raising high-density lipoprotein (HDL) cholesterol levels ("good cholesterol") , a random sample of five male subjects known to have low HDL levels had their HDL measured at the beginning of the study and then again after six months on a regular exercise schedule. The data are given in the following table: $\begin{array} { | l | l | l | l | l | l | } \hline \text { Subject } & 1 & 2 & 3 & 4 & 5 \\\hline \text { Initial HDL } & 31 & 36 & 32 & 28 & 37 \\\hline \text { HDL after 6 months } & 39 & 32 & 44 & 34 & 39 \\\hline \text { Difference } & 8 & - 4 & 12 & 6 & 2 \\\hline\end{array}$
The data are to be analyzed using the Wilcoxon signed rank test. If the responses have a continuous distribution that is not affected by the different treatments, what is the mean of W⁺?

Question

In a study to investigate the effects of regular exercise on raising high-density lipoprotein (HDL) cholesterol levels ("good cholesterol") , a random sample of five male subjects known to have low HDL levels had their HDL measured at the beginning of the study and then again after six months on a regular exercise schedule. The data are given in the following table:

\begin{array} { | l | l | l | l | l | l | } \hline \text { Subject } & 1 & 2 & 3 & 4 & 5 \\\hline \text { Initial HDL } & 31 & 36 & 32 & 28 & 37 \\\hline \text { HDL after 6 months } & 39 & 32 & 44 & 34 & 39 \\\hline \text { Difference } & 8 & - 4 & 12 & 6 & 2 \\\hline\end{array}

The data are to be analyzed using the Wilcoxon signed rank test. If the responses have a continuous distribution that is not affected by the different treatments, what is the mean of W⁺?

Quizplus · Accepted Answer

First, we need to calculate the signed ranks using the formula $W_i = \text{sign}(x_i-y_i)\cdot R_i$, where $x_i$ is the initial HDL level, $y_i$ is the HDL level after 6 months, and $R_i$ is the rank of the absolute value of the difference $|x_i-y_i|$. The signed ranks are: $$\begin{array} { | l | l | l | l | l | l | } \hline \text { Subject } & 1 & 2 & 3 & 4 & 5 \ \hline \text { Difference } & 8 & - 4 & 12 & 6 & 2 \ \hline \text { Rank } & 4 & 1.5 & 5 & 3 & 2 \ \hline \text { Sign (-/+) } & + & - & + & + & + \ \hline \text { Signed Rank } & 4 & -1.5 & 5 & 3 & 2 \ \hline \end{array}$$ The next step is to sum the absolute values of the signed ranks for $\text{WS}_1$ and $\text{WS}_2$. Since there are no ties, we can use the formula $\text{WS} = \text{min}(\sum |W_i^+|, \sum |W_i^-|)$, where $W_i^+$ are the positive signed ranks and $W_i^-$ are the negative signed ranks. In this case, we have $\text{WS}_1 = 15.5$ and $\text{WS}_2 = 1.5$, so we take $\text{WS} = \text{WS}_2 = 1.5$. The next step is to find the corresponding critical value from the Wilcoxon signed rank table for $n=5$ at the $\alpha=0.05$ significance level. The table shows that the critical value is 1. Since $\text{WS} < 1$, we cannot reject the null hypothesis that there is no difference between the initial HDL levels and the HDL levels after 6 months on a regular exercise schedule. Therefore, the mean of W⁺ is not relevant for this problem, and the answer is only (B) 7.5.

In a Study to Investigate the Effects of Regular Exercise