Question

Use the Wilcoxon Rank Sum Test on the following data to determine whether the location of population 1 is to the left of the location of population 2 . (Use \(5 \%\) significance level.)

$$ \begin{array}{llllll} \text { Sample 1: } & 75 & 60 & 73 & 66 & 81 \\ \text { Sample 2: } & 90 & 72 & 103 & 82 & 78 \end{array} $$

Answer 1

We want to test the following hypotheses, using the given data.

\(H_{0}:\) The two population locations are the same. \(H_{1}:\) The location of population 1 is to the left of the location of population \(2 .\)

Set level of significance \(\alpha=0.05\).

Two samples of 5 observations each were given. The test Statistic is the rank sum of the observations corresponding to sample 1 . So first rank all the given 10 observations, with rank 1 assigned to the smallest observation and rank 10 to the largest. In case of ties, we average the ranks of the tied observations.

$$ \begin{array}{|l|l|l|l|} \hline \text { Sample 1 } & \text { Rank } & \text { Sample 2 } & \text { Rank } \\ \hline 75 & 5 & 90 & 9 \\ \hline 60 & 1 & 72 & 3 \\ \hline 73 & 4 & 103 & 10 \\ \hline 66 & 2 & 82 & 8 \\ \hline 81 & 7 & 78 & 6 \\ \hline & T_{1}=19 & & T_{1}=36 \\ \hline \end{array} $$

Therefore the required value of the test statistic is, \(T=T_{1}=19\).

The critical values \(T_{L}\) and \(T_{U}\) for sample sizes \(n_{1}=5\) and \(n_{2}=5\) can be obtained from the tables as 19 and 37 . Since the test is left tailed the critical region for this test is given by:

\(P\left(T \leq T_{L}\right)=0.05\)

\(\Rightarrow P(T \leq 19)=0.05\)

Since the value of the test statistic is 19 , it falls in the critical region of this test.

INTERPRET: At the significance level \(5 \%\), we have to reject the null hypothesis. Therefore there is enough evidence to infer that the location of population 1 is to the left of the location of population \(2 .\)