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The kruskal- wallis test

Test for testing equality of the \[{\mu _i}'s\]. The test static is

\[\begin{array}{l}K = \frac{{12}}{{N\left( {N + 1} \right)}}\sum\limits_{i = 1}^I {{J_i}} {\left( {{{\bar R}_i} - \frac{{N + 1}}{2}} \right)^2}\\ = \frac{{12}}{{N\left( {N + 1} \right)}}\sum\limits_{i = 1}^I {\frac{{R_i^2}}{{{J_i}}} - 3} \left( {N + 1} \right)\end{array}\]

When the null hypothesis is true, and either

\[\begin{array}{l}I = 3,{J_i} \ge 6\left( {i = 1,2,3} \right)\\or,\\I > 3,{J_i} \ge 5\left( {i = 1,2,I} \right)\end{array}\]

Test statistic K has chi-squared distribution with I-1 degrees of freedom. The P-value is corresponding are to the right of k under the \[X_{I - 1}^2\]curve

Two table goes together- point\[{x_{ij}}\]in the first table corresponds to the\[{r_{ij}}\]rank in the second table.

The test statistic value is,

\[\begin{array}{l}k = \frac{{12}}{{N\left( {N + 1} \right)}}\sum\limits_{i = 1}^I {{J_i}} {\left( {{{\bar R}_i} - \frac{{N + 1}}{2}} \right)^2}\\ = \frac{{12}}{{N\left( {N + 1} \right)}}\sum\limits_{i = 1}^I {\frac{{R_i^2}}{{{J_i}}}} - 3\left( {N + 1} \right)\\ = \frac{{12}}{{36.\left( {36 + 1} \right)}}\left( {\frac{{{{104}^2} + {{160}^2} + {{176}^2} + {{226}^2}}}{9}} \right) - 3 \cdot 37\\ = 7.587\end{array}\]

Degrees of freedom are

\[{d_f} = I - 1 = 4 - 1 = 3\]

Critical value at significant level 0.5 is

\[\begin{array}{l}X_{0,0.5,3}^2 = 7.815\\X_{0,0.5,3}^2 = 7.815 > 7.587 = K\end{array}\]

Do not reject null hypothesis

Hence, do not reject null hypothesis.