91影视

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,

\(\begin{array}{l}k = \frac{{12}}{{N\left( {N + 1} \right)}}\sum\limits_{i = 1}^I {{J_i}} {\left( {{{\bar R}_i} - \frac{{N + 1}}{2}} \right)^2}\\ = \frac{{12}}{{N\left( {N + 1} \right)}}\sum\limits_{i = 1}^I {\frac{{R_i^2}}{{{J_i}}}} - 3\left( {N + 1} \right)\\ = \frac{{12}}{{20.\left( {20 + 1} \right)}}\left( {\frac{{{{31}^2} + {{68}^2} + {{26}^2} + {{85}^2}}}{5}} \right) - 3 \cdot 21\\ = 14.06\end{array}\)

Degrees of freedom are

\({d_f} = I - 1 = 4 - 1 = 3\)

Critical value at significant level 0.1 is

\(\begin{array}{l}X_{0,1,3}^2 = 6.25\\X_{0,1,3}^2 = 6.25 < 14.06 = K\end{array}\)

Reject null hypothesis

Hence, reject null hypothesis.