91影视

The Friedman鈥檚 test for rendomized block experiment

Test for teststatistic is

\(\begin{array}{l}{F_r} = \frac{{12J}}{{I\left( {I + 1} \right)}}\sum\limits_{i = 1}^I {{{\left( {{{\bar R}_i} - \frac{{I + 1}}{2}} \right)}^2}} \\ = \frac{{12J}}{{IJ\left( {I + 1} \right)}}\sum\limits_{i = 1}^I {\bar R_i^2 - 3J} \left( {I + 1} \right)\end{array}\)

When the null hypothesis is true test statistic \({F_r}\), has approximately chi-squared distribution with I-1 degrees of freedom.

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,

\(\begin{array}{l}{f_r} = \frac{{12J}}{{I\left( {I + 1} \right)}}{\sum\limits_{i = 1}^I {\left( {{{\bar R}_i} - \frac{{I - 1}}{2}} \right)} ^2}\\ = \frac{{12}}{{IJ\left( {I + 1} \right)}}\sum\limits_{i = 1}^I {\bar R_i^2} - 3J\left( {I + 1} \right)\\ = \frac{{12}}{{4 \cdot 10 \cdot 5}} \cdot \left( {400 + 100 + 1521 + 961} \right) - 3 \cdot 10 \cdot 5\\ = 28.91\end{array}\)

Degrees of freedom are

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

Critical value at significant level 0.01 is

\(\begin{array}{l}X_{0.01,3}^2 = 11.344\\X_{0.01,3}^2 = 11.344 < 28.91 = {f_r}\end{array}\)

Reject null hypothesis

Hence, Reject null hypothesis.