A test based on critical region \(W\) is said to be unbiased for testing \(H_0: \theta = \theta_0\) against \(H_1: \theta \ne \theta_0\) if (where \(\alpha\) and \(\beta\) are Type I and Type II errors respectively)
From a set of data involving four "tropical feed stuffs A, B, C and D", tried on 20 chicks, the following information was extracted:
\[ \begin{array}{|l|c|c|} \hline \textbf{Source of variation} & \textbf{Sum of squares} & \textbf{Degrees of freedom} \\ \hline \text{Treatment} & 26000 & 3 \\ \text{Error} & 11500 & 16 \\ \hline \end{array} \]
All the 20 chicks were treated alike, except for the feeding treatment, and each feeding treatment was given to 5 chicks. Then, the critical difference between any two means is:
It is given that there are six treatments and four blocks,
\[ \begin{array}{|l|cccccc|} \hline \textbf{Treatment totals} & T_1 & T_2 & T_3 & T_4 & T_5 & T_6 \\ & 63 & 65 & 57 & 64 & 65 & 66 \\ \hline \textbf{Block totals} & B_1 & B_2 & B_3 & B_4 & & \\ & 90 & 85 & 106 & 98 & & \\ \hline \end{array} \]
and that \( G = \sum_i \sum_j y_{ij} = 380 \), then the sum of squares due to treatment is:
For the given ANOVA table:
\[ \begin{array}{|l|c|c|} \hline \textbf{Source of variation} & \textbf{Sum of squares} & \textbf{Degrees of freedom} \\ \hline \text{Service station} & 6810 & 9 \\ \text{Rating} & 400 & 4 \\ \text{Total} & 9948 & 49 \\ \hline \end{array} \]
The test statistic to test that there is no significant difference between the service stations is: