In an experiment on immunization of cattle from Tuberculosis the following result were obtained

Use $χ^2$ Test to determine the efficacy of vaccine in preventing tuberculosis

(i) The null hypothesis $H_0$: The efficacy of vaccine in preventing tuberculosis.

Alternative hypothesis $H_a$: The efficacy of vaccine is not in preventing tuberculosis.

(ii) Calculation of test statistic: Table of calculated frequencies $(using formula (A×B)N)$

$ \begin{array}{|c|c|c|c|} \hline & \text{Affected} & \text{Not Affected} & \text{Total} \\ \hline \text{Inoculated} & 250 & 44 & 294 \\ \hline \text{Not Inoculated} & 774 & 138 & 912 \\ \hline \text{Total} & 1024 & 182 & 1206 \\ \hline \end{array} $

Calculation of $\frac{(O – E)^2}{E}$ :

$ \begin{array}{|c|c|c|c|} \hline \text{O} & \text{E} & \text{(O - E)$^2$} & \text{$\frac{(O - E)^2}{E}$} \\ \hline 267 & 250 & 289 & 1.156 \\ \hline 27 & 44 & 289 & 6.568 \\ \hline 757 & 774 & 289 & 0.374 \\ \hline 155 & 138 & 289 & 2.094 \\ \hline & & \text{Total} & 10.192 \\ \hline \end{array} $

(iii) Level of significance : $α = 0.05$

(iv) Degrees of freedom = $(r - 1)(c - 1)$ = $(2 - 1)(2 - 1)$ = 1

(v) Critical value : The table value of $χ^2$ at 5% level of significance for 1 degree of freedom is 3.81

(vi) Decision : Since the calculated value of $χ^2$ = 10.192 is greater than the table value $χ^2$ = $3.81$, the null hypothesis is rejected.