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

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

Affected Not Affected Total
Inoculated 267 27 294
Not Inoculated 757 155 912
Total 1024 182 1206

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

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(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.