Answer:
Step-by-step explanation:
From the given information:
The number of children that were randomly allocated to each vaccination group; n₁ = 200,000
No of polio cases X₁ = 57
Now, in the vaccine group:
the proportion of polio cases is:
![\hat p_1 = \dfrac{57}{200000}](https://tex.z-dn.net/?f=%5Chat%20p_1%20%3D%20%5Cdfrac%7B57%7D%7B200000%7D)
= 0.000285
The number of children that were randomly allocated to the placebo group, n₂ = 200,000
No of polio cases X₂ = 142
In the placebo group
the proportion of polio cases is:
![\hat p_2 = \dfrac{142}{200000}](https://tex.z-dn.net/?f=%5Chat%20p_2%20%3D%20%5Cdfrac%7B142%7D%7B200000%7D)
Null and alternative hypothesis is computed as follows:
H₀: There is no difference in the proportions of polio cases between both groups.
H₁: There is a difference in the proportions of polio cases between both groups.
Let assume that the level of significance ∝ = 0.05
The test statistic can be computed as:
![Z = \dfrac{\hat p_1-\hat p_2}{\sqrt{\dfrac{\hat p_1 \hat q_1}{n_1}+ \dfrac{\hat p_2 \hat q_2}{n_2}}}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cdfrac%7B%5Chat%20p_1-%5Chat%20p_2%7D%7B%5Csqrt%7B%5Cdfrac%7B%5Chat%20p_1%20%5Chat%20q_1%7D%7Bn_1%7D%2B%20%5Cdfrac%7B%5Chat%20p_2%20%5Chat%20q_2%7D%7Bn_2%7D%7D%7D)
![Z = \dfrac{0.000285-0.000710}{\sqrt{\dfrac{0.000285(1-0.000285)}{200000}+ \dfrac{0.000710(1-0.000710)}{200000}}}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cdfrac%7B0.000285-0.000710%7D%7B%5Csqrt%7B%5Cdfrac%7B0.000285%281-0.000285%29%7D%7B200000%7D%2B%20%5Cdfrac%7B0.000710%281-0.000710%29%7D%7B200000%7D%7D%7D)
![Z = \dfrac{-4.25\times 10^{-4}}{\sqrt{\dfrac{0.000285(0.999715)}{200000}+ \dfrac{0.000710(0.99929)}{200000}}}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cdfrac%7B-4.25%5Ctimes%2010%5E%7B-4%7D%7D%7B%5Csqrt%7B%5Cdfrac%7B0.000285%280.999715%29%7D%7B200000%7D%2B%20%5Cdfrac%7B0.000710%280.99929%29%7D%7B200000%7D%7D%7D)
Z = - 6.03
P-value = 2P(Z < -6.03)
From the Z - tables
P-value = 2 × 0.0000
= 0.000
We reject the H₀ provided that P-value is very less.
Therefore, we may conclude that there is a difference in the proportions of polio cases between the vaccine group and placebo group not due to chance.