91影视

The study is conducted on 401974 children divided into two groups:

Treatment: Of 201229, 33 developed polio.

Placebo: Of 200,745, 115 developed polio.

The general notations are expressed as,

$n_1=$size of first sample

$n_2=$size of second sample

${\hat{p}}_1=$sample proportion of success in first sample

${\hat{q}}_1=$complement of sample successes in second sample.

${\hat{p}}_2=$sample proportion of success in second sample

${\hat{q}}_2=$complement of sample successes in first sample

$\stackrel{炉}{p}=$pooled sample proportion

$\stackrel{炉}{q}=1-\stackrel{炉}{p}$

Let the treatment group be defined as group 1 and Placebo as group 2.

Then,

$$\begin{gathered} {\mathbf{n}}_{\mathbf{1}}\mathbf{=}\mathbf{201229} \\ {\mathbf{x}}_{\mathbf{1}}\mathbf{=}\mathbf{33} \\ {\mathbf{n}}_{\mathbf{2}}\mathbf{=}\mathbf{200745} \\ {\mathbf{x}}_{\mathbf{2}}\mathbf{=}\mathbf{115} \end{gathered}$$

Sample proportions are calculated as,

$$\begin{gathered} {\hat{p}}_1=\frac{x_1}{n_1} \\ =\frac{33}{201229} \\ =0.000164 \end{gathered}$$

$$\begin{gathered} {\hat{q}}_1=1-{\hat{p}}_1 \\ =1-0.000164 \\ =0.999836 \end{gathered}$$

Similarly,

$$\begin{gathered} {\hat{p}}_2=\frac{x_2}{n_2} \\ =\frac{115}{200745} \\ =0.000573 \end{gathered}$$

$$\begin{gathered} {\hat{q}}_2=1-{\hat{p}}_2 \\ =1-0.000573 \\ =0.999427 \end{gathered}$$

Now, the pooled sample proportion can be calculated as,

$$\begin{gathered} \stackrel{炉}{p}=\frac{x_1+x_2}{n_1+n_2} \\ =\frac{33+115}{201229+200745} \\ =0.000368 \end{gathered}$$

Complement of pooled sample proportion can be calculated as,

$$\begin{gathered} \stackrel{炉}{q}=1-\stackrel{炉}{p} \\ =1-0.000368 \\ =0.999632 \end{gathered}$$