Question

Notation for the sample data given in exercise 1, consider the salk vaccine treatment group to be the first sample. Identify the values of ${\mathbf{n}}_{\mathbf{1}}\mathbf{,}{\hat{\mathbf{p}}}_{\mathbf{1}}\mathbf{,}{\hat{\mathbf{q}}}_{\mathbf{1}}\mathbf{,}{\mathbf{n}}_{\mathbf{2}}\mathbf{,}{\hat{\mathbf{p}}}_{\mathbf{2}}\mathbf{,}{\hat{\mathbf{q}}}_{\mathbf{2}}\mathbf{,}\overline{\mathbf{p}}$and $\overline{\mathbf{q}}$. Round all values so that they have six significant digits.

Short answer

The values of notations are as follows:

$$\begin{gathered} n_1=201,229 \\ {\hat{p}}_1=0.000164 \\ {\hat{q}}_1=0.999836 \\ {n}_2=200745 \end{gathered}$$

$$\begin{gathered} {\hat{p}}_2=0.000573 \\ {\hat{q}}_2=0.999427 \\ \overline{p}=0.000368 \\ \overline{q}=0.999632 \end{gathered}$$

Step-by-step solution

Step-1: Given information

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

Treatment: Of 201229, 33 developed polio.

Placebo: Of 200,745, 115 developed polio.

Step-2: Interpretation of notations in two proportions test

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

$\overline{p}=$pooled sample proportion

$\overline{q}=1-\overline{p}$

Step-3: Identify the values from the given information

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

Then,

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

Step-4:  Compute measure of sample proportions 

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,

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

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

Find sample pooled proportion 

Now, the pooled sample proportion can be calculated as,

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

Complement of pooled sample proportion can be calculated as,

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