Problem
In a large clinical trial, 397,278 children were randomly assigned to two groups. The treatment group consisted of 198,417 children given a vaccine for a certain disease, and 30 of those children developed the disease. The other 198,861 children were given a placebo, and 105 of those children developed the disease. Consider the vaccine treatment group to be the first sample. Identify the values of $n_{1}, \hat{p}_{1}, \hat{q}_{1}, n_{2}, \hat{p}_{2}, \hat{q}_{2}, \bar{p}$, and $\bar{q}$. \[ \begin{array}{l} n_{1}=198,417 \\ \hat{p}_{1}=0.00015120 \end{array} \] (Type an integer or a decimal rounded to eight decimal places as needed.) \[ \hat{q}_{1}=0.9998488 \] (Type an integer or a decimal rounded to eight decimal places as needed.) \[ \begin{array}{l} \mathrm{n}_{2}=198861 \\ \hat{p}_{2}=0.00052801 \end{array} \] (Type an integer or a decimal rounded to eight decimal places as needed.) \[ \hat{q}_{2}=\square \] (Type an integer or a decimal rounded to eight decimal places as needed.)
Solution
Step 1 ：Given that the proportion of children in the second group who developed the disease, denoted as \( \hat{p}_{2} \), is 0.00052801.
Step 2 ：The proportion of children in the second group who did not develop the disease, denoted as \( \hat{q}_{2} \), can be calculated as 1 - \( \hat{p}_{2} \).
Step 3 ：Substitute \( \hat{p}_{2} \) into the equation: \( \hat{q}_{2} = 1 - 0.00052801 \).
Step 4 ：Calculate the value to get \( \hat{q}_{2} = 0.99947199 \).
Step 5 ：So, the final answer is \( \hat{q}_{2} = \boxed{0.99947199} \).
