Questions: In a large clinical trial, 397,892 children were randomly assigned to two groups. The treatment group consisted of 198, 235 children given a vaccine for a certain disease, and 29 of those children developed the disease The other 199.657 children were given a placebo, and 109 of those children developed the disease. Consider the vaccine treatment group to be the first sample. Identify the values of n1, p̂1, q̂1, n2, p̂2, q̂2, p⃗, and q⃗.

Solution Steps

To identify the values of \( n_{1}, \hat{p}_{1}, \hat{q}_{1}, n_{2}, \hat{p}_{2}, \hat{q}_{2}, \vec{p} \), and \(\vec{q}\), we need to follow these steps:

1. \( n_{1} \) is the number of children in the treatment group.
2. \( \hat{p}_{1} \) is the proportion of children in the treatment group who developed the disease.
3. \( \hat{q}_{1} \) is the proportion of children in the treatment group who did not develop the disease.
4. \( n_{2} \) is the number of children in the placebo group.
5. \( \hat{p}_{2} \) is the proportion of children in the placebo group who developed the disease.
6. \( \hat{q}_{2} \) is the proportion of children in the placebo group who did not develop the disease.
7. \( \vec{p} \) is the combined proportion of children who developed the disease in both groups.
8. \( \vec{q} \) is the combined proportion of children who did not develop the disease in both groups.

The number of children in the treatment group is: \[ n_{1} = 198235 \]

The number of children in the placebo group is: \[ n_{2} = 199657 \]

The proportion of children in the treatment group who developed the disease is: \[ \hat{p}_{1} = \frac{29}{198235} \approx 0.0001463 \]

The proportion of children in the placebo group who developed the disease is: \[ \hat{p}_{2} = \frac{109}{199657} \approx 0.0005459 \]

The proportion of children in the treatment group who did not develop the disease is: \[ \hat{q}_{1} = 1 - \hat{p}_{1} \approx 0.9998537 \]

The proportion of children in the placebo group who did not develop the disease is: \[ \hat{q}_{2} = 1 - \hat{p}_{2} \approx 0.9994541 \]

The total number of children in both groups is: \[ n_{1} + n_{2} = 198235 + 199657 = 397892 \]

The total number of children who developed the disease in both groups is: \[ 29 + 109 = 138 \]

The combined proportion of children who developed the disease is: \[ \vec{p} = \frac{138}{397892} \approx 0.0003468 \]

The combined proportion of children who did not develop the disease is: \[ \vec{q} = 1 - \vec{p} \approx 0.9996532 \]

Final Answer