In a large clinical​ trial, 392,762 children were randomly assigned to two groups. The treatment group consisted of 197,982 children given a vaccine for a certain​ disease, and 31 of those children developed the disease. The other 194,780 children were given a​ placebo, and 104 of those children developed the disease. Consider the vaccine treatment group to be the first sample. Identify the values of n 1n1​, ModifyingAbove p with caret 1p1​, ModifyingAbove q with caret 1q1​, n 2n2​, ModifyingAbove p with caret 2p2​, ModifyingAbove q with caret 2q2​, p overbarp​, and q overbarq.

To analyze the given clinical trial data, let's identify the values of \( n_1 \), \( \hat{p}_1 \), \( \hat{q}_1 \), \( n_2 \), \( \hat{p}_2 \), \( \hat{q}_2 \), \( \bar{p} \), and \( \bar{q} \).

1. **Sample sizes**:
   - \( n_1 \): The number of children in the treatment group.
   - \( n_2 \): The number of children in the placebo group.

From the problem, we have:
\[ n_1 = 197,982 \]
\[ n_2 = 194,780 \]

2. **Proportion of children who developed the disease**:
   - \( \hat{p}_1 \): The proportion of children in the treatment group who developed the disease.
   - \( \hat{q}_1 \): The proportion of children in the treatment group who did not develop the disease.
   - \( \hat{p}_2 \): The proportion of children in the placebo group who developed the disease.
   - \( \hat{q}_2 \): The proportion of children in the placebo group who did not develop the disease.

We calculate these as follows:
\[ \hat{p}_1 = \frac{\text{Number of children in the treatment group who developed the disease}}{\text{Total number of children in the treatment group}} = \frac{31}{197,982} \]
\[ \hat{q}_1 = 1 - \hat{p}_1 = 1 - \frac{31}{197,982} \]

\[ \hat{p}_2 = \frac{\text{Number of children in the placebo group who developed the disease}}{\text{Total number of children in the placebo group}} = \frac{104}{194,780} \]
\[ \hat{q}_2 = 1 - \hat{p}_2 = 1 - \frac{104}{194,780} \]

3. **Combined proportions**:
   - \( \bar{p} \): The combined proportion of children who developed the disease in both groups.
   - \( \bar{q} \): The combined proportion of children who did not develop the disease in both groups.

We calculate these as follows:
\[ \bar{p} = \frac{\text{Total number of children who developed the disease in both groups}}{\text{Total number of children in both groups}} = \frac{31 + 104}{197,982 + 194,780} \]
\[ \bar{q} = 1 - \bar{p} = 1 - \frac{31 + 104}{197,982 + 194,780} \]

Let's compute these values step-by-step.

1. Calculate \( \hat{p}_1 \) and \( \hat{q}_1 \):
\[ \hat{p}_1 = \frac{31}{197,982} \approx 0.0001566 \]
\[ \hat{q}_1 = 1 - 0.0001566 \approx 0.9998434 \]

2. Calculate \( \hat{p}_2 \) and \( \hat{q}_2 \):
\[ \hat{p}_2 = \frac{104}{194,780} \approx 0.0005341 \]
\[ \hat{q}_2 = 1 - 0.0005341 \approx 0.9994659 \]

3. Calculate \( \bar{p} \) and \( \bar{q} \):
\[ \bar{p} = \frac{31 + 104}{197,982 + 194,780} = \frac{135}{392,762} \approx 0.0003438 \]
\[ \bar{q} = 1 - 0.0003438 \approx 0.9996562 \]

So, the identified values are:
- \( n_1 = 197,982 \)
- \( \hat{p}_1 \approx 0.0001566 \)
- \( \hat{q}_1 \approx 0.9998434 \)
- \( n_2 = 194,780 \)
- \( \hat{p}_2 \approx 0.0005341 \)
- \( \hat{q}_2 \approx 0.9994659 \)
- \( \bar{p} \approx 0.0003438 \)
- \( \bar{q} \approx 0.9996562 \)

Would you like any details or further explanations on these calculations? 

Here are five questions you might ask next:
1. How do we interpret the values of \( \hat{p}_1 \) and \( \hat{p}_2 \)?
2. What is the significance of the combined proportion \( \bar{p} \)?
3. How can we use these proportions to test the effectiveness of the vaccine?
4. What statistical tests are appropriate for comparing these two proportions?
5. How can confidence intervals be calculated for these proportions?

**Tip:** When dealing with proportions in clinical trials, always ensure your sample sizes are large enough to yield statistically significant results.

Explore the analysis of clinical trial data involving 392,762 children, focusing on proportions of disease development in the treatment and placebo groups. Learn how to calculate and interpret proportions, including detailed step-by-step solutions.

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