Clinical trials involved treating flu patients with Tamiflu. Among the 586 patients treated with Tamiflu, 37 experienced nausea as an adverse reaction. You will be constructing a 96% confidence interval to estimate the true proportion of patients that experience nausea. 

To construct a 96% confidence interval for the true proportion of patients who experience nausea as an adverse reaction to Tamiflu, we'll use the formula for a confidence interval for a proportion. Here are the steps:

### Step 1: Define the given values
- **Sample size (n)** = 586
- **Number of patients with nausea (x)** = 37
- **Sample proportion (p̂)** = \( \frac{x}{n} = \frac{37}{586} \approx 0.0631 \)

### Step 2: Determine the confidence level
Since we need a 96% confidence interval, our **confidence level (C)** is 96%, which corresponds to a **z-score** for a 96% confidence level. To find this, we look up the z-score that leaves 2% in each tail (since 96% confidence leaves 4% outside the interval). The **z-score** for a 96% confidence level is approximately **2.05**.

### Step 3: Calculate the confidence interval
The confidence interval for a proportion is given by:
\[
p̂ \pm z \cdot \sqrt{\frac{p̂ (1 - p̂)}{n}}
\]
Plugging in our values:
1. **Calculate the standard error (SE):**
   \[
   SE = \sqrt{\frac{0.0631 \cdot (1 - 0.0631)}{586}} \approx 0.0101
   \]
2. **Calculate the margin of error (ME):**
   \[
   ME = z \cdot SE = 2.05 \cdot 0.0101 \approx 0.0207
   \]
3. **Construct the interval:**
   \[
   0.0631 \pm 0.0207 = (0.0424, 0.0838)
   \]

### Conclusion
The 96% confidence interval for the true proportion of patients experiencing nausea as an adverse reaction to Tamiflu is approximately:
\[
(0.0424, 0.0838)
\]

This means we are 96% confident that the true proportion of patients who experience nausea is between 4.24% and 8.38%.

Would you like more details on any of the steps?

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**Further Questions to Explore:**
1. How is the z-score determined for various confidence levels?
2. What does a confidence interval really mean in the context of clinical studies?
3. How would the confidence interval change if the sample size was smaller?
4. Why do we use a normal approximation for the confidence interval of a proportion?
5. How does the confidence interval interpretation change if more adverse effects were recorded?

**Tip:** For proportions near 0 or 1, larger sample sizes give more accurate confidence intervals due to reduced variability in extreme proportions.

Clinical trials involved treating flu patients with Tamiflu. Among the 586 patients treated with Tamiflu, 37 experienced nausea as an adverse reaction. You will be constructing a 96% confidence interval to estimate the true proportion of patients that experience nausea.

Learn how to construct a 96% confidence interval to estimate the true proportion of nausea cases among Tamiflu-treated flu patients. Given a sample of 586 patients, with 37 experiencing nausea, this example walks through the steps using key statistical concepts and formulas.

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