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.