In a clinical trial of a certain drug, 25 subjects experience headaches among the 216 subjects treated with the drug. Construct a 99% confidence interval estimate for the proportion of treated subjects who experience headaches, completing parts (a) through (d) below. write a statement that correctly interprets the confidence interval.

To construct a 99% confidence interval estimate for the proportion of treated subjects who experience headaches, we follow the steps for computing a confidence interval for a population proportion. Let’s go through parts (a) to (d):

### Given:
- \( n = 216 \) (number of subjects treated with the drug)
- \( x = 25 \) (number of subjects who experience headaches)
- Confidence level = 99%

The proportion of treated subjects who experience headaches is given by:

\[
\hat{p} = \frac{x}{n} = \frac{25}{216}
\]

### Step 1: Find the sample proportion \( \hat{p} \)
\[
\hat{p} = \frac{25}{216} \approx 0.1157
\]

### Step 2: Find the critical value \( z_{\alpha/2} \) for a 99% confidence level
For a 99% confidence interval, the critical value \( z_{\alpha/2} \) corresponds to the area in the tails being 0.5% on each side, so \( z_{\alpha/2} \) is approximately 2.576.

### Step 3: Calculate the standard error
The standard error (SE) of the proportion is given by:

\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]
Substituting the values:

\[
SE = \sqrt{\frac{0.1157(1 - 0.1157)}{216}} \approx 0.0213
\]

### Step 4: Construct the confidence interval
The confidence interval is calculated as:

\[
\hat{p} \pm z_{\alpha/2} \times SE
\]
Substituting the values:

\[
0.1157 \pm 2.576 \times 0.0213
\]
\[
0.1157 \pm 0.0549
\]
Thus, the confidence interval is approximately:

\[
(0.0608, 0.1706)
\]

### Step 5: Interpretation of the Confidence Interval
We are 99% confident that the true proportion of subjects who experience headaches after being treated with the drug is between 6.08% and 17.06%.

---

### Questions You Might Ask:
1. How is the critical value \( z_{\alpha/2} \) determined?
2. What happens to the confidence interval if we change the confidence level to 95%?
3. Why do we use a standard error for the proportion?
4. What is the meaning of a 99% confidence interval in practical terms?
5. How do we interpret the confidence interval if the value 0 is included in it?

### Tip:
The width of a confidence interval depends on both the confidence level and the sample size—higher confidence levels or smaller sample sizes lead to wider intervals.

In a clinical trial of a certain drug, 25 subjects experience headaches among the 216 subjects treated with the drug. Construct a 99% confidence interval estimate for the proportion of treated subjects who experience headaches, completing parts (a) through (d) below. Write a statement that correctly interprets the confidence interval.

Calculate a 99% confidence interval for the proportion of subjects who experience headaches in a clinical drug trial, with detailed steps and interpretation. Suitable for students studying college-level statistics.

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