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%.

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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.