Questions: In a trial of 108 patients who received 10 -mg doses of a drug daily, 28 reported headache as a side effect. Use this information to complete parts (a) and (b) below. (a) Verify that the requirements for constructing a confidence interval about p are satisfied. Are the requirements for constructing a confidence satisfied? A. Yes, the requirements for constructing a confidence interval are satisfied. B. No, the requirement that np(1-p) is greater than 10 is not satisfied. C. No, the requirement that the sample size is no more than 5% of the population is not satisfied. D. No, the requirement that each trial be independent is not satisfied. (b) Construct and interpret a 90% confidence interval for the population proportion of patients who receive the drug and report a headache as a side effect. One can be 90% confident that the proportion of patients who receive the drug and report a headache as a side effect is between and . (Round to three decimal places as needed. Use ascending order.)

Solution Steps

To construct a confidence interval for the population proportion \( p \), we need to verify the following requirements:

1. Sample Size Requirement: We calculate \( n \hat{p} (1 - \hat{p}) \): \[ n = 108, \quad \hat{p} = \frac{28}{108} \approx 0.259 \] \[ n \hat{p} (1 - \hat{p}) = 108 \cdot 0.259 \cdot (1 - 0.259) \approx 108 \cdot 0.259 \cdot 0.741 \approx 27.1 > 10 \] Thus, the requirement is satisfied.

2. Population Size Requirement: The sample size should be no more than 5% of the population. Assuming the population is large, this requirement is typically satisfied.

3. Independence Requirement: Each trial should be independent. We assume this condition is satisfied in the context of the problem.

We will calculate the 90% confidence interval for the population proportion using the formula: \[ \hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \] where \( z \) is the z-score corresponding to the desired confidence level. For a 90% confidence level, \( z \approx 1.645 \).

Calculating the margin of error: \[ \text{Margin of Error} = z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = 1.645 \cdot \sqrt{\frac{0.259(1 - 0.259)}{108}} \approx 1.645 \cdot \sqrt{\frac{0.259 \cdot 0.741}{108}} \approx 1.645 \cdot \sqrt{0.001847} \approx 1.645 \cdot 0.04294 \approx 0.070 \]

Thus, the confidence interval is: \[ \hat{p} \pm \text{Margin of Error} = 0.259 \pm 0.070 \] Calculating the bounds: \[ \text{Lower Bound} = 0.259 - 0.070 \approx 0.189 \] \[ \text{Upper Bound} = 0.259 + 0.070 \approx 0.329 \]

Final Answer