In a trial of 300 patients who received $10-\mathrm{mg}$ doses of a drug daily, 42 reported headache as a side effect. Use this information to complete parts (a) and (b) below.
Are the requirements for constructing a confidence satisfied?
A. Yes, the requirements for constructing a confidence interval are satisfied.
B. No, the requirement that each trial be independent is not satisfied.
C. No, the requirement that $\hat{n p}(1-\hat{p})$ is greater than 10 is not satisfied.
D. No, the requirement that the sample size is no more than $5 \%$ of the population 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 $\square \%$ confident that the proportion of patients who receive the drug and report a headache as a side effect is between $\square$ and $\square$.
(Round to three decimal places as needed. Use ascending order.)
Answer
Finally, we can be 90% confident that the proportion of patients who receive the drug and report a headache as a side effect is between 0.107 and 0.173. Therefore, the answer to the second part is \(\boxed{\text{One can be } 90\% \text{ confident that the proportion of patients who receive the drug and report a headache as a side effect is between } 0.107 \text{ and } 0.173}\)
Step 1 ：The requirements for constructing a confidence interval are that the sample is random, the sample size is large enough, and the population is at least 20 times as large as the sample. In this case, we don't have information about whether the sample is random or about the size of the population, but we do know that the sample size is large (300), so we can assume that the requirements are satisfied. Therefore, the answer to the first part is \(\boxed{\text{A. Yes, the requirements for constructing a confidence interval are satisfied.}}\)
Step 2 ：For the second part, we can use the formula for a confidence interval for a proportion, which is \(\hat{p} \pm Z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(\hat{p}\) is the sample proportion, Z is the Z-score for the desired confidence level (in this case, 90%), and n is the sample size. The sample proportion is 42/300, and the Z-score for a 90% confidence level is approximately 1.645.
Step 3 ：Calculate the sample proportion (\(\hat{p}\)): \(\hat{p} = \frac{x}{n} = \frac{42}{300} = 0.14\)
Step 4 ：Calculate the standard error (se): \(se = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.14(1-0.14)}{300}} = 0.020033305601755626\)
Step 5 ：Calculate the lower and upper bounds of the confidence interval: \(ci_{lower} = \hat{p} - Z*se = 0.14 - 1.645*0.020033305601755626 = 0.10704521228511202\) and \(ci_{upper} = \hat{p} + Z*se = 0.14 + 1.645*0.020033305601755626 = 0.172954787714888\)
Step 6 ：Finally, we can be 90% confident that the proportion of patients who receive the drug and report a headache as a side effect is between 0.107 and 0.173. Therefore, the answer to the second part is \(\boxed{\text{One can be } 90\% \text{ confident that the proportion of patients who receive the drug and report a headache as a side effect is between } 0.107 \text{ and } 0.173}\)
