Question

use the sample data and confidence level given below to complete parts (a) through (d) a drug is used to help prevent blood clots in certain patients. in clinical trials, among 4861 patients treated with the drug, 195 developed the adverse reaction of nausea. construct a 95% confidence interval for the proportion of adverse reactions a) find the best point estimate of the population proportion p (round to three decimal places as needed.) b) identify the value of the margin of error e e= (round to three decimal places as needed.) c) construct the confidence interval <p< (round to three decimal places as needed.) d) write a statement that correctly interprets the confidence interval. choose the correct answer below a. 95% of sample proportions will fall between the lower bound and the upper bound b. there is a 95% chance that the true value of the population proportion will fall between the lower bound and the upper bound c. one has 95% confidence that the sample proportion is equal to the population proportion d. one has 95% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion

Explanation:

Step1: Calculate the best point estimate (\(\hat{p}\))

The formula for the sample proportion \(\hat{p}\) is \(\hat{p}=\frac{x}{n}\), where \(x = 195\) (number of successes) and \(n=4601\) (sample size).
\(\hat{p}=\frac{195}{4601}\approx0.042\)

Step2: Find the critical value \(z_{\alpha/2}\)

For a 95% confidence level, \(\alpha=1 - 0.95=0.05\), and \(\alpha/2=0.025\).
From the standard normal distribution table, \(z_{\alpha/2}=z_{0.025} = 1.96\)

Step3: Calculate the margin of error \(E\)

The formula for \(E\) is \(E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)
Substitute \(\hat{p}=0.042\), \(n = 4601\), and \(z_{\alpha/2}=1.96\)
\(\hat{p}(1-\hat{p})=0.042\times(1 - 0.042)=0.042\times0.958 = 0.040236\)
\(\sqrt{\frac{0.040236}{4601}}\approx\sqrt{8.745\times10^{-6}}\approx0.00296\)
\(E=1.96\times0.00296\approx0.006\)

Step4: Construct the confidence interval

The confidence interval is \(\hat{p}-ESubstitute \(\hat{p}=0.042\) and \(E = 0.006\)
\(0.042- 0.006\(0.036

Answer:

a) \(0.042\)
b) \(0.006\)
c) \(0.036d) D. One has 95% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.