Question

Please, help me to resolve the following: 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 4868 patients treated with the​ drug, 126 developed the adverse reaction of nausea. Construct a 90​% confidence interval for the proportion of adverse reactions. ​a) Find the best point estimate of the population proportion p, identify the value of the margin of error E, construct the confidence interval, and interprets the confidence interval.​(Round to three decimal places as​ needed.)

Answer 1

The best point estimate of the population proportion p is 0.0259. The 90% confidence interval for the proportion is (0.0229, 0.0289), meaning we are 90% confident that the true proportion of adverse reactions lies between 2.29% and 2.89%.

Step-by-step explanation:

To resolve the problem, we first need to identify the sample data and the confidence level given. The sample size (n) is 4868 patients, and number of adverse reactions (x) is 126. The confidence level is 90%. The steps to construct the confidence interval are as follows:

1. Find the best point estimate of the population proportion p', which is equal to x/n. p' = 126/4868 = 0.0259.
2. Next, we need to identify the value of the margin of error E. The formula for E when estimating a proportion is E = Z*(sqrt(p'(1-p')/n)), where Z is the Z-score corresponding to the confidence level. For 90% confidence, Z is approximately 1.645.
3. Calculate E: E = 1.645*(sqrt(0.0259*(1-0.0259)/4868)) = 0.003.
4. Construct the confidence interval, which is p' ± E. Therefore, the 90% confidence interval is (0.0259 - 0.003, 0.0259 + 0.003), which simplifies to (0.0229, 0.0289).
5. Interpret the confidence interval: We are 90% confident that the true proportion of adverse reactions in the population lies between 2.29% and 2.89%.

The confidence interval provides an estimated range of values which is likely to include the population parameter with a certain degree of confidence. It is not a guarantee, but rather an indication of where the true population proportion is expected to be based on the sample data.