Answer:
The best point estimate of the population proportion p is 0.0251.
The 99% confidence interval for the proportion of adverse reactions is (0.0191, 0.0311).
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of
, and a confidence level of
, we have the following confidence interval of proportions.
![\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}](https://tex.z-dn.net/?f=%5Cpi%20%5Cpm%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D)
In which
z is the zscore that has a pvalue of
.
In clinical trials, among 4547 patients treated with the drug, 114 developed the adverse reaction of nausea.
This means that ![n = 4547, \pi = \frac{114}{4547} = 0.0251](https://tex.z-dn.net/?f=n%20%3D%204547%2C%20%5Cpi%20%3D%20%5Cfrac%7B114%7D%7B4547%7D%20%3D%200.0251)
The best point estimate of the population proportion p is 0.0251.
99% confidence level
So
, z is the value of Z that has a pvalue of
, so
.
The lower limit of this interval is:
![\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.0251 - 2.575\sqrt{\frac{0.0251*0.9749}{4547}} = 0.0191](https://tex.z-dn.net/?f=%5Cpi%20-%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D%20%3D%200.0251%20-%202.575%5Csqrt%7B%5Cfrac%7B0.0251%2A0.9749%7D%7B4547%7D%7D%20%3D%200.0191)
The upper limit of this interval is:
![\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.0251 + 2.575\sqrt{\frac{0.0251*0.9749}{4547}} = 0.0311](https://tex.z-dn.net/?f=%5Cpi%20-%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D%20%3D%200.0251%20%2B%202.575%5Csqrt%7B%5Cfrac%7B0.0251%2A0.9749%7D%7B4547%7D%7D%20%3D%200.0311)
The 99% confidence interval for the proportion of adverse reactions is (0.0191, 0.0311).