Answer:
a) 0.00698
b) 0.00127
c) 0.00249
d) (0.00449,0.00947)
Step-by-step explanation:
We are given the following in the question:
Number of people vaccinated, n = 4300
Number of subjects that developed illness after vaccination, x = 30
a) point estimate of the population proportion that were vaccinated with the vaccine but still developed the illness.
[tex]p = \displaystyle\frac{x}{n} = \frac{30}{4300} = 0.00698[/tex]
b) standard error of this estimate
[tex]\text{Standard error} = \sqrt{\displaystyle\frac{pq}{n}} = \sqrt{\displaystyle\frac{p(1-p)}{n}} = \sqrt{\displaystyle\frac{0.00698\times 0.99302}{4300}} = 0.00127[/tex]
c) margin of error for a 95% confidence interval
[tex]z_{critical}\text{ at}~\alpha_{0.05} = 1.96[/tex]
[tex]\text{Margin of error} = z_\text{critical}\times \text{Standard error}\\= \pm 1.96\times 0.00127 = \pm 0.00249[/tex]
d) 95% confidence interval
[tex]p\pm \text{Margin of error}\\= 0.00698\pm 0.00249\\=(0.00449,0.00947)[/tex]
