Respuesta :
Answer:
The 80% confidence interval for the population proportion of oil tankers that have spills each month is (0.199, 0.257).
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
Suppose a sample of 333 tankers is drawn. Of these ships, 257 did not have spills.
333 - 257 = 76 have spills.
This means that [tex]n = 333, \pi = \frac{76}{333} = 0.228[/tex]
80% confidence level
So [tex]\alpha = 0.2[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.2}{2} = 0.9[/tex], so [tex]Z = 1.28[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.228 - 1.28\sqrt{\frac{0.228*0.772}{333}} = 0.199[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.228 + 1.28\sqrt{\frac{0.228*0.772}{333}} = 0.257[/tex]
The 80% confidence interval for the population proportion of oil tankers that have spills each month is (0.199, 0.257).
