Respuesta :
Answer:
The 95%-confidence interval for the percentage of history majors at U.S. universities who knew Abraham Lincoln's first Vice President is (10.5%, 12.1%).
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 z-score that has a p-value of [tex]1 - \frac{\alpha}{2}[/tex].
6250 students, 11.3% knew Abraham Lincoln's first vice-president.
This means that [tex]n = 6250, \pi = 0.113[/tex]
95% confidence level
So [tex]\alpha = 0.05[/tex], z is the value of Z that has a p-value of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.113 - 1.96\sqrt{\frac{0.113*0.887}{6250}} = 0.105[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.113 + 1.96\sqrt{\frac{0.113*0.887}{6250}} = 0.121[/tex]
As percentages:
0.105*100% = 10.5%
0.121*100% = 12.1%
The 95%-confidence interval for the percentage of history majors at U.S. universities who knew Abraham Lincoln's first Vice President is (10.5%, 12.1%).
