Respuesta :
Answer:
a. .92
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].
The margin of error of the interval is:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
The lower bound is the point estimate [tex]\pi[/tex] subtracted by the margin of error.
The upper bound is the point estimate [tex]\pi[/tex] added to the margin of error.
Point estimate:
The confidence interval is symmetric, so it is the mean between the two bounds.
In this problem:
[tex]\pi = \frac{0.372 + 0.458}{2} = 0.415[/tex]
Sample of 400, which means that [tex]n = 400[/tex]
Margin of error is the estimate subtracted by the lower bound. So [tex]M = 0.415 - 0.372 = 0.043[/tex]
We have to find z.
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]0.043 = z\sqrt{\frac{0.415*0.585}{400}}[/tex]
[tex]z = \frac{0.043\sqrt{400}}{\sqrt{0.415*0.585}}[/tex]
[tex]z = 1.745[/tex]
[tex]z = 1.745[/tex] has a pvalue of 0.96.
This means that:
[tex]1 - \frac{\alpha}{2} = 0.96[/tex]
[tex]\frac{\alpha}{2} = 1 - 0.96[/tex]
[tex]\frac{\alpha}{2} = 0.04[/tex]
[tex]\alpha = 0.08[/tex]
Confidence level:
[tex]1 - \alpha = 1 - 0.08 = 0.92[/tex]
So the correct answer is:
a. .92
