Respuesta :
Answer:
[tex]n = 191[/tex]
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 interval [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 is:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
95% confidence level
So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].
In this problem, we have that:
[tex]M = 0.07, \pi = 0.42[/tex]
We have to find n
[tex]0.07 = 1.96*\sqrt{\frac{0.42*0.58}{n}}[/tex]
[tex]0.07\sqrt{n} = 1.96*\sqrt{0.42*0.58}[/tex]
[tex]0.07\sqrt{n} = 0.9674[/tex]
[tex]\sqrt{n} = \frac{0.9674}{0.07}[/tex]
[tex]\sqrt{n} = 13.82[/tex]
[tex]\sqrt{n}^{2} = (13.82)^{2}[/tex]
[tex]n = 190.9[/tex]
So, rounded to the nearest integer
[tex]n = 191[/tex]
