Respuesta :
Answer:
0.071 to 0.129
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].
For this problem, we have that:
In a sample of 400 voters, 360 indicated they favor the incumbent governor. This means that 400-360 = 40 do not favor the incumbent governor. So [tex]n = 400, \pi = \frac{40}{400} = 0.1[/tex]
95% confidence interval
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].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.10 - 1.96\sqrt{\frac{0.10*0.90}{400}} = 0.071[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.10 + 1.96\sqrt{\frac{0.10*0.90}{400}}= 0.129[/tex]
The correct answer is:
0.071 to 0.129
