Answer:
The 95% confidence interval for the true proportion of all voters in the state who favor approval is (0.4384, 0.5050).
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of
, and a confidence level of
, we have the following confidence interval of proportions.
![\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}](https://tex.z-dn.net/?f=%5Cpi%20%5Cpm%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D)
In which
z is the zscore that has a pvalue of
.
For this problem, we have that:
![n = 865, \pi = \frac{408}{865} = 0.4717](https://tex.z-dn.net/?f=n%20%3D%20865%2C%20%5Cpi%20%3D%20%5Cfrac%7B408%7D%7B865%7D%20%3D%200.4717)
95% confidence level
So
, z is the value of Z that has a pvalue of
, so
.
The lower limit of this interval is:
![\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.4717 - 1.96\sqrt{\frac{0.4717*0.5283}{865}} = 0.4384](https://tex.z-dn.net/?f=%5Cpi%20-%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D%20%3D%200.4717%20-%201.96%5Csqrt%7B%5Cfrac%7B0.4717%2A0.5283%7D%7B865%7D%7D%20%3D%200.4384)
The upper limit of this interval is:
![\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.4717 + 1.96\sqrt{\frac{0.4717*0.5283}{865}} = 0.5050](https://tex.z-dn.net/?f=%5Cpi%20%2B%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D%20%3D%200.4717%20%2B%201.96%5Csqrt%7B%5Cfrac%7B0.4717%2A0.5283%7D%7B865%7D%7D%20%3D%200.5050)
The 95% confidence interval for the true proportion of all voters in the state who favor approval is (0.4384, 0.5050).