Answer:
The 95% confidence interval for the population proportion of times that the bats would follow the point is (0.505, 0.995).
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of
, and a confidence interval
, 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 ![1 - \frac{\alpha}{2}](https://tex.z-dn.net/?f=1%20-%20%5Cfrac%7B%5Calpha%7D%7B2%7D)
For this problem, we have that:
There are 12 bats, and 9 would follow the feeder. This means that
and
.
Find the 95% confidence interval for the population proportion of times that the bats would follow the point.
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}} = 0.75 - 1.96\sqrt{\frac{0.75*0.25}{12}} = 0.505](https://tex.z-dn.net/?f=%5Cpi%20-%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D%20%3D%200.75%20-%201.96%5Csqrt%7B%5Cfrac%7B0.75%2A0.25%7D%7B12%7D%7D%20%3D%200.505)
The upper limit of this interval is:
![\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.75 + 1.96\sqrt{\frac{0.75*0.25}{12}} = 0.995](https://tex.z-dn.net/?f=%5Cpi%20%2B%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D%20%3D%200.75%20%2B%201.96%5Csqrt%7B%5Cfrac%7B0.75%2A0.25%7D%7B12%7D%7D%20%3D%200.995)
The 95% confidence interval for the population proportion of times that the bats would follow the point is (0.505, 0.995).