Answer:
The 99% confidence interval for the proportion of FM residents whose favorite season is summer is between (0.376, 0.524).
The lower bound of this interval is 0.376.
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 = 300, \pi = \frac{135}{300} = 0.45](https://tex.z-dn.net/?f=n%20%3D%20300%2C%20%5Cpi%20%3D%20%5Cfrac%7B135%7D%7B300%7D%20%3D%200.45)
99% 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}} = 0.45 - 2.575\sqrt{\frac{0.45*0.55}{300}} = 0.376](https://tex.z-dn.net/?f=%5Cpi%20-%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D%20%3D%200.45%20-%202.575%5Csqrt%7B%5Cfrac%7B0.45%2A0.55%7D%7B300%7D%7D%20%3D%200.376)
The upper limit of this interval is:
![\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.45 + 2.575\sqrt{\frac{0.45*0.55}{300}} = 0.524](https://tex.z-dn.net/?f=%5Cpi%20%2B%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D%20%3D%200.45%20%2B%202.575%5Csqrt%7B%5Cfrac%7B0.45%2A0.55%7D%7B300%7D%7D%20%3D%200.524)
The 99% confidence interval for the proportion of FM residents whose favorite season is summer is between (0.376, 0.524).
The lower bound of this interval is 0.376.