Confidence interval is written as
Sample proportion +/- margin of error
The formula for determining margin of error is
[tex]\begin{gathered} \text{margin of error = z }\times\text{ }\sqrt[\square]{\frac{pq}{n}} \\ \end{gathered}[/tex]where
z represents the z score corresponding to the confidence interval
p represents the sample proportion. It also means the probability of success
q represents the probability of failure
q = 1 - p
p = x/n
where
n represents the number of samples
x represents the number of success
From the information given,
n = 325
x = 260
p = 260/325 = 0.8
q = 1 - p = 1 - 0.8 = 0.2
To determine the z score, we would subtract the confidence level from 100 to get alpha, a
Thus
a = 1 - 0.99 = 0.01
a/2 = 0.01/2 = 0.005
This is the area in each tail. Since we want the area in the middle, it becomes
1 - 0.005 = 0.995
The z score corresponding to the area on the z scores table is 2.576. Thus, the z score for a 99% confidence interval is 2.576
Thus, the 99% confidence interval is
[tex]\begin{gathered} 0.8\text{ }\pm\text{ 2.576 }\frac{\sqrt[\square]{0.8\times0.2}}{325} \\ 0.8\text{ }\pm\text{ 2.576 x 0.022} \\ 0.8\text{ }\pm0.057 \end{gathered}[/tex]Thus,
the lower limit is
0.8 - 0.057 = 0.743
the upper limit is
0.8 + 0.057 = 0.857
