Question 1199178
sample size = 225
p = 69/225 
q = (1 - 69) / 225 = 225/225 - 69/225 = (225 - 69) / 225 = 156/225
mean = 69/225
standard error = sqrt(p*q/n) = sqrt((69/225*156/225)/225) = .0307406515.
z = (x-m)/s
z is the z-score
(x-m) is the margin of error.
s is the standard error.


your confidence interval is 95%.
at 95% confidence interval, the critical z-score is plus or minus 1.96


to find the margin of error, you only need to use the hi side z-score because the confidence interval is symmetric about the mean and the margin of error above the mean is the same as the margin of error below the mean.


formula becomes 1.96 = (x-m) / standard error which becomes 1.96 = (x-m) / .0307406515.


solve for (x-m) to get (x-m) = 1.96 * .0307406515 = .0602516769.


that should be the margin of error on either side of the mean at 95% confidence interval.


the mean is equal to 69/225
that minus the margin of error yields a low side value of .2464149897.
that plus the margin of error yields a high side value of .3669183436.


this is tested using the calculator at <a href = "https://davidmlane.com/hyperstat/z_table.html" target = "_blank">https://davidmlane.com/hyperstat/z_table.html</a>


below are the results.


<img src = "http://theo.x10hosting.com/2022/121608.jpg">


<img src = "http://theo.x10hosting.com/2022/121609.jpg">


your solution is shown below:


the margin of error is .0602516769 at 95% confidence interval.


the 95% confidence interval is from .2464149897 to .3669183436.


let me know if you have any questions.
theo