Question 1206695
test size = 390
12 out of 390 failed the test
failure proportion = 12/390


p = 12/390
q = 1 - p = 378/390


standard error of a proportion = sqrt(p*q/n) = sqrt(12/390 * 378/390 / 390) = .0087445932.


95% two tail confidence interval required a critical z-score of plus or minus 1.959963986.


on the low side of the confidence interval, the z-score formula of z = (x - m) / s becomes -1.959963986 = (x - 12/390) / .0087445932.
solve for x to get x = .013630143.


on the high side of the confidence interval, the z-score formula of z = (x - m) / s becomes 1.959963986 = (x - 12/390) / .0087445932.
solve for x to get x = .0479083186.


that should be your 95% confidence interval.
i used the calculator at <a href = "https://davidmlane.com/hyperstat/z_table.html" target = "_blank">https://davidmlane.com/hyperstat/z_table.html</a>


here's what it looks like using that calculator.


<img src = "http://theo.x10hosting.com/2024/032705.jpg">


the calculator truncates inputs to 6 or 7 decimal digits, so i didn't get the full value in there.
the rounding / truncating is usually good enough to get an accurate answer.


from the calculator, you can see that the confidence interval is the area between those two x-scores = .95.