Michael D. answered • 06/21/25
Maths, Stats, and CompSci Tutoring from a former University Professor
This uses some standard formulas for confidence intervals; you might have slightly different variations or steps.
Assuming this came from a Simple Random Sample of size n (usually the case, but almost never stated explicitly in these sort problems), the Standard Error of phat is given by:
SEP = sqrt(phat*qhat/n) = sqrt(phat*(1-phat)/n) = ... = 0.02958 for the given data.
Since this is a confidence interval for proportions, the critical value is ALWAYS a z-score. You can look this up in a table or use technology to compute it (although if you've done as many of these as I have, you'll know it from memory). Regardless of method, the 99% critical z-score is 2.576 (to three decimal places)
The general formula for this Confidence Interval is
(phat) ± (critical z-score)*(SEP)
Thus 0.65 ± 0.07619 for the given data. If you prefer an interval, manually add/subtract to get 0.57381< phat < 0.72619.
If you're being completely rigorous about your methods, you should check that the number of successes and failures in the sample is at least 5 (or at least 10, in some textbooks). The number of successes is 0.65 * 260 = 169, and the number of failures is 260 - 169 = 91, so we're good in this case.
