A confidence interval for a proportion is used when estimating the percentage of a population that possesses a certain characteristic based on sample data. It is valid under these key conditions:

• Random Sampling: The data must be collected through a simple random sample from the target population.
• Normality Condition: The sample size must be large enough. A common guideline is that both the number of successes ($n cdot p ge 10$) and failures ($n cdot (1-p) ge 10$) must be at least 10.
• Finite Population Correction (FPC): When sampling without replacement from a finite population where sample size $n$ is more than 5% of the total population $N$ (i.e. $n/N > 0.05$), the FPC factor should be applied to account for the reduced variance.

Understanding what a confidence interval represents is critical for correct statistical inference:

• Correct Interpretation: A 95% confidence level means that if we repeated the sampling process many times, approximately 95% of the calculated intervals would contain the true population proportion.
• Common Misconception: It does not mean there is a 95% probability that the true population proportion lies within this specific interval. The true proportion is a fixed, constant value, so it is either in the interval (probability 1) or it is not (probability 0).
• Effect of Sample Size: Increasing the sample size $n$ reduces the standard error, making the interval narrower and more precise.
• Effect of Confidence Level: Demanding a higher confidence level (e.g. 99% vs 95%) increases the critical value $z^*$, which widens the interval to ensure greater certainty.