A confidence interval provides a RANGE of plausible values for an unknown population parameter (here, proportion $p$), built directly from Lesson 3.5's sampling distribution theory — instead of a single point-estimate, you get an estimate PLUS a margin of uncertainty.
General structure of every confidence interval in this course: (point estimate) $\pm$ (margin of error), where margin of error $=$ (critical value) $\times$ (standard error) — this template recurs throughout Unit 4, just with different critical values and standard error formulas per situation.
For a proportion specifically: $\hat p\pm z^*\sqrt{\frac{\hat p(1-\hat p)}{n}}$ — the critical value $z^*$ comes from the standard normal distribution (e.g., $z^*=1.96$ for 95% confidence — memorize this common value), and the standard error estimates $\sigma_{\hat p}$ from Lesson 3.5 using the SAMPLE proportion $\hat p$ (since the true $p$ is unknown — that's the whole point of estimating it!).
Conditions to verify before constructing ANY confidence interval (a required checklist, not optional): RANDOM sample (or randomized experiment); INDEPENDENCE (sample size is less than 10% of the population, if sampling without replacement); NORMALITY/success-failure condition ($n\hat p\ge10$ and $n(1-\hat p)\ge10$) — skipping this verification is a common and heavily-penalized error on exams.
Correct interpretation of a confidence interval is a precise, testable skill: 'We are [C]% confident that the TRUE POPULATION PROPORTION lies within this interval' — NOT 'there's a [C]% probability the true proportion is in this interval' (the true proportion is a fixed, though unknown, number — it either IS or ISN'T in any particular computed interval; the [C]% describes the PROCEDURE'S long-run success rate across repeated sampling, not this one specific interval).
Margin of error trade-offs: increasing confidence level (say 95% → 99%) INCREASES the critical value $z^*$, WIDENING the interval (more confidence requires more room for error); increasing sample size $n$ SHRINKS the standard error, NARROWING the interval — this connects directly to sample-size PLANNING (choosing $n$ in advance to guarantee a maximum desired margin of error).

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