Precision

Precision is how tightly an estimate pins a parameter down, and for a confidence interval you read it straight off the width: the narrower the interval, the more precise the estimate. It is a property of the procedure, fixed by the sample size, the confidence level, and how variable the data are. It says nothing about whether the estimate is centered on the right value.

Take $\hat{p} = 0.40$ (p-hat) from a random sample of 400. The standard error is $\sqrt{0.40(0.60)/400} = 0.024495$, so a 95% interval reaches $1.96(0.024495) = 0.0480$ on each side: 0.352 to 0.448, a width of 0.0960. Run the study again with 1600 people and the same $\hat{p}$ and the standard error halves to 0.012247, the interval becomes 0.376 to 0.424, and the width halves to 0.0480. Four times the data bought twice the precision, because the margin of error shrinks like $1/\sqrt{n}$.

The sentence to distrust is "the interval is narrow, so the estimate must be close to the truth." Precision is not accuracy. Suppose the survey reaches only the people who pick up unknown numbers, and that group runs 8 points high. The estimate from 1600 of them is still 0.048 wide, and it is still centered about 8 points off. Precision describes how tightly repeated estimates cluster around their own center. Accuracy is about where that center sits, and no sample size moves it, because bias appears nowhere in the arithmetic above.

There is also a cheap way to look precise. Keep the same 400 people and the same $\hat{p} = 0.40$, then drop the confidence level from 95% to 90%: the width falls from 0.0960 to 0.0806, about 16% narrower, bought entirely by accepting that 10 intervals in 100 miss instead of 5. Sample size buys precision, while a lower confidence level only relabels it. That is why a width means nothing unless the confidence level is quoted alongside it.