Confidence interval

A confidence interval turns a single estimate into a range of plausible parameter values by attaching an explicit allowance for sampling error. Every interval in AP Statistics has the same shape:

The product of the last two pieces is the margin of error, and the standard error estimates how much the statistic varies from one sample to the next.

Say 520 of 1000 randomly sampled voters back a measure. The estimate is $\hat{p} = 0.52$ (p hat), the standard error is $\sqrt{\frac{0.52(0.48)}{1000}} = 0.0158$, and a 95 percent interval uses the critical value $z^* = 1.960$ (z star). The margin of error is $1.960 \times 0.0158 = 0.031$, so the interval runs from 0.489 to 0.551, or 48.9 percent to 55.1 percent.

The interpretation that loses marks is what the 95 percent attaches to. There is not a 95 percent chance that the true proportion lies between 0.489 and 0.551. The true proportion is a fixed number: it is either inside those endpoints or it is not, and no probability is left over once the sample is drawn. What varies from sample to sample is the interval itself, since a fresh sample of 1000 voters produces different endpoints. The 95 percent describes the method, which captures the parameter in about 95 percent of all possible samples. The interval also says nothing about where 95 percent of individual voters sit.

Width is the other half of the idea. The margin of error shrinks like $1/\sqrt{n}$, so quadrupling the sample from 1000 to 4000 only halves it, from 0.0310 to 0.0155. Raising confidence widens the interval instead ($z^* = 1.645$ at 90 percent, $2.576$ at 99 percent), so precision and confidence trade against each other at any fixed sample size.

One reading comes free. This interval contains 0.50, so the data do not rule out an even split, which is the same verdict a two-sided test of $H_0: p = 0.5$ at $\alpha = 0.05$ returns on these counts.