confidence interval explained
A point estimate from a sample is one number, so it is almost sure to sit a little off the true value. A confidence interval is the method that admits that uncertainty. Attach a margin of error to both sides of the estimate and you make the interval estimate ± z·SE. The hard part is the interpretation. A 95% confidence is not a statement about this interval now. It is the rate at which the procedure catches the true value when samples are drawn again and again. The true value is fixed, and what is random is the interval. You see how a margin builds an interval, how 95% of repeated samples cover the true value, the trade of confidence level against width, how the error shrinks as 1/√n when the sample grows, and the right reading beside the common misunderstanding. The interval shakes and the true value stays still.
A confidence interval starts from a point estimate and attaches a margin of error to both sides. The margin = z·SE, and the multiplier z is set by the confidence level. At 90% , z = 1.645. At 95% , z = 1.96. At 99% , z = 2.576. Change the level and the brackets widen around the same estimate, while the fixed true value stays where it was. The level sets the width of the brackets.
This picture is the real meaning of 95%. The estimate differs from sample to sample, so the interval differs every time. Draw many times and about 95% of them cover the fixed true value, and the intervals that miss are marked in another color. The more you draw, the share that actually covers converges to the confidence level. The true value never moved once. What moved was the interval. Only the interval is the moving side.
Greater confidence is not free. To cover the true value more often you have to throw a wider net. Raise the level and the middle area under the curve grows, and at the same time the interval width 2·z·SE grows. 95% and 99% spread in width so you can see it, and this trade cannot be avoided. The greater the confidence, the wider the net.
The margin is z·σ/√n, so it shrinks as the sample grows, but only at the speed 1/√n. It is the same standard error seen in the sampling distribution. To double the precision, that is to halve the margin, you have to grow the sample fourfold. Diminishing returns sit inside the formula from the start. Precision asks for four times the sample.
Last is a careful reading. The sentence that, repeated, this procedure catches the true value with 95% probability is right. But the sentence that the probability the true value lies inside this interval is 95% is wrong. Once the interval is drawn once, the fixed true value is either inside it or not. Probability lives in the random drawing of samples, not on one interval already realized. A drawn interval has no probability left on it.