Interval estimate

An interval estimate is any report that gives a range of parameter values instead of a single one. The useful versions state how reliable the range is, and attaching a confidence level is exactly what turns an interval estimate into a confidence interval. In this course the shape is always $\text{estimate} \pm \text{margin of error}$, so the range sits symmetrically around the point estimate and the margin of error is half its width.

Twenty-five randomly selected seedlings have mean height $\bar{x} = 68.4$ mm with $s = 4.2$ mm. The standard error is $\frac{4.2}{\sqrt{25}} = 0.84$, the 95% critical value at $25 - 1 = 24$ degrees of freedom is $t^* = 2.064$, and the margin of error is $2.064 \times 0.84 = 1.73$. The interval estimate is $68.4 \pm 1.73$ mm, or 66.67 to 70.13 mm. 68.4 is still the single best guess; the interval reports which other values the data leave standing.

"The true mean is more likely to be near 68.4 than near 70.1" does not follow from that interval. An interval estimate is a set, not a distribution over the parameter. It separates values the method does not rule out at the stated level from values it does, and it says nothing finer. Its edges behave the same way: 70.13 is inside and 70.20 is outside, but no property of the seedlings changes between those two numbers.

An interval estimate inherits every flaw of the estimate at its center. Measure 25 seedlings from the sunniest tray and you get a narrow range around the wrong mean, and no confidence level will warn you, because the capture rate assumes the sample was random. Precision and accuracy are separate questions and only the first is in the arithmetic.

Symmetry is a property of the $\pm$ form rather than a law. For a proportion close to 0 or 1 the same construction can push an endpoint below 0 or above 1, which signals that the normal approximation behind the margin of error has broken rather than saying anything about impossible proportions.