The axis is marked in true standard errors, so every margin of error can be measured against the same ruler. You have one sample, and you do not have the quantity the true standard error is built from — you have an estimate of it. Pick a recipe for the margin of error and watch whether the interval still reaches the parameter.
Before you draw: the second recipe uses the same 1.96, just with s in place of σ. Over many samples, would you expect it to capture μ more than 95% of the time, less, or exactly 95%?
A confidence interval for a mean is x̄ ± (multiplier) × (standard error). The honest margin is z*σ/√n — but σ is a fact about the population, and you do not have it. You have s, computed from the same sample that gave you x̄. This page is about what that substitution costs and how t* pays for it.
t* does not repair the s you happen to have. In a sample where s came in low, the t interval is still shorter than it should be; in a sample where s came in high, it is longer than it needed to be. What t* fixes is the rate: over all samples, exactly the advertised fraction capture μ. Drag the s slider to see one sample get it wrong in each direction, then draw a few hundred and watch the rate come right anyway.
Raise n and t* slides down toward z*: with a lot of data, s is a reliable stand-in for σ and the correction stops mattering. That is why the two recipes agree in large samples and why the distinction is a small-sample one.
?ci= 90 / 95 / 99 · ?n= sample size · ?s= this sample’s s as a fraction of σ · ?xbar= where the mean landed, in true SE · ?me= 1, 2 or 3 to pick the recipe · ?seed= · ?mode=present
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