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0 of 0 intervals captured the true mean (μ = 50). Aiming for 95%.
μ = 50
Each horizontal bar is one 95% CI from one sample. Green = it captured the true mean; red = it missed. The black line is the true μ.
The true mean is fixed and known here. Draw sample after sample, build a CI for each, and watch what "95% confidence" actually means.
Each horizontal bar is one 95% CI from one sample. Green = it captured the true mean; red = it missed. The black line is the true μ.
Not quite — and this simulator shows why. The true mean is fixed; each interval either contains it or doesn't. "95% confidence" describes the long-run success rate of the method: if you repeated the sampling many times, about 95% of the intervals you build would capture the true mean. That's exactly the capture rate you watch converge here.
By chance, some samples land unusually far from the population mean, so their intervals — centred on that unlucky sample mean — sit entirely to one side of μ. With a 95% level, about 1 in 20 intervals will miss. Raise the level to 99% and fewer miss, but the intervals get wider.
The intervals get narrower (the standard error σ/√n shrinks), so each estimate is more precise — but the capture rate stays around the confidence level. Bigger n buys precision, not a higher chance of capture. See SD vs standard error.