t or z — and why it matters at small n
If you knew the population's true standard deviation you would use the normal distribution. You almost never do; you estimate it from the same sample, and that extra uncertainty has to be paid for. The t distribution charges it, with fatter tails that widen as the sample shrinks.
At n=25 the 95% multiplier is 2.064 instead of 1.960 — about 5% wider. At n=5 it is 2.776, over 40% wider. Above roughly n=30 the two converge and the choice stops mattering. Tools that always use 1.96 quietly report intervals that are too narrow on small samples.
What '95% confident' does not mean
It does not mean there is a 95% chance the true value lies in this particular interval — the true value is fixed, and this interval either contains it or does not. It means the procedure produces intervals that capture the true value 95% of the time across repeated sampling.
The practical reading is simpler: a narrow interval means your estimate is well pinned down, a wide one means it is not, and the width shrinks with the square root of the sample size — so quartering the margin of error costs sixteen times the data.
Frequently asked questions
Why did my interval get wider when I raised the confidence level?
Because more confidence requires a bigger net. A 99% interval must catch the truth more often than a 95% one, and the only way to do that with the same data is to be less precise.
Can I rely on this for coursework or research?
The t-values are checked against published tables at every degrees-of-freedom step. Whether to use z or t depends on sample size and whether the population deviation is known.
Is my data uploaded?
No. Nothing you paste is transmitted — useful when the sample is real study data.
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