Confidence Interval Calculator

If your sample is small (roughly under 30) and you estimated the standard deviation from the sample rather than knowing the population value, yes, strictly you should use t. This calculator uses z, which produces a narrower interval than t does. At n = 10 the 95 percent multiplier should be 2.262 but z gives 1.960, so the interval here is about 15 percent too tight. Above n = 30 the two converge and the difference is not practically important.

Yes. Choose Custom and type any level strictly between 0 and 100. The critical value is computed from an inverse normal CDF, not read from a lookup table, so 80, 97.5, and 99.9 all give an exact z. The tool rejects 0 and 100 because both correspond to a degenerate or infinite interval.

Either your sample standard deviation is large relative to the mean, or your sample is small, or both. The margin of error is z times s divided by the square root of n, so the only lever that reliably narrows it is more data, and it narrows only with the square root of the sample size: to halve the width you need four times as many observations.

It is a property of the procedure, not of the one interval you are looking at. If you drew many samples the same way and computed an interval from each, about 95 percent of those intervals would contain the true population mean. The true mean is a fixed number; it is the interval that moves from sample to sample. Saying "there is a 95 percent probability the mean is in this range" is the common misstatement, and it is a Bayesian claim that this frequentist procedure does not support.

Because you cannot have 12.5 observations. The tool requires n to be a whole number of at least 1, and it requires the standard deviation to be non-negative. Those checks catch the most common data-entry mistakes before they turn into a plausible-looking but meaningless interval.