Conditions for a One-Sample t-Test

A t-test estimates the population standard deviation from the sample itself, which makes it shakier than a z-test. That extra uncertainty is exactly why the conditions matter so much: if the data are messy going in, the t-distribution can't fix that on the way out. Three things need to hold. First, randomness — the sample must come from a random process, like an SRS, so the results generalize. Second, independence — knowing one value shouldn't tell you much about another; with sampling without replacement, check that the sample is less than 10% of the population. Third, normality of the population, which you can never check directly, so you check what you can see: the sample.

To judge normality, look at a dotplot, boxplot, or normal probability plot of the sample. If n is small, say under 15, the data should show no strong skew and no outliers, since one extreme value can drag the sample mean and inflate the sample standard deviation, distorting the whole test. As n grows past about 15, moderate skew becomes tolerable, and once n is around 30 or more, the central limit theorem takes over and even fairly skewed populations produce a nearly normal sampling distribution for the mean.

This condition check sits right before you compute the test statistic and p-value. Skipping it doesn't just weaken the argument — it can make the stated p-value meaningless, because the t-distribution's accuracy depends on these assumptions. The same three-part check reappears for every quantitative-means procedure that follows, including confidence intervals and two-sample t-tests, so getting comfortable with it here pays off all unit long.