The t-Test for a Mean

The t-test is simply the z-test from the last lesson for the realistic case where you don't know the population's standard deviation. Look at the formula: it's the same z-score with one swap, $s$ (the standard deviation you measured in your sample) in place of $\sigma$ (the true one you don't have). The only other new piece is $df = n - 1$ (degrees of freedom, which tells you which t curve to use, and grows with the sample size). It's also called Student's t, after the pen name William Gosset used when he published it, and "t" is simply the letter later attached to his statistic.

Why a different curve? Think about what happens when you estimate $\sigma$. In a small sample, $s$ could easily come out too small by luck, and dividing by a too-small number makes the statistic look more extreme than it really is. The t distribution accounts for that: it has fatter tails than the normal, so you need a more extreme statistic before you call it significant. With more data $s$ becomes a reliable estimate, and the t curve turns into the normal.

Decision by critical value. Instead of a p-value, compare $|t|$ with the critical value $t^*$ from a table: reject $H_0$ when the statistic falls beyond it. The two approaches always agree, since "beyond $t^*$" is exactly "tail area less than $\alpha$."

Assumptions: a random sample from a roughly normal population, or a large enough sample. Check for strong skew or outliers with a quick plot first; the t-test is sensitive to them in small samples.

A common mistake. It's tempting to use the z cutoff 1.96 whenever you have a t statistic, since it's the number you remember. With small samples that's too lenient. For $df = 8$ the two-tailed cutoff is $2.306$, and a $t$ of $2$ that would pass the z cutoff fails here.

Manufacturers test whether products meet labeled amounts, and researchers test whether a treatment changes an average outcome.