One-Sample T-Test Statistic

The t-statistic looks identical to the z-statistic but replaces the known σ with s, the standard deviation estimated from the same small sample. That substitution adds a second source of uncertainty, so the reference distribution has fatter tails than the normal — you need a bigger t to claim significance. It was invented in the Guinness brewery in Dublin: William Sealy Gosset, working on barley and yeast batches where a sample of four or five was all he could get, derived the distribution in 1908 and published it as "Student" because Guinness forbade its staff from publishing under their own names, having lost trade secrets that way before.

The trap is reading t against the normal cutoff of 1.96. With 36 observations that is nearly right, but with n = 5 (4 degrees of freedom) the two-sided 5% cutoff is 2.776 — quote 1.96 there and you will declare victory on noise. A worked case: a supplier claims a mean fill of 12.0, and 36 units average 12.5 with s = 1.2, so t = 0.5/(1.2/6) = 2.5 against a cutoff of about 2.03 at 35 degrees of freedom — significant. Reversed, the formula does power planning: detecting a 1-unit difference with s = 2 at t = 3 needs n = (3 × 2/1)² = 36.