t-Test

Every t-test follows the same architecture: take the estimate, subtract the value assumed true under the null hypothesis, and divide by the estimated standard error of that estimate. For a population mean, the numerator is the sample mean minus the hypothesized mean and the denominator is the sample standard deviation divided by the square root of the sample size. The resulting ratio expresses the distance between what was observed and what was assumed, measured in units of sampling noise.

The decision rule compares this ratio to a critical value drawn from the t-distribution at the chosen significance level and degrees of freedom, or equivalently compares the p-value to alpha. A two-tailed test splits alpha across both tails and rejects for large absolute values; a one-tailed test places the full rejection region on the side specified by the alternative hypothesis.

The t-distribution is symmetric and bell-shaped but has fatter tails than the standard normal, because the denominator of the test statistic is itself estimated rather than known. That extra uncertainty pushes critical values further from zero, making the test more demanding at small sample sizes. As degrees of freedom rise, the tails thin and the distribution converges on the normal; the two become numerically almost indistinguishable in large samples.

Degrees of freedom count the independent pieces of information left after estimating parameters from the same data. Estimating one mean consumes one, so a single-mean test uses n minus one. A regression consumes one degree of freedom per estimated coefficient, intercept included. Tables and software require this input, and using the wrong figure shifts the critical value enough to flip borderline conclusions.

Beyond the single mean, the same framework handles a paired comparison, in which the differences between matched observations are reduced to one series and tested against zero. This is the natural tool for before-and-after studies, such as tracking error on the same portfolios across two regimes. Two independent samples can also be compared, with the standard error built from both samples' variances and degrees of freedom adjusted accordingly.

The test also governs correlation: the sample correlation is scaled by the square root of n minus two and divided by the square root of one minus the squared correlation, then compared to the t-distribution with n minus two degrees of freedom. In regression, every reported coefficient carries a t-statistic equal to the estimate divided by its standard error, testing the hypothesis that the true coefficient is zero.

Regression and statistical output typically reports the estimate, its standard error, the t-statistic and a p-value. A candidate should be able to reconstruct any one of these from the others, invert the test into a confidence interval, and state the conclusion in the language of the null hypothesis rather than claiming proof. Failing to reject is not evidence that the parameter equals zero; it means the data cannot distinguish it from zero at the chosen confidence level.

Validity rests on assumptions: observations drawn independently, and either an approximately normal population or a sample large enough for the central limit theorem to apply. Statistical significance is also distinct from economic significance. A coefficient can clear a threshold on a very large sample while being far too small, after transaction costs, to support any investment decision.