The T-Test › The Student's T-Distribution

When we replace $\sigma$ with $s$, the test statistic becomes:

$$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$$

This ratio follows a t-distribution with $n - 1$ degrees of freedom (df). The degrees of freedom represent the amount of independent information available to estimate the variance. When computing $s$ from $n$ observations, we use up one degree of freedom estimating the mean, leaving $n - 1$ degrees of freedom for estimating variance.

The t-distribution has several key properties:

1. Bell-shaped and symmetric: Like the normal distribution, centered at zero under the null hypothesis
2. Heavier tails: More probability in the extreme values than the normal distribution
3. Parameterized by degrees of freedom: The shape depends on df, with smaller df meaning heavier tails
4. Converges to normal: As $df \to \infty$, the t-distribution approaches the standard normal

The heavier tails account for the uncertainty added by estimating variance: with small samples, $s$ can deviate substantially from $\sigma$, so extreme t-values become more probable. The t-distribution therefore requires larger critical values to reject the null hypothesis.

Consider what happens when $s$ happens to underestimate $\sigma$:

• The denominator $s/\sqrt{n}$ is too small
• The t-statistic is inflated
• We might incorrectly reject $H_0$

When $s$ overestimates $\sigma$:

• The denominator is too large
• The t-statistic is deflated
• We might miss a real effect

These errors don't favor either direction on average, but they add variability. The t-distribution captures exactly this additional spread. With only 5 observations (df = 4), the sample standard deviation is quite unreliable, so the t-distribution has very heavy tails. With 100 observations (df = 99), $s$ is a much more reliable estimate of $\sigma$, and the t-distribution is nearly identical to the normal.

The heavier tails translate directly into larger critical values. For a two-sided test at $\alpha = 0.05$:

With df = 5, you need a t-statistic 31% larger than the z-critical value to reject $H_0$. This penalty appropriately reflects the unreliability of variance estimates from small samples. By df = 30, the difference is only about 4%, which is why older textbooks sometimes said "use z for n > 30." Modern practice uses the t-distribution regardless, as computational tools make it trivially easy.