Hypothesis Testing: T-Test

When to use the t-test (vs z-test)

Use the t-test when the population standard deviation $\sigma$ is unknown and you use the sample standard deviation $s$ instead. The t-distribution has heavier tails than the normal, especially for small samples.

Step-by-step: paired t-test

Step 1 — Hypotheses:

$H_0: \mu_d = 0$ (no improvement)

$H_a: \mu_d > 0$ (improvement — one-tailed)

Step 2 — Test statistic:

$t = \frac{\bar{d} - 0}{s_d / \sqrt{n}} = \frac{72 - 65}{10 / \sqrt{15}} = \frac{7}{2.582} = 2.711$

Step 3 — Degrees of freedom: $df = n - 1 = 14$.

Step 4 — Critical value: For one-tailed $\alpha = 0.05$, $df = 14$: $t_{\text{crit}} = 1.761$.

Step 5 — Decision: $t = 2.711 > 1.761$. Reject $H_0$.

$\boxed{\text{The tutoring program produced a statistically significant improvement.}}$

t-distribution vs normal

The t-distribution is wider (heavier tails) than the normal, reflecting extra uncertainty from estimating $\sigma$. As $n$ increases, $t$ approaches the normal.

Try it in the visualization

The t-distribution with the appropriate df is drawn. The test statistic and critical value are marked. The rejection region is shaded.