One-Sample T-Test
Hypothesis Testing
The Everyday Workhorse for Mean Comparisons
The one-sample t-test is the most commonly used hypothesis test, applicable whenever σ is unknown — which is nearly always in practice. It provides reliable inference even with small samples.
- Quality Assurance — Testing whether product dimensions meet specifications
- Education Research — Comparing class performance against national averages
- Healthcare — Evaluating whether patient measurements differ from healthy norms
When σ is unknown, the t-test is always the right choice.
The one-sample t-test tests whether a population mean μ equals a hypothesized value μ₀, when σ is unknown (the typical real-world case).
The t-distribution has heavier tails than the normal, accounting for uncertainty in estimating σ with s.
Assumptions
Checking Normality
Complete Worked Example
T-Distribution: Degrees of Freedom Effect
x = np.linspace(-5, 5, 500)
fig, ax = plt.subplots(figsize=(10, 5))
ax.plot(x, stats.norm.pdf(x), 'k-', linewidth=2, label='Normal (df=∞)')
for df in [1, 2, 5, 10, 30]:
ax.plot(x, stats.t.pdf(x, df=df), linewidth=1.5, label=f't (df={df})')
ax.set_xlim(-5, 5)
ax.set_title('T-Distribution vs Normal: Heavy Tails Decrease with df')
ax.legend()
ax.set_xlabel('t')
ax.set_ylabel('Density')
plt.savefig('t_distribution.png', dpi=150)
plt.show()