T-Distribution Formula
The Student's t-distribution formula for hypothesis testing with small samples when population standard deviation is unknown.
The T-Statistic Formula
t = (x̄ - μ₀) / (s / √n)
Variables
| Symbol | Meaning |
|---|---|
| t | The t-statistic (test statistic) |
| x̄ | Sample mean |
| μ₀ | Hypothesized population mean |
| s | Sample standard deviation |
| n | Sample size |
| s / √n | Standard error of the mean |
Two-Sample T-Test
t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)
Compares the means of two independent groups (Welch's t-test).
Degrees of Freedom
- One-sample: df = n - 1
- Two-sample (Welch's): df = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁-1) + (s₂²/n₂)²/(n₂-1)]
When to Use T vs Z
- Use t-test when population σ is unknown (most real-world situations)
- Use z-test when population σ is known or n is very large (>30)
- The t-distribution approaches the z-distribution as sample size increases
Example
A sample of 25 students scored a mean of 78 with s = 10. Is this different from the expected mean of 75?
t = (78 - 75) / (10 / √25) = 3 / 2 = 1.5
df = 25 - 1 = 24
Two-tailed p-value ≈ 0.147
Since p > 0.05, we fail to reject the null hypothesis. The difference is not statistically significant.