Inference using normal and t-distributions
The t-distribution
- When a sample is small and the population variance is unknown, use the $t$-distribution instead of the normal.
- A confidence interval for the mean:
$$\bar{x} \pm t\,\frac{s}{\sqrt{n}}$$
- where $t$ comes from the $t$-tables with $n - 1$ degrees of freedom.
Practice
You use the t-distribution (rather than the normal) when the sample is:
The t-distribution handles small samples with an unknown population variance.
Practice
A sample n = 10 has s = 4. With t = 2.262, what is the margin t × s/√n? (2 dp)
2.262 × 4/√10 = 2.262 × 1.265 ≈ 2.86.
Practice
For a single-sample t-interval with n = 10, the degrees of freedom are:
Degrees of freedom = n − 1 = 10 − 1 = 9.
Comparing populations
- Use a 2-sample or paired-sample $t$-test to compare two populations.
- Find a pooled estimate of the shared variance when appropriate.
You've got it
Key idea
- small sample + unknown variance → use the $t$-distribution ($n-1$ degrees of freedom)
- CI for the mean: $\bar{x} \pm t\dfrac{s}{\sqrt{n}}$
- compare two populations with a 2-sample or paired $t$-test