t-Test
Test means with unknown σ.
Key idea
Test means with unknown σ.
In depth
When σ is unknown, estimate it with s and use t = (x̄ − μ₀)/(s/√n) with n − 1 degrees of freedom. The t distribution has heavier tails, accounting for the extra uncertainty.
Worked example
x̄ = 52, s = 8, n = 16: t = 2/2 = 1.0, df = 15, p ≈ 0.33.
Formula
Real-world uses
- Comparing average exam scores of two classes with small samples in education research
Common mistake
Using z critical values with small samples.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is t-Test?
Test means with unknown σ. It is a topic at the expert level in Probability & Statistics.
What is the formula for t-Test?
The key formula is t = (x̄−μ₀)/(s/√n).
Where is t-Test used in real life?
Real-world examples: Comparing average exam scores of two classes with small samples in education research.
What should I learn before t-Test?
Make sure you are comfortable with z-Test, the previous topic in Probability & Statistics.
What is a common mistake with t-Test?
Using z critical values with small samples.
How can I practise t-Test?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after t-Test?
Move on to Two-Sample Tests next.
Related expert topics in Probability & Statistics
- Skewness and Kurtosis – Measures of asymmetry and tail heaviness.
- Residuals and R² – Error of a fit and proportion of variance explained.
- Simpson’s Paradox – A trend reverses when groups are combined.
- Law of Total Probability – Split an event over a partition.
- Bayes’ Theorem – Update beliefs with evidence.
- Medical Tests and Base Rates – Why positive results can still be probably false.