z-Test
Test a mean with known σ.
Key idea
Test a mean with known σ.
In depth
Used to test a mean (σ known) or proportion with large samples: z = (x̄ − μ₀)/(σ/√n). Compare to the standard normal distribution.
Worked example
x̄ = 52, μ₀ = 50, σ = 8, n = 64: z = 2/1 = 2, p ≈ 0.046 (two-sided).
Formula
Real-world uses
- Testing whether a batch’s average weight matches the label when the standard deviation is known
Common mistake
Using it when σ is unknown and n is small; use a t-test.
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 z-Test?
Test a mean with known σ. It is a topic at the expert level in Probability & Statistics.
What is the formula for z-Test?
The key formula is z = (x̄−μ₀)/(σ/√n).
Where is z-Test used in real life?
Real-world examples: Testing whether a batch’s average weight matches the label when the standard deviation is known.
What should I learn before z-Test?
Make sure you are comfortable with Type I and Type II Errors, the previous topic in Probability & Statistics.
What is a common mistake with z-Test?
Using it when σ is unknown and n is small; use a t-test.
How can I practise z-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 z-Test?
Move on to t-Test 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.