Z-test Calculator
One-sample hypothesis testing for population means
Test Configuration
Sample Statistics
The arithmetic mean of your sample
The mean specified in the null hypothesis
Number of observations in your sample
Known population standard deviation (σ > 0)
Z-test Results
Test Statistics
Critical Values
Sample Information
Hypothesis Statements
Interpretation
At the 95% confidence level, we reject the null hypothesis. There is significant evidence that the population mean is not equal to 1000.
Example: Bottle Filling Machine
Problem
Claim: Average volume = 1000 ml
Suspicion: Average volume < 1000 ml
Sample: 9 bottles
Sample mean: 980 ml
Population σ: 30 ml
Solution
H₀: μ = 1000
H₁: μ < 1000 (left-tailed)
Z: (980-1000)/(30/√9) = -2
p-value: 0.0228
Conclusion
When to Use Z-test
Known Population σ
Population standard deviation is known
Normal Distribution
Data follows normal distribution
Large Sample
n ≥ 30 (Central Limit Theorem applies)
Independent Observations
Each data point is independent
Common Critical Values
Two-tailed (α)
One-tailed (α)
Understanding Z-tests
What is a Z-test?
A Z-test is a statistical hypothesis test used to determine whether a sample mean significantly differs from a population mean when the population standard deviation is known. It's based on the standard normal distribution.
Key Assumptions
- •Independence: Observations are independent
- •Normality: Data follows normal distribution or large sample (n ≥ 30)
- •Known σ: Population standard deviation is known
Test Statistic Formula
Z-test Formula
Z = (x̄ - μ₀) / (σ/√n)
x̄ = sample mean
μ₀ = hypothesized population mean
σ = population standard deviation
n = sample size
Standard Error
SE = σ / √n
Standard error of the mean
P-value Interpretation
- •p < α: Reject null hypothesis (significant result)
- •p ≥ α: Fail to reject null hypothesis
- •Small p-value: Strong evidence against H₀
- •Large p-value: Weak evidence against H₀
Z-test vs t-test
Use Z-test when:
- • Population σ is known
- • Large sample size (n ≥ 30)
- • Data is normally distributed
Use t-test when:
- • Population σ is unknown
- • Small sample size (n < 30)
- • Sample standard deviation used