p-value Calculator
Calculate p-values from test statistics for Z, t, chi-square, and F distributions
Calculate p-value from Test Statistics
Choose the distribution of your test statistic
The computed test statistic from your data
Type of alternative hypothesis
Threshold for statistical significance
p-value Results
Test Information
Interpretation
Moderate evidence against the null hypothesis
Decision: Reject the null hypothesis H₀ at the 5% significance level.
Step-by-Step Example
Z-test Example
Problem: Testing if population mean differs from 100
Sample: n=36, x̄=97, σ=12
Null hypothesis: H₀: μ = 100
Alternative: H₁: μ ≠ 100 (two-tailed)
Solution Steps
1. Calculate Z-score: Z = (x̄ - μ₀)/(σ/√n) = (97-100)/(12/√36) = -1.5
2. For two-tailed test: p-value = 2 × Φ(-|Z|) = 2 × Φ(-1.5)
3. p-value = 2 × 0.0668 = 0.1336
4. Since p = 0.1336 > α = 0.05, fail to reject H₀
p-value Interpretation
Common Test Types
Z-test
• Large samples (n ≥ 30)
• Known population σ
• Population means
t-test
• Small samples or unknown σ
• df = n - 1 (one sample)
• Population means
Chi-square
• Goodness of fit tests
• Independence tests
• Variance tests
F-test
• ANOVA tests
• Variance equality
• Regression significance
Quick Tips
p-value measures evidence against null hypothesis
Lower p-values indicate stronger evidence
Choose significance level before testing
Consider practical significance too
Understanding p-values
What is a p-value?
The p-value is the probability that the test statistic would produce values at least as extreme as the observed value, assuming the null hypothesis is true. It measures the strength of evidence against the null hypothesis.
How to Interpret
- •Small p-value: Strong evidence against H₀
- •Large p-value: Weak evidence against H₀
- •p ≤ α: Reject null hypothesis
- •p > α: Fail to reject null hypothesis
Calculation Methods
p-values are calculated using cumulative distribution functions (CDFs) of the appropriate statistical distribution. The formula depends on whether you're conducting a one-tailed or two-tailed test.