Critical Value Calculator
Calculate critical values for Z, t, chi-square, and F distributions for hypothesis testing
Critical Value Parameters
Critical Value Results
Rejection Region
(-∞, 0.8874] ∪ [-0.8874, ∞)
Interpretation
For a two-tailed test with standard normal (Z) distribution at α = 0.05 (95.0% confidence level), reject the null hypothesis if the test statistic falls in the rejection region: (-∞, 0.8874] ∪ [-0.8874, ∞).
Example Calculation
Two-tailed t-test
α: 0.05
df: 15
Distribution: t-Student
Critical Values
±2.1314
Rejection region: (-∞, -2.1314] ∪ [2.1314, ∞)
Decision Rule
Reject H₀ if |t| ≥ 2.1314
Distribution Guide
Z (Normal)
Use when σ is known and large sample size
t-Student
Use when σ is unknown or small sample size
Chi-Square
Use for variance tests and goodness of fit
F-Distribution
Use for comparing variances and ANOVA
Common Alpha Levels
Understanding Critical Values
What are Critical Values?
Critical values are cut-off points that define the rejection region in hypothesis testing. They represent the threshold beyond which we reject the null hypothesis. Critical values depend on the chosen significance level (α), the type of test (one-tailed or two-tailed), and the underlying probability distribution.
How to Use Critical Values
- •Calculate your test statistic from sample data
- •Compare it to the critical value(s)
- •If test statistic falls in rejection region, reject H₀
- •Otherwise, fail to reject H₀
Test Type Selection
Two-tailed Test
H₁: μ ≠ μ₀
Tests for difference in either direction
Right-tailed Test
H₁: μ > μ₀
Tests for increase
Left-tailed Test
H₁: μ < μ₀
Tests for decrease
Z-Distribution
- •Population σ known
- •Large sample size (n ≥ 30)
- •Normal population
- •Standard: N(0,1)
t-Distribution
- •Population σ unknown
- •Small sample size
- •Normal population
- •df = n - 1
Chi-Square
- •Variance tests
- •Goodness of fit
- •Independence tests
- •Right-skewed
F-Distribution
- •Variance comparison
- •ANOVA tests
- •Regression analysis
- •Two df parameters