Table of Contents
What Is a Critical Value?
A critical value is the boundary that defines the rejection region in hypothesis testing. If the test statistic exceeds the critical value, we reject the null hypothesis. Critical values depend on the significance level (alpha), test direction, and distribution used.
They translate the abstract alpha into concrete thresholds, enabling researchers to make binary decisions about hypotheses. The choice of alpha (typically 0.05) represents the acceptable risk of a Type I error (false positive).
Test Types
| Type | Rejection Region | Critical Value |
|---|---|---|
| Two-Tailed | Both extremes | ±z(α/2) |
| Left-Tailed | Left extreme | -z(α) |
| Right-Tailed | Right extreme | +z(α) |
Common Critical Values
| α | Two-Tailed | One-Tailed |
|---|---|---|
| 0.10 | ±1.645 | 1.282 |
| 0.05 | ±1.960 | 1.645 |
| 0.01 | ±2.576 | 2.326 |
How to Use Critical Values
1) State null and alternative hypotheses. 2) Choose significance level. 3) Determine test direction. 4) Find critical value. 5) Compare test statistic to critical value. 6) Reject null if statistic exceeds critical value.
Frequently Asked Questions
P-value vs critical value approach?
The critical value approach sets a fixed boundary; the p-value approach calculates exact probability of the observed data. Both lead to the same conclusion, but p-values provide more information about evidence strength.
When to use t instead of z?
Use t-distribution when sample size is small (n < 30) and population standard deviation is unknown. For large samples, t approximates z closely.
Does smaller alpha mean larger critical value?
Yes. Smaller alpha makes rejection harder, requiring more extreme test statistics. This reduces Type I errors but increases Type II errors (missed true effects).