The critical value in inference is a threshold that's compared to a test statistic to decide whether to reject the null hypothesis.
Here's a breakdown:
-
Definition: The critical value is a point (or points) on the scale of the test statistic beyond which we reject the null hypothesis. It defines the rejection region.
-
How it works:
- Calculate the Test Statistic: You calculate a test statistic (e.g., t-statistic, z-statistic, F-statistic, chi-square statistic) from your sample data.
- Determine the Critical Value(s): Based on the chosen significance level (alpha, often 0.05), the type of test (one-tailed or two-tailed), and the degrees of freedom (if applicable), you find the corresponding critical value(s) from a statistical table or using statistical software.
- Compare: You compare the absolute value of your calculated test statistic to the critical value.
- Decision:
- If the absolute value of the test statistic is greater than the critical value, you reject the null hypothesis. This indicates that the observed result is statistically significant, and it's unlikely to have occurred by chance alone if the null hypothesis were true.
- If the absolute value of the test statistic is less than or equal to the critical value, you fail to reject the null hypothesis. This means there's not enough evidence to conclude that the null hypothesis is false.
-
Significance Level (Alpha): The significance level (α) represents the probability of rejecting the null hypothesis when it's actually true (Type I error). The critical value is directly related to the significance level. A lower significance level (e.g., 0.01 instead of 0.05) will result in a larger critical value, making it harder to reject the null hypothesis.
-
One-tailed vs. Two-tailed Tests: The critical value depends on whether you're conducting a one-tailed or two-tailed test:
- Two-tailed test: Tests for differences in either direction (e.g., "is the mean different from X?"). It has two critical values, one on each tail of the distribution.
- One-tailed test: Tests for differences in a specific direction (e.g., "is the mean greater than X?" or "is the mean less than X?"). It has one critical value on the tail corresponding to the direction of the hypothesis.
-
Example: Suppose you're conducting a two-tailed t-test with α = 0.05 and 20 degrees of freedom. The critical values would be approximately ±2.086. If your calculated t-statistic is 2.5, you would reject the null hypothesis because 2.5 > 2.086.
-
In Summary: The critical value acts as a cutoff point. If the test statistic falls beyond this cutoff, the result is considered statistically significant, suggesting that the null hypothesis should be rejected.
