This table provides critical values for the $F_{m, \infty}$ distribution, which is commonly used in statistical tests, particularly in ANOVA (Analysis of Variance) and regression analysis. Here’s how to use it:

1. Determine the Degrees of Freedom:

   • The degrees of freedom (DF) is a parameter that you need to locate the correct critical value in the table. In an $F$-test, the degrees of freedom come from the sample sizes in the groups you're comparing.
   • In this table, only one set of degrees of freedom (for the numerator, $m$) is provided. The table assumes the denominator degrees of freedom approach infinity, which simplifies calculations in many cases.

2. Choose a Significance Level:

   • Decide the significance level for your test, typically 10%, 5%, or 1%, representing the probability of rejecting the null hypothesis when it is actually true (Type I error).
   • In the table, each significance level has its own column. A lower significance level (e.g., 1%) means a stricter criterion for significance.

3. Locate the Critical Value:

   • Once you have the degrees of freedom and the significance level, find the intersection of the appropriate row and column to get the critical value.
   • For example, if you have 5 degrees of freedom and are testing at the 5% significance level, the critical value would be 2.37.

4. Compare with the Calculated $F$-Statistic:

   • After finding the critical value, compare it with the $F$-statistic calculated from your data.
   • If the $F$-statistic is greater than or equal to the critical value, you reject the null hypothesis at the chosen significance level. Otherwise, you fail to reject the null hypothesis.

Example Usage

Suppose we have:

• Degrees of freedom $m = 4$
• Significance level $\alpha = 0.05$
• $F$-statistic calculated from the data = 3.0

From the table:

• For $m = 4$ and a 5% significance level, the critical value is 2.37.
• Since 3.0 > 2.37, we reject the null hypothesis at the 5% significance level.

Would you like more details on a specific part, or do you have any questions?

Related Questions

1. How do we interpret the $F$-statistic in ANOVA?
2. How does the choice of significance level affect hypothesis testing?
3. What happens when the denominator degrees of freedom is not infinite?
4. How are $F$-tables constructed for finite degrees of freedom?
5. Why do we reject the null hypothesis when the $F$-statistic exceeds the critical value?

Tip

For small sample sizes or non-infinite denominator degrees of freedom, use a table or software that provides the $F$ distribution critical values for both numerator and denominator degrees of freedom.