In Exercises 9–12, find the critical F-value for a right-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.


α=0.05,d.f.N=20,d.f.D=25

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding the critical F-value for a right-tailed test using the given level of significance (α = 0.05) and degrees of freedom for the numerator (d.f.N = 20) and denominator (d.f.D = 25). The critical F-value is the value that separates the rejection region from the non-rejection region in an F-distribution.
2. Step 2: Recall the formula and concept. The F-distribution is used in hypothesis testing, particularly in ANOVA and regression analysis. The critical F-value depends on the level of significance (α), the degrees of freedom for the numerator (d.f.N), and the degrees of freedom for the denominator (d.f.D).
3. Step 3: Use an F-distribution table or statistical software. Locate the row corresponding to d.f.N = 20 and the column corresponding to d.f.D = 25 in the F-distribution table for α = 0.05. Alternatively, use statistical software like R, Python, or Excel to compute the critical F-value.
4. Step 4: Interpret the result. The critical F-value represents the threshold above which the test statistic would lead to rejecting the null hypothesis in a right-tailed test. Ensure you understand its role in hypothesis testing.
5. Step 5: Verify your result. Double-check the table or software output to ensure accuracy. If using software, confirm that the inputs (α, d.f.N, d.f.D) are correctly entered.

Key Concepts

Critical F-value

The critical F-value is a threshold used in hypothesis testing to determine whether to reject the null hypothesis in an F-test. It is derived from the F-distribution, which is used to compare variances between two groups. The critical value is based on the chosen significance level (α) and the degrees of freedom for the numerator (d.f.N) and denominator (d.f.D). If the calculated F-statistic exceeds this critical value, the null hypothesis is rejected.

Degrees of Freedom

Degrees of freedom (d.f.) refer to the number of independent values or quantities that can vary in an analysis without violating any constraints. In the context of an F-test, d.f.N represents the degrees of freedom associated with the numerator (typically the group with more variance), while d.f.D represents the degrees of freedom for the denominator (the group with less variance). These values are crucial for determining the shape of the F-distribution and finding the critical F-value.

Significance Level (α)

The significance level (α) is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. It is a threshold set by the researcher, commonly at 0.05, which indicates a 5% risk of concluding that a difference exists when there is none. In hypothesis testing, the significance level helps determine the critical value needed to assess the results of the test, guiding the decision-making process regarding the null hypothesis.