"In Exercises 13–16, find the critical F-value for a two-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.


α=0.10,d.f.N=15,d.f.D=27"

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding the critical F-value for a two-tailed test. The level of significance (α) is 0.10, and the degrees of freedom for the numerator (d.f.N) and denominator (d.f.D) are 15 and 27, respectively.
2. Step 2: Recognize that for a two-tailed test, the significance level (α) is split equally between the two tails of the F-distribution. This means each tail will have an area of α/2 = 0.10/2 = 0.05.
3. Step 3: Use an F-distribution table or statistical software to find the critical F-values. For a two-tailed test, you will need to find two critical values: one for the upper tail and one for the lower tail. The degrees of freedom for the numerator (d.f.N = 15) and denominator (d.f.D = 27) are used to locate the critical values.
4. Step 4: For the upper critical value, look up the F-value corresponding to α/2 = 0.05 in the upper tail with d.f.N = 15 and d.f.D = 27. For the lower critical value, use the reciprocal property of the F-distribution: F_lower = 1 / F_upper.
5. Step 5: Verify your results by ensuring the total area under the F-distribution curve outside the critical values equals the significance level (α = 0.10). This confirms the critical values are correct for the two-tailed test.

Key Concepts

Critical F-value

The critical F-value is a threshold used in hypothesis testing to determine whether to reject the null hypothesis. It is derived from the F-distribution, which is used when comparing variances between two groups. In a two-tailed test, the critical F-value is found at both ends of the distribution, corresponding to the specified level of significance (α).

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 the F-test, there are two types of degrees of freedom: d.f.N (numerator) and d.f.D (denominator), which correspond to the number of groups being compared and the total number of observations, respectively.

Level of Significance (α)

The level of significance (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 or 0.10, which defines the cutoff for determining whether the observed data is statistically significant. In this case, α=0.10 indicates a 10% risk of making a Type I error.