Question

Obtain as much information as you can about the \(P\) -value for an upper-tailed \(F\) test in each of the following situations: a. \(\mathrm{df}_{1}=3, \mathrm{df}_{2}=15,\) calculated \(F=4.23\) b. \(\mathrm{df}_{1}=4, \mathrm{df}_{2}=18,\) calculated \(F=1.95\) c. \(\quad \mathrm{df}_{1}=5, \mathrm{df}_{2}=20,\) calculated \(F=4.10\) d. \(\mathrm{df}_{1}=4, \mathrm{df}_{2}=35,\) calculated \(F=4.58\)

Short answer

P-values will vary according to the F distribution table used. If the calculated F value falls beyond the critical F value in the table, we infer that P-value is less than the level of significance used to look up the critical F value. If our calculated F value is less than the critical F value in the table, then we infer that our P value is greater than the level of significance used to look up the critical F value. Without the actual F distribution table, it is not possible to give a specific P-value estimate.

Step-by-step solution

Understand the F distribution

This step involves understanding that the F distribution is skewed to the right, and not symmetrical. Its shape depends on the degrees of freedom, so it differs from situation to situation. There are F distribution tables that show the critical value for different alpha levels and different degrees of freedom.

Locate critical value in the table

For each of the scenarios, the F distribution table must be consulted to find the P-value. From the table, identify the row corresponding to \(df_1\) and the column indexed by \(df_2\). The location where the row and column intersect is the critical value of the F distribution, denoted \(F_{crit}\).

Calculate P-value

From the table value obtained in step 2, determine if the calculated F value is greater or less than the critical value \(F_{crit}\). If the computed F value is greater than \(F_{crit}\), this suggests that the calculated p-value is less than the level of significance used to look up \(F_{crit}\). If the computed F is less than \(F_{crit}\), then the p-value is greater than the level of significance used. Note not all F distribution tables show actual probability (P-value), however using the method mentioned can give an estimate.

Key concepts

F Distribution

The F distribution is a probability distribution that appears in analysis of variance (ANOVA) and tests of equality of variances. One key feature of the F distribution is that it is not symmetrical—it is skewed to the right. This means there is a longer tail on the right side, affecting how values are distributed across the range.

The shape of the F distribution is determined by two different types of "degrees of freedom" which are associated with the numerator and the denominator in the F-test calculations. Because of this, the curve can vary greatly based on the context of the data being analyzed. It can be helpful to visualize the F distribution as a family of curves that change shape depending on these degrees of freedom. Understanding the unique characteristics of the F distribution helps us interpret statistical tests effectively, especially as it is used to compare two variability sources in the context of hypothesis testing.

Degrees of Freedom

Degrees of freedom (df) are integral to understanding the F distribution and its behavior. In an F-test, degrees of freedom are specified in two ways: one for the numerator (df1) and one for the denominator (df2).

Essentially, degrees of freedom refer to the number of independent values or quantities available to estimate another statistic or parameter. They play a crucial role in determining how spread out the F distribution is. For example:

• High degrees of freedom in the numerator result in a less peaked distribution.
• High degrees of freedom in the denominator make the distribution closer to normal.

Degrees of freedom arise from the sample sizes and the structure of the data, impacting both the critical value from the F distribution and the interpretation of statistical tests.

Critical Value

The critical value is a point on the scale of the test statistic beyond which we reject the null hypothesis in favor of the alternative. For F-tests, critical values are extracted from F distribution tables based on the selected significance level and degrees of freedom.

Critical values serve as a threshold to decide if our sample data indicates a significant effect or not. Let's break down the process of finding a critical value:

• First, identify the significance level (usually denoted by \( \alpha \), such as 0.05 or 5%).
• Next, locate the row for the numerator degrees of freedom (df1) in the F distribution table, and a column for the denominator degrees of freedom (df2).
• The point where these intersect provides the critical value \( F_{crit} \).

If the calculated F statistic from a test exceeds this critical value, we may conclude that there is evidence to reject the null hypothesis, showing that a significant difference or effect exists.

Significance Level

In hypothesis testing, the significance level is a threshold risk we are willing to take of incorrectly rejecting a true null hypothesis, this is often represented by \( \alpha \). Common significance levels are 0.05, 0.01, or 0.10, meaning 5%, 1%, or 10% risk respectively.

The significance level is chosen by the researcher before the test begins and is crucial because it influences the critical value extracted from the F distribution table. This level decides the **stringency** of the test:

• A lower significance level (e.g., 0.01) means stricter testing criteria, reducing the probability of false positives.
• A higher significance level (e.g., 0.10) allows for more false positives, increasing the chance of detecting a true effect when one exists.

Understanding the significance level helps explain how and why we decide to reject or not reject the null hypothesis during statistical testing.