Hypothesis Testing Framework for F-test

Using hypothesis testing, the f test is performed to verify that the variances are equivalent. For various hypothesis tests, the f test formula is provided as follows:

Left Tailed Test:

Null Hypothesis: H₀ :
Alternate Hypothesis: H₁ : 
Decision-Making Standard: The null hypothesis is to be rejected if the f statistic is less than the f critical value.

Right Tailed Test:

Null Hypothesis: H₀ :
Alternate Hypothesis: H₁ : \sigma_{2}^2 " title="Rendered by QuickLaTeX.com" height="29" width="89" style="vertical-align: 26px;">
Decision-Making Standard: Dismiss the null hypothesis if the f test statistic is greater than the f test critical value.

Two Tailed Test:

Null Hypothesis: H₀ : \sigma_{1}^2 = \sigma_{2}^2
Alternate Hypothesis: H₁ :
Decision-Making Standard: When the f test statistic surpasses the f test critical value, the null hypothesis is declared invalid.

F Test Statistic Formula Assumptions

Several assumptions are used in the F Test equation. For the F-test Formula to be utilized, the population distribution needs to be normal. Independent events should be the basis for the test samples. Apart from this, the following considerations should also be taken into consideration.

• It is simpler to calculate right-tailed tests. By pushing the bigger variance into the numerator, the test is forced to be right tailed.
• Before the critical value is determined in two-tailed tests, alpha is divided by two.
• Squares of standard deviations equal variances.

Steps to calculate F-Test

Step 1: Use Standard deviation (σ) and find variance (σ2) of the data. (if not already given)

Step 2: Determine the null and alternate hypothesis.

•   H0: no difference in variances.
•   H1: difference in variances.

Step 3: Find Fcalc using Equation 1 (F-value).

NOTE : While calculating F_calc, divide the larger variance with small variance as it makes calculations easier.

Step 4: Find the degrees of freedom of the two samples.

Step 5: Find Ftable value using d1 and d2 obtained in Step-4 from the F-distribution table. Take learning rate, α = 0.05 (if not given) 

Looking up the F-distribution table: 

In the F-Distribution table (Link here), refer the table as per the given value of α in the question. 

• d1 (Across) = df of the sample with numerator variance.  (larger)
• d2 (Below) = df of the sample with denominator variance. (smaller)

Consider the F-Distribution table given below, while performing One-Tailed F-Test.

GIVEN: 
α = 0.05
d₁ = 2
d₂ = 3

Then, F_table = 9.55

Step 6: Interpret the results using Fcalc and Ftable.

Interpreting the results:

If Fcalc < Ftable :
    Cannot reject null hypothesis.
    ∴ Variance of two populations are similar. 
    
If Fcalc > Ftable :
    Reject null hypothesis. 
    ∴ Variance of two populations are not similar.

F test is a statistical test that is used in hypothesis testing, that determines whether or not the variances of two populations or two samples are equal. An f distribution is what the data in a f test conforms to. By dividing the two variances, this test compares them using the f statistic. Depending on the details of the situation, a f-test can be one-tailed or two-tailed. The article will provide further information on the f test, the f statistic, its calculation, critical value, and how to use it to test hypotheses.