statistics hypothesis testing

F-test is named after the more prominent analyst R.A. Fisher. F-test is utilized to test whether the two autonomous appraisals of populace change contrast altogether or whether the two examples may be viewed as drawn from the typical populace having the same difference. For doing the test, we calculate F-statistic is defined as:


${F} = \frac{Larger\ estimate\ of\ population\ variance}{smaller\ estimate\ of\ population\ variance} = \frac{{S_1}^2}{{S_2}^2}\ where\ {{S_1}^2} \gt {{S_2}^2}$


Its testing procedure is as follows:

  1. Set up null hypothesis that the two population variance are equal. i.e. ${H_0: {\sigma_1}^2 = {\sigma_2}^2}$
  2. The variances of the random samples are calculated by using formula:

    ${S_1^2} = \frac{\sum(X_1- \bar X_1)^2}{n_1-1}, \\[7pt]
    \ {S_2^2} = \frac{\sum(X_2- \bar X_2)^2}{n_2-1}$

  3. The variance ratio F is computed as:

    ${F} = \frac{{S_1}^2}{{S_2}^2}\ where\ {{S_1}^2} \gt {{S_2}^2}$

  4. The degrees of freedom are computed. The degrees of freedom of the larger estimate of the population variance are denoted by v1 and the smaller estimate by v2. That is,
    1. ${v_1}$ = degrees of freedom for sample having larger variance = ${n_1-1}$
    2. ${v_2}$ = degrees of freedom for sample having smaller variance = ${n_2-1}$
  5. Then from the F-table given at the end of the book, the value of ${F}$ is found for ${v_1}$ and ${v_2}$ with 5% level of significance.
  6. Then we compare the calculated value of ${F}$ with the table value of ${F_.05}$ for ${v_1}$ and ${v_2}$ degrees of freedom. If the calculated value of ${F}$ exceeds the table value of ${F}$, we reject the null hypothesis and conclude that the difference between the two variances is significant. On the other hand, if the calculated value of ${F}$ is less than the table value, the null hypothesis is accepted and concludes that both the samples illustrate the applications of F-test.
  7. Example

    Problem Statement:

    In a sample of 8 observations, the entirety of squared deviations of things from the mean was 94.5. In another specimen of 10 perceptions, the worth was observed to be 101.7 Test whether the distinction is huge at 5% level. (You are given that at 5% level of centrality, the basic estimation of ${F}$ for ${v_1}$ = 7 and ${v_2}$ = 9, ${F_.05}$ is 3.29).


    Let us take the hypothesis that the difference in the variances of the two samples is not significant i.e. ${H_0: {\sigma_1}^2 = {\sigma_2}^2}$

    We are given the following:

    ${n_1} = 8 , {\sum {(X_1 – \bar X_1)}^2} = 94.5, {n_2} = 10, {\sum {(X_2 – \bar X_2)}^2} = 101.7, \\[7pt]
    {S_1^2} = \frac{\sum(X_1- \bar X_1)^2}{n_1-1} = \frac {94.5}{8-1} = \frac {94.5}{7} = {13.5}, \\[7pt]
    {S_2^2} = \frac{\sum(X_2- \bar X_2)^2}{n_2-1} = \frac {101.7}{10-1} = \frac {101.7}{9} = {11.3}$

    Applying F-Test

    ${F} = \frac{{S_1}^2}{{S_2}^2} = \frac {13.5}{11.3} = {1.195}$

    For ${v_1}$ = 8-1 = 7, ${v_2}$ = 10-1 = 9 and ${F_.05}$ = 3.29. The Calculated value of ${F}$ is less than the table value. Hence, we accept the null hypothesis and conclude that the difference in the variances of two samples is not significant at 5% level.

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