F-distribution Properties Analysis

Let's evaluate each statement regarding the F-distribution:

Statement (a): F-distribution has 2 sets of degrees of freedom

This statement is true. The F-distribution is characterized by two independent parameters known as degrees of freedom: the numerator degrees of freedom ($df_1$) and the denominator degrees of freedom ($df_2$).

Statement (b): F-distribution has 1 set of degree of freedom

This statement is false. As established in statement (a), it requires two sets of degrees of freedom.

Statement (c): F-distribution is used to test the null-hypothesis that the population variances are equal

This statement is true. A primary application of the F-distribution is in hypothesis testing, specifically for comparing the variances of two populations. It's also fundamental in Analysis of Variance (ANOVA) for testing the equality of multiple population means, which indirectly involves variance comparisons.

Statement (d): F-distribution is an all positive distribution

This statement is true. The F-statistic is calculated as a ratio of variances (or mean squares), and variances are always non-negative. Therefore, the values generated by the F-distribution are always greater than or equal to zero ($F \ge 0$).

Statement (e): F-distribution is both positive and negative distribution

This statement is false. Since the F-statistic cannot be negative, the F-distribution does not include negative values.

Conclusion on F-distribution Statements

Based on the analysis, the true statements about the F-distribution are:

• (a) It has 2 sets of degrees of freedom.
• (c) It is used to test the null-hypothesis that the population variances are equal.
• (d) It is an all positive distribution.

Therefore, the correct combination includes statements (a), (c), and (d).