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gFor an F distribution, the number of degrees of freedom for the numerator Question 9 options: must be larger than the number of degrees of freedom for the denominator. must be smaller than the number of degrees of freedom for the denominator. must be equal to the number of degrees of freedom for the denominator. can be larger, smaller, or equal to the number of degrees of freedom for the denominator.

Sagot :

Answer: Choice D

can be larger, smaller, or equal to the number of degrees of freedom for the denominator.

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Explanation:

The F distribution is useful to see if two groups have the same variance or not. By extension, we can compare standard deviations between two groups.

Let [tex]n_1[/tex] and [tex]n_2[/tex] be the sample sizes of the two groups. Those two n values aren't necessarily equal in value. All that matters is that they are positive whole numbers. The associated degrees of freedom are [tex]n_1 - 1[/tex] and [tex]n_2 - 1[/tex]. Convention usually has the numerator with the larger variance, but we don't have to worry about that when addressing this particular question.

All we care about is if the quantities [tex]n_1 - 1[/tex] and [tex]n_2 - 1[/tex] are either:

  • A) equal
  • B) the first is larger than the second
  • C) the first is smaller than the second

It turns out that there aren't any restrictions on the values of n, which in turn means there aren't any restrictions on [tex]n_1 - 1[/tex] and [tex]n_2 - 1[/tex]. Therefore, the degrees of freedom for the numerator can be larger, smaller, or equal to the number of degrees of freedom for the denominator.

This is why the answer is choice D

Further confirmation of this is to look at a standard F distribution table. Note that any cell is possible and we could have either of the three cases mentioned happen.

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