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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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