Hypothesis Testing - F Stat about, change, either, includes

lxwarr30

I am confused about the F-Statistic:

Does the test statistic [ F = (Larger Variance) / (Smaller Variance) ] or the rejection point change depending on if the Null Hypothese includes either a <= or a >=?

Thanks

Mechanic

My question really gets to what would the rejection decision be on these two null hypotheses given two samples of 26 and 23 with sample variances of 10 and 4 respectively at 5% significance.

Pop Var 1 <= Pop Var 2
Pop Var 1 >= Pop Var 2

Wouldn't the test statistic be the same for both? (10/4 = 2.5) along with the rejection point of 2.02? I feel like I'm missing something as it would appear that the decision would be to reject both nulls.

homie

I'm just looking at the example null hypotheses listed in the curriculum and trying to wrap my head around the rationale for potential decisions based on the test. Text shows both one- and two-tailed hypotheses.

Londonrocks

you raise a very good point. frankly i am not too sure about this either, but one way to look at this would be since the higher variance is used in the numerator one is effectively forcing the test statistic into the right tail which then is the relevant null rejection region for a "greater than" alternative hypothesis. the way its worded in the curriculum, it appears that the same test statistic / rejection region is used to reject the null for both the greater than and less than alternatives. as you point out, that really doesnt make sense.

dcfox83

RMontgomery Wrote:
-------------------------------------------------------
> My question really gets to what would the
> rejection decision be on these two null hypotheses
> given two samples of 26 and 23 with sample
> variances of 10 and 4 respectively at 5%
> significance.
>
> Pop Var 1 <= Pop Var 2
> Pop Var 1 >= Pop Var 2
>
> Wouldn't the test statistic be the same for both?
> (10/4 = 2.5) along with the rejection point of
> 2.02? I feel like I'm missing something as it
> would appear that the decision would be to reject
> both nulls.

My two cents:

PV1 = 10, dof1 = 25
PV2 = 4, dof2 = 22

Case 1:
H0 : PV1 < = PV2 , Ha : PV1 > PV2 (makes sense since 10>4 but we want to see if it is statistically greater)
F stat = PV1/PV2 = 10/4 = 2.5
Critical Value = 2.02
So reject null. We get PV1 > PV2 (it is in fact greater statistically)

Case 2:
H0 : PV1 >= PV2, Ha : PV1 < PV2 (doesn't make sense to check since 10 > 4 so you will not frame the hypothesis like this)

Whenever you have two variances and you want to check if one is statistically greater than the other, you do it only if the sample suggests so.