# Two-tailed inverse CDF of F distribution

1. Aug 11, 2011

### Rasalhague

Two-tailed "inverse CDF" of F distribution

I'm working through Koosis: Statistics: A Self Teaching Guide, 4th edition. In Chapter 5, Koosis describes how to use a function which goes by the name of F.INV.RT(probability,deg_freedom1,deg_freedom2) in Excel 2010 to find the critical region for a given significance level, for a statistical test where the alternative hypothesis is that the standard deviation of the numerator population is greater than that of the denominator population. I've tried this on the example in the book, in section 16, and get the same result.

I've now come to sections 5.19-22 where he introduces the idea of a statistical test for the alternative that the standard deviations of a pair of populations are not equal.

In 5.19 he says the method is the same, except that "you double the probabilities when you use the F table." When I try this on the example in 5.20, I get a different result from the book. In this example, the size of both samples is 10. The significance level is 2%. In Excel 2010, I get F.INV.RT(0.04,9,9) = 3.438684. In Mathematica, I get InverseCDF[FRatioDistribution[9, 9], 1 - .04] = 3.43868. (This function in Mathematica produces the same results as the book for the one-tailed case.) The book's answer is 5.35; the critical region is the region greater than or equal to 3.35.

Is 5.35 a typo, or am I making a mistake? If the latter, how do I find the correct result in Excel and Mathematica?

2. Aug 11, 2011

### Rasalhague

Re: Two-tailed "inverse CDF" of F distribution

Correction: "The book's answer is 5.35; according to the book, the critical region is greater than or equal to 5.35."

3. Aug 11, 2011

### Rasalhague

Re: Two-tailed "inverse CDF" of F distribution

Epiphany! When he says "double the probabilities" to perform a 2-tailed F test, he means: to find the critical value for a given significance level, $\alpha$, you should input half of the number you would have used if this was a 1-tailed F test!

More precisely, suppose you have a function $g:(0,1)\rightarrow \mathbb{R}$, which you can use for a specific 1-tailed F test as follows: you input your desired significance level, $\alpha$, and it outputs the critical value, $g(\alpha)$, which, for this 1-tailed F test, corresponds to that significance level, $\alpha$. And let $f:(0,1)\rightarrow (0,1) \; | \; f(x)=x/2$. Then, if you input $\alpha$ into the composite function $g\circ f$, its value $g\circ f (\alpha)$ will be the critical value of the 2-tailed F test which has the same parameters as your original 1-tailed F test. (That is, the 1-tailed F test for which $g(\alpha)$ was the critical value.)

In other words, when performing a 2-tailed F test: if you input a given number, $\alpha$, into a function $g$, such that, for a 1-tailed F test, $g(\alpha)$ would be the critical value corresponding to significance level, $\alpha$, then - in this 2-tailed test - $g(\alpha)$ be the critical value which corresponds to the significance level $2\alpha$. So you should either "halve the input" xor "interpret the output as corresponding to a doubled input".

Last edited: Aug 11, 2011
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