Calculating the "critical value" for F test

Question

I'm following along here: http://reliawiki.org/index.php/Multiple_Linear_Regression_Analysis#Calculation_of_the_Statistic

I've successfully made all of the calculations up to $f_0=211.9$. Now I need to calculate $f_{\alpha, k, n-(k+1)}$. On the page, this is simply given as:

The critical value for this test, corresponding to a significance level of 0.1, is:

$$f_{\alpha, k, n-(k+1)} = f_{0.1, 2, 14} = 2.726$$

I understand that $n$ is the number of observations and $k$ is the degrees of freedom. But, how is this number, $2.726$, calculated?

[[Edit: more questions]]

Is this value the result of the CDF for the F-distribution?:

$$F(x; d_1, d_2) = I_{\frac{d_1x}{d_1x+d_2}}(\frac{d_1}{2}, \frac{d_2}{2})$$

Where $x$ is $0.1$, $d_1$ is $2$ an $d_2$ is $14$? and $I$ is:

$$I_v(a,b) = \frac{B(v; a, b)}{B(a, b)}$$

And $B$ is:

$$B(v; a, b) = \int_{0}^{v}t^{a-1}(1-t)^{b-1}dt$$

and

$$B(a, b) = \int_{0}^{1}t^{a-1}(1-t)^{b-1}dt$$

Answer 1

Some general information:

The calculation of critical values is essentially that of finding tail area of a density or probability function:

This involves finding $x$ that solves the integral equation $\int_x^\infty f(x) dx$.

It used to be that the main way people found these critical values were from tables of the cdf or tail areas (generally combined with interpolation). Nowadays it's mostly done using functions in programs.

When the indefinite integral is doable, this may be easy. Otherwise numerical approximations of various kinds may be invoked, and then some kind of search procedure to find where the tail area is $\alpha$.

Sometimes either the cdf is available which saves the integration step (but not the 'solve' step). Most often, however, the inverse cdf is available either as an algebraic function or as some approximating function, whereupon you just evaluate that function with argument $1-\alpha$:

In the case of the F distribution, one can evaluate the cdf from the incomplete beta function (which is often available in packages of numeric functions), and there's also functions (e.g. ASA 109) that compute approximations to the inverse incomplete beta function, which can be used for finding the inverse cdf.

The result is that in packages you can do something simple like this (here done in R):

 qf(.1,2,14,lower.tail=FALSE)
[1] 2.726468

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Is this value the result of the CDF for the F-distribution?:
$F(x; d_1, d_2) = I_{\frac{d_1x}{d_1x+d_2}}(\frac{d_1}{2}, \frac{d_2}{2})$
Where x is 0.1, d1 is 2 an d2 is 14?

No, $F(2.726;2,14)=0.1$; it's $F^{-1}_{2,14}(0.1)$ that is $2.726$