3.5.3.1.25 Srangecdf

Definition:

\(double prob = srangecdf(q, v, group)\) computes the probability associated with the lower tail of the distribution of the Studentized range statistic.

The externally Studentized range, \(q\), for a sample, \(x_1,x_2,\cdots,x_r\) is defined as:

\[ q=\frac{\max (x_i)-\min (x_i)}{\hat{\sigma _e}} \]

where \(\hat{\sigma _e}\) is an independent estimate of the standard error of the \(x_i\) 's.

For a Studentized range statistic the probability integral,\(P(q)\) , for \(\nu\) degrees of freedom and \(r\) groups, can be written as:

\[P(q)=C\int_0^{+\infty }x^{\nu -1}e^{-\nu x^2/2}\{r\int_{-\infty }^{+\infty }\Phi (y)[\Phi (y)-\Phi (y-qx)]^{r-1}dy\}dx\]

where \(C=\frac{\nu ^{\nu /2}}{\Gamma (\nu /2)2^{\nu /2-1}}\), \(\Phi (y)=\int_{-\infty }^y\frac 1{\sqrt{2\pi }}e^{-t^2/2}dt\)

Parameters:

\(q\) (input, double)

the Studentized range statistic,\(q.\) \( q>0 \)

\(v\) (input, double)

the number of degrees of freedom for the experimental error. \(\nu . \) \( \nu \geq 1.0\)

\(group\) (input,int)

the number of groups,\( r.group\geq 2\)

\(prob \)(output, double)

the probability.