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Student[Statistics]

 CriticalTable
 return the critical table of values for a given distribution

 Calling Sequence CriticalTable( distribution ) CriticalTable( distribution, options )

Parameters

 distribution - name; name of the distribution chosen from ChiSquare, FRatio, Gamma, or StudentT options - one or more keyword options as described below

Options

 • alpha : float or list; the level of significance
 • parameter1 : list; list of parameter values
 • parameter2 : list; list of parameter values
 • noheaders : truefalse; specifies if column and row header information is included. By default this is set to false.
 • output : identical( matrix, embed ); controls the output of the resulting matrix. If output = embed, then a datatable component containing the matrix of values is returned. By default, output is set to matrix, and a matrix is returned.

Description

Critical values are used in hypothesis testing to specify a cut-off value for computed test statistics, which defines the statistical significance of the test. The $\mathrm{\alpha }$ level is the maximum probability that would result in the null hypothesis being rejected if the null hypothesis is true. Critical values can be found by solving the equation $\mathrm{\alpha }$ = CDF of a given distribution in terms of x for x, or directly from the Quantile command.

For example, the critical t-distribution table gives $x$ for given values of the distribution function $F\left(x\right)$ for a t-distribution with $\mathrm{\nu }$ degrees of freedom.

 • For $x<0$ values of $F\left(x\right)$ can be obtained from $F\left(-x\right)=1-F\left(x\right)$. The values in the critical t-distribution table can be found by solving the equation $\mathrm{\alpha }$ = CDF for the t-distribution for x:

$\mathrm{\alpha }=\frac{1}{2}+\frac{x\mathrm{\Gamma }\left(\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)\mathrm{hypergeom}\left(\left[\frac{1}{2},\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right],\left[\frac{3}{2}\right],-\frac{{x}^{2}}{\mathrm{\nu }}\right)}{\sqrt{\mathrm{\pi }\mathrm{\nu }}\mathrm{\Gamma }\left(\frac{\mathrm{\nu }}{2}\right)}$

 • The CriticalTable command returns the critical table of values as a matrix. If embed = true, then a datatable component containing the matrix of values is returned.

Examples

 > $\mathrm{with}\left(\mathrm{Student}:-\mathrm{Statistics}\right):$

The values of the critical t-distribution table can be found directly from the CDF of the t-distribution:

 > $\mathrm{CDF}\left(\mathrm{StudentTRandomVariable}\left(\mathrm{ν}\right),x\right)$
 $\frac{{1}}{{2}}{+}\frac{{x}{}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){}{\mathrm{hypergeom}}{}\left(\left[\frac{{1}}{{2}}{,}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right]{,}\left[\frac{{3}}{{2}}\right]{,}{-}\frac{{{x}}^{{2}}}{{\mathrm{ν}}}\right)}{\sqrt{{\mathrm{π}}{}{\mathrm{ν}}}{}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{ν}}\right)}$ (1)

For a significance level of 5% we have $\mathrm{\alpha }=0.95$, and with 2 degrees of freedom, $\mathrm{\nu }=2$, then:

 > $\mathrm{fsolve}\left(0.95=\frac{1}{2}+\frac{x\mathrm{GAMMA}\left(\frac{1\cdot 2}{2}+\frac{1}{2}\right)\mathrm{hypergeom}\left(\left[\frac{1}{2},\frac{1\cdot 2}{2}+\frac{1}{2}\right],\left[\frac{3}{2}\right],-\frac{{x}^{2}}{2}\right)}{\sqrt{\mathrm{Pi}\cdot 2}\mathrm{GAMMA}\left(\frac{1\cdot 2}{2}\right)},x\right)$
 ${2.919985580}$ (2)

The CriticalTable command computes the values in the critical t-distribution table numerically solving the above CDF. In contrast to traditional printed tables of values in text books, this can return the results with any level of precision, and also makes it possible to return more accurate results in cases where one would otherwise need to interpolate values for entries that are not in the table.

 > $\mathrm{CriticalTable}\left('\mathrm{StudentT}',\mathrm{α}=0.95,\mathrm{parameter1}=\left[2\right],\mathrm{noheaders}\right)$
 $\left[\begin{array}{c}{2.91998558035372}\end{array}\right]$ (3)
 > $\mathrm{CriticalTable}\left('\mathrm{StudentT}',\mathrm{α}=0.80,\mathrm{parameter1}=\left[100\right]\right)$
 $\left[\begin{array}{cc}{\mathrm{ν}}& {0.80}\\ {100.}& {0.845230424491434}\end{array}\right]$ (4)

To return a section from the critical t-distribution table as a matrix:

 > $\mathrm{CriticalTable}\left('\mathrm{StudentT}',\mathrm{α}=\left[0.95,0.99\right],\mathrm{parameter1}=\left[\mathrm{seq}\left(1..10\right)\right]\right)$
 $\left[\begin{array}{ccc}{\mathrm{ν}}& {0.95}& {0.99}\\ {1.}& {6.31375151467504}& {31.8205159537739}\\ {2.}& {2.91998558035372}& {6.96455673428327}\\ {3.}& {2.35337931741224}& {4.54070317981486}\\ {4.}& {2.13182283743277}& {3.74695414933971}\\ {5.}& {2.01504256032808}& {3.36482991478358}\\ {6.}& {1.94317841848300}& {3.14263811305830}\\ {7.}& {1.89457788693877}& {2.99794031723641}\\ {8.}& {1.85954771984131}& {2.89645460931077}\\ {9.}& {1.83311277671022}& {2.82143560256431}\\ {\mathrm{:}}& {\mathrm{:}}& {"11 × 3 Matrix"}\end{array}\right]$ (5)

To return the typical critical t-distribution table as a matrix:

 > $\mathrm{CriticalTable}\left('\mathrm{StudentT}'\right)$
 $\left[\begin{array}{ccccc}{\mathrm{ν}}& {0.95}& {0.975}& {0.99}& {0.995}\\ {1}& {6.31375151467504}& {12.7062047361747}& {31.8205159537739}& {63.6567411628715}\\ {2}& {2.91998558035372}& {4.30265272974946}& {6.96455673428327}& {9.92484320091829}\\ {3}& {2.35337931741224}& {3.18244902417659}& {4.54070317981486}& {5.84090941075811}\\ {4}& {2.13182283743277}& {2.77634649058963}& {3.74695414933971}& {4.60409670525569}\\ {5}& {2.01504256032808}& {2.57055895700023}& {3.36482991478358}& {4.03215862887992}\\ {6}& {1.94317841848300}& {2.44690471689581}& {3.14263811305830}& {3.70735135904456}\\ {7}& {1.89457788693877}& {2.36462154608823}& {2.99794031723641}& {3.49945522587367}\\ {8}& {1.85954771984131}& {2.30600295299711}& {2.89645460931077}& {3.35537539111690}\\ {9}& {1.83311277671022}& {2.26215658811496}& {2.82143560256431}& {3.24982986145654}\\ {\mathrm{:}}& {\mathrm{:}}& {\mathrm{:}}& {\mathrm{:}}& {"31 × 5 Matrix"}\end{array}\right]$ (6)

To return a single column from the critical chi-square distribution table for the alpha level of 0.95:

 > $\mathrm{CriticalTable}\left('\mathrm{ChiSquare}',\mathrm{α}=0.95\right)$
 $\left[\begin{array}{cc}{\mathrm{ν}}& {0.95}\\ {1}& {3.84145606580278}\\ {2}& {5.99146454710798}\\ {3}& {7.81472828803626}\\ {4}& {9.48772903678116}\\ {5}& {11.0704974062099}\\ {6}& {12.5915872437440}\\ {7}& {14.0671405764057}\\ {8}& {15.5073130558655}\\ {9}& {16.9189774487099}\\ {\mathrm{:}}& {"31 × 2 Matrix"}\end{array}\right]$ (7)

To return a section from the FRatio distribution table with no headers:

 > $\mathrm{CriticalTable}\left('\mathrm{FRatio}',\mathrm{α}=0.975,\mathrm{parameter1}=\left[\mathrm{seq}\left(1..5\right)\right],\mathrm{parameter2}=\left[\mathrm{seq}\left(1..2\right)\right],\mathrm{noheaders}\right)$
 $\left[\begin{array}{ccccc}{647.789011477866}& {799.500000000017}& {864.162972163493}& {899.583310177867}& {921.847903299674}\\ {38.5063291139240}& {39.0000000000000}& {39.1654945640138}& {39.2484176581314}& {39.2982277754032}\end{array}\right]$ (8)

The Gamma distribution depends on two parameters, b, the scale parameter, and c, the shape parameter. To return a sub-matrix of values from the Gamma distribution:

 > $\mathrm{CriticalTable}\left('\mathrm{Gamma}',\mathrm{α}=0.95,\mathrm{parameter1}=\left[\mathrm{seq}\left(1..5\right)\right],\mathrm{parameter2}=\left[1,2\right]\right)$
 $\left[\begin{array}{cccccc}{\mathrm{c\b}}& {1.}& {2.}& {3.}& {4.}& {5.}\\ {1.}& {2.99573227355399}& {5.99146454710798}& {8.98719682066197}& {11.9829290942160}& {14.9786613677699}\\ {2.}& {4.74386451839058}& {9.48772903678116}& {14.2315935551717}& {18.9754580735623}& {23.7193225919529}\end{array}\right]$ (9)

The output option can be used to create a datatable for any of the critical value tables:

 > $\mathrm{CriticalTable}\left('\mathrm{Gamma}',\mathrm{α}=0.975,\mathrm{output}=\mathrm{embed}\right)$
 ${\mathrm{DT}}$ (10)

 > 

Compatibility

 • The Student[Statistics][CriticalTable] command was introduced in Maple 2015.
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