User:Asitgoes/t-Tester

Calculator appearance t-Tester
Developer(s)	Institute for Land Reclamation and Improvement (ILRI)
Written in	Delphi
Operating system	Microsoft Windows
Available in	English
Type	Statistical software
License	Proprietary Freeware
Website	t-Tester

In statistics and data analysis the application software t-Tester is a free and user-friendly calculator for t-testing using Student's t-distribution of a variable X .

The calculator determines the cumulative probability Pc(T) for any t-test value T. Here, the cumulative probability Pc(T) stands for the probability P that X is less than a reference value T of X. Biefly : Pc(T) = P(X<T).

Reversely, the calculator can give the value of T given Pc. Hence, it is a two-way calculator. The data required are the degrees of freedom.

Intervals

The probability (Pi) that X occurs in an interval between an upper limit (U) and a lower limit (L) can be found from:

Pi = P(L<X<U) = Pc(U) - Pc(L) .

Thus, using the calculator twice, namely for T=U and T=L, and subtracting the results, one finds the value of Pi that L<X<U.

Numerical method

The cumulative distribution function of the t-distribution can be calculated numerically as follows.

When T is positive or zero :

$Pc(T)=Po+{\frac {1-Po}{2}}$

When T is negative :

$Pc(T)={\frac {1-Po}{2}}$

where Po is an auxiliary probability value as elaborated below.

Using

$Z=arctan({\frac {T}{\sqrt {N}}})$

where N is the number of degrees of freedom,

the following auxiliary probability equations are applicable according to 3 different conditions:

1 - N (degrees of freedom) even :

$Po=sin(Z)\cdot [1+{\frac {cos^{2}(Z)}{2}}+{\frac {3\cdot cos^{4}(Z)}{2\cdot 4}}+{\frac {3\cdot 5\cdot cos^{6}(Z)}{2\cdot 4\cdot 6}}+\cdots ]$

2 - N (degrees of freedom) uneven and >1 :

$Po={\frac {2Z}{\pi }}+{\frac {2}{\pi }}\cdot cos(Z)\cdot sin(Z)\cdot [1+{\frac {2\cdot cos^{2}(Z)}{3}}+{\frac {2\cdot 4\cdot cos^{4}(Z)}{3\cdot 5}}+\cdots ]$

3 - N (degrees of freedom) = 1 :

$Po={\frac {2Z}{\pi }}$

The number of terms between the parentheses [ ] to be used is N / 2 when N is even and (N-1) / 2 when N is uneven.

Cumulative distribution function of the t-distribution with 10 degrees of freedom and t-test application

Probability area between a lower an upper interval in a probability density function of the t-distribution with 10 degrees of freedom as used in t-test applications.

Graphics

The t-Tester software provides graphics for the various values computed with the calculator. See the examples to left and right.

References

Category:Statistical software Category:Data analysis software Category:Freeware