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Student's t distribution



This algorithm (the Matlab code developed below) calculates right-tail values for points on a t-distribution curve. You must provide the value of t and the degrees of freedom.

The t-distribution is a continuous distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small.

Student's t-distribution curve
Student's t-distribution curve

The shaded area represents the right-tail value for t. 

The right-tail value is approximated using the following formula:

right tail values for Students t-distribution

where

a1 = 0.196854
a2 = 0.115194
a3 = 0.000344
a4 = 0.019527

x value for Students t distrubution

d = degrees of freedom

epsilon value for t-distribution

The Matlab program is:

% Clears screen and deletes all the variables in the workspace
clc; clear; 

% Asks the user for input
t = input('T-value: ');
d = input(
'Degrees of freedom: '); 

x = 1;
y = 1;
t = t^2; 

% Computes using inverse for small T-values
if t < 1
    s = d;
    r = y;
    z = 1/t;

else
    s = y;
    r = d;
    z = t;

end

j = 2/(9*s);
k = 2/(9*r); 

% Computes using approximation formulas
l = abs((1 - k)*z^(1/3) - 1 + j)/sqrt(k*z^(2/3) + j); 

if r < 4
    l = l*(1 + 0.08*l^4/r^3);

end 

a1 = 0.196854;
a2 = 0.115194;
a3 = 0.000344;
a4 = 0.019527;
x = 0.25/(1 + l*(a1 + l*(a2 + l*(a3 + l*a4))))^4;
x = floor(x*10000 + 0.5)/10000; 

% Adjusts if inverse was calculated
if t < 1
    x = 1 - x;

end 

% Displays result
str = ['Rigth tail value: ' num2str(x)];
disp(str)

Example 1:


 
What is the right-tail value when the t-value is 2.921 and there are 16 degrees of freedom?

We run the code above and enter data: 

T-value: 2.921
Degrees of freedom: 16 

Matlab answer is 

Rigth tail value: 0.0049


Example 2:


T-value is 11.178, with 5 degrees of freedom. What’s the tail value?

Running the program and entering appropriate data, we get:

T-value: 11.178
Degrees of freedom: 5
Rigth tail value: 0.0002


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