http://www.graphpad.com/quickcalcs/PValue1.cfm  stat computation

Student t analysis
Assume a group of measurements assumed to have a mean value of u0
A sample of nn items is assumed to be from this population
The mean value of this sample is determined by measurement=xb.
The degree of freedom= is  n = nn-1.
The sample standard deviation sx is measured.

t= (xbar-mean)/(s/sqrt(n))
mean= assumed population mean
xbar=sample mean
s=sample standard deviation
n=sample size

The formula computintg curve generating the probability data is

    Gamma((n-1)/2)
---------------------*Integral(t=x to oo) ( (1+t^2/n) ^ (-(n+1)/2) dt)
Gamma(n/2)*Sqrt(n*Pi)

Gamma[n]= Integral[ ( 0 to oo) e^(-x) * x^(n+1) dx ]

Example usng TI-89 Titanium
SelectStat/List Editor Select  2nd F6    Select 2:T-Test
Data Input Method    Stats
u0=40    assumed mean value of u0 in the total population
xb=42.6  The mean value of this sample
sx=3.7   The sample standard deviatio
n=15     number of samples
Select Calculate 
Select u>u0 assumption probability mean of sample is greater then mean of
population
Enter

See Display
u>u0
u0=      40
t=       2.72156 computed independent variable of curve
P value   .008271= area under curve from t= 2.72156 to oo= probability
sample is drawn from total population with mean=u0
xb=42.6  mean value of sample
sx=3.7  The sample standard deviation
n=15 number of samples

compute ts=t for sample (xb-uo)/(sx/Sqrt[nn]) = 2.72156

Find t for 0.01 area under curve = assumed rejection critical value

Select F5
Inverse
2:Inverse t
Area .01   Deg  of Freedom df= 14
Enter 
Observe Inverse   = -2.62449 = ti = t value where area under curve is .01
Observe Area = .01
df=14 = 15-1

Since ts > ti, 2.72156>2.62449 , the probability the sample is representative
of the standard population is less than .01

Clear y1= menu
Go to 2nd F6 and select Draw

Original data is still present

See bell curve with t=2.72156     p=.008271= area under curve = probability
of sample being drawn from parent population
.
If sample were close to standard population, the right fraction of the bell
curve is darkened in region between z (instead of t) to oo
u=xb = mean of sample
If u<u0
P value= .991729

u not equal uo      P=.016543 = 2*008271