What does area under t distribution give? My question is really simple but i have not been able to find any satisfactory answer anywhere, hence asking here. What does area under t distribution mean?
Example : for normal distribution, P(z< a) will be given by F(a), the area under curve from -infinity to a. But if i want to find P(t< a) where t is t distributed, how do i find it? What does t table value give?
 A: There is only one standard normal distribution, so a full page of print is often used to show the distribution in detail.
By contrast, there are many different t distributions, one for each number of degrees of freedom (DF). In a typical printed table of t distributions, there is one row for each DF.
The highly selected information about the distribution is provided in that one row.
Using a t table. In a typical normal table, probabilities are given in the body of the table and $z$-values are given in the margins. In a typical t table, probabilities are given along the top margin and t values are given along each row of the body of the table.
Connections to normal table. For orientation, you might begin by looking at the very last row of your "t table." It may be marked Inf or with an $\infty$-symbol. Values in that one row are for the normal distribution. Along that row you may find 1.96in the column marked 0.025. That means for standard normal $Z$ you have $P(Z > 1.96) - 0.025.$

*

*Now see if you can find that same information in your printed normal table: From the margins of the table, find 1.96, the in the body
of the table you may find a probability for $P(Z \le 1.96)= 0.9750$ or $P(0 < Z < 1.96)$ $= 0.9750 - 0.5000$ $= 0.4750,$ depending on the style of your normal table.


*Try to match several numbers on the bottom row of the t table with corresponding values in the normal table. (You may have
to do some rounding to get approximate values.)
Back to the t table: Look at the row for DF = 20 and the column marked .05. You should find 1.725 which means
$P(T > 1.725) = 0.05.$
Notes: (1) You usually can't find exact P-values from
a printed normal table. Exact P-values are usually obtained from computer printouts of statistical tests.
Approximate P-values for t table. If DF = 20, and the t statistic is $1.762$ you can
look along row 20 of your t table to find that
$1.725 < 1.762 <2.086.$ The column headers corresponding
to $1.725$ and $2.086$ are $.05$ and $.025,$ respectively.
So you know that the P-value corresponding to $1.762$ is between $0.025$ and $0.05,$ but you'd need software
to find the exact value. For example, using R you could see
that the P-value is $0.0467$ to four decimal places.
[In R 'pt` denotes the CDF of a t distribution.]
1 - pt(1.762, 20)
[1] 0.04667406

(2) Connections with R. Here are some additional bits of R output matching
earlier parts of this Answer. [In R, qt is an inverse CDF of a t distribution; pnorm is a normal CDF, and qnorm is a normal inverse CDF. (An inverse CDF is called a 'quantile' function.]
pnorm(1.96)
[1] 0.9750021
qnorm(.975)
[1] 1.959964
pt(1.725, 20)
[1] 0.9500259
qt(.95, 20)
[1] 1.724718

(3) Yours may be the last generation of students who will use books that have printed probability tables. [R is excellent statistical software available free of
charge for Windows, Mac, and UNIX operating systems from www.r-project.org. It does more than any one person
will ever need; if you try it, focus just on the parts you need.]
