T distribution. Find $P(t \ge -0.879)$ for $24$ degree of freedom. It is at the inner part of the t distribution graph. T table only show the outer quartile of it. I got no idea how to get this without using any t distribution calculator.
 A: In R statistical software for $df=24,$ one has $P(t \ge -0.879) = 1 - P(t \le -0.879) = 0.8059.$ By symmetry, $P(t \le 0.879) = 0.8056$ also. 
 1 - pt(-0.879, 24)
 ## 0.8059413

 pt(.879, 24)
 ## 0.8059413

Typically, a printed table of the Student's t has probabilities to
the right of a few selected values. The one I'm looking at right now
has headings for .25 (the probability above .685) and for .10 (the
probability above 1.318). With luck, you might find a table
with heading .20 (the probability above 0.857), but that is as
close as you are likely to get for the answer to your problem in a printed t table.
 qt(.80, 24)
 ## 0.8568555

 pt(.8568, 24)
 ## 0.799985

Printed t tables are just fine for getting critical values at standard significance
levels such as 10%, 5%, or 1%, or for getting numbers to use
in formulas for 95% or 99% confidence intervals, but they are
not generally useful for problems like yours. So for that kind
of problem, you are correct that you do need a statistical calculator, or statistical
software. Maybe you are using some kind of statistical software
in class. 
R is excellent statistical software available without
charge and with easy installation from www.r-project.org.
(Windows, Mac, and UNIX operating systems are supported.) 
SAS, Minitab, SPSS. and so on, will also do this kind of
computations, and they may be available in your department,
but they would be expensive to buy for home use. You might
investigate whether Excel now does this kind of computations; my
ancient version doesn't. 
