The t distribution
is symmetric but comparatively flatter
(see the solid line in the graph below)
than the standard normal distribution
(the dashed line below).
The shape of particular t
-distribution is determined by
the degrees of freedom
When the sample mean
and the sample standard deviation
are obtained from the data of
it is often assumed that the test statistic
has the t
degrees of freedom
with true population mean
If the true standard deviation
= +Inf (the infinity
We can calculate the critical region corresponding to the level .
The appropriateness of this calculation can be ensured if
(a) the sample distribution is approximately normal
(the use of QQ plot is recommended),
or (b) the sample size is adequately large
(as a rule of thumb it is desirable to have ).
Conversely when the statistic
we can find the corresponding so that the value belongs
to the critical region,
and call it p-value.