Student's t-distribution calculation How to calculate this Student's t-distribution?
$X $~$ t(17)$
I have to find $P(X>8)$ and $a$ where $P(X>a)=0.1$.
I used standardizing for calculating similar probability for normal distribution $X $~$ N(18,3)$. How can I do that kind of thing here in Student's?
I know there exists the formula of $Z$ but don't know how to use it.
$\frac{\bar X-\mu}{S/\sqrt n}$
Thanks.
 A: If your distribution is a t with 17 d.o.f. you can directly use the tables. With 17 d.o.f. your distribution is very close to a Standard Gaussian thus $P(X>8)\to 0$
Here is a table to  understand what I mean. The student t is table C-4
$$P(X>a)=0.1$$
Is $a=1.333$
A: In R, where pt is the CDF of a designated t distribution: for $X \sim \mathsf{T}(\nu=17),$ we have $$P(X > 8) = 1 - P(X\le 8) \approx 0,$$ as shown below. From most printed tables of t distributions one cannot get an exact value,
but from mine, one can see that $P(X > 3.985) < 0.0005.$
1 - pt(8, 17)
[1] 1.824967e-07

To find $a$ with $P(X > a)=.01,$ one can use the t quantile function (inverse CDF) qt to get $a =2.566934$ or look on line DF=17 of a printed t table
to find the value $2.567.$
qt(.99, 17) 
[1] 2.566934

The first question might be to find the P-value of a right-sided t test, the second to find the critical value for such a test at level $\alpha = 0.01 = 1\%.$
Most statistical computer programs give P-values, making it unnecessary to input a fixed significance level. Printed t tables give critical values for
tests at frequently-used significance levels, but cannot ordinarily be used to find exact P-values.
The following graph shows the density function of
$\mathsf{T}(\nu = 17)$ with value that cuts 1% of
probability from its upper tail.
hdr = "Density of T(17)"
curve(dt(x,17), -8, 8, ylab="PDF", xlab="t", 
       col="blue", lwd=2, main=hdr)
 abline(v = 0, col="green2")
 abline(h = 0, col="green2")
 abline(v = qt(.99, 17), col="red", lwd=2, lty="dotted")


