Find alpha when we know valuse of student distribution

Question

Assume, we have $t_{1-\alpha/2,39} = 2.252401$.

I don't know how can find $\alpha$.

Please help me and how find it in R.

Answer 1

In R, the syntax is pt(2.252401,39) which gives $0.985 = 1-\alpha/2$, hence $\alpha = 0.03$. In Mathematica, the syntax is CDF[StudentTDistribution[39],2.252401], which, although lengthier to type, has the advantage of being far more readable and directly related to the underlying mathematical relationship: $F_{T_\nu} (t) = \Pr[{T_\nu} \le t] = 1-\alpha/2$, where $T_\nu$ is the Student's $T$ distribution with $\nu$ degrees of freedom.