In hypothesis testing..

Question

... we use two tail test and divide the area by $\frac{\alpha}{2}$ each. Why don't we divide it into some other propotion say $\frac{3}{4} \alpha$ and $\frac{\alpha}{4}$? Here $\alpha$ means type 1 error.

Answer 1

Sometimes when dealing with asymmetrical distributions one does split the Type I error unevenly between the tails for a two-sided test. Also, when dealing with discrete distributions like the binomial or Poisson (for which probability comes in discrete chunks) one sometimes splits Type I error probability unevenly to try to come closer to the exact number $\alpha$ required.

However, for continuous, symmetrical distributions (e.g., normal and Student's t) the way to get the greatest power against alternatives to the null hypothesis is to make an even split. Also, it is easier to make an even split because then there is only one number to be looked up in tables. For example, if +1.96 cuts 2.5% from the upper tail of a standard normal distribution, then by symmetry -1.96 cuts 2.5% from the lower tail.