▁estim ators , ▁etc .. ▁As ▁an ▁example ▁consider ▁the ▁problem ▁of ▁prediction . ▁Suppose ▁ ▁is ▁some ▁point ▁within ▁the ▁domain ▁of ▁distribution ▁of ▁the ▁re gress ors , ▁and ▁one ▁wants ▁to ▁know ▁what ▁the ▁response ▁variable ▁would ▁have ▁been ▁at ▁that ▁point . ▁The ▁mean ▁response ▁is ▁the ▁quantity ▁, ▁whereas ▁the ▁predicted ▁response ▁is ▁. ▁Clear ly ▁the ▁predicted ▁response ▁is ▁a ▁random ▁variable , ▁its ▁distribution ▁can ▁be ▁derived ▁from ▁that ▁of ▁: ▁▁▁ ▁which ▁allows ▁construct ▁confidence ▁intervals ▁for ▁mean ▁ ▁response ▁ ▁to ▁be ▁constructed : ▁▁ ▁  ▁at ▁the ▁ ▁confidence ▁level . ▁▁ ▁Hyp oth esis ▁testing ▁▁ ▁Two ▁hypothesis ▁tests ▁are ▁particularly ▁widely ▁used . ▁First , ▁one ▁wants ▁to ▁know ▁if ▁the ▁estimated ▁regression ▁equation ▁is ▁any ▁better ▁than ▁simply ▁predict ing ▁that ▁all ▁values ▁of ▁the ▁response ▁variable ▁equal ▁its ▁sample ▁mean ▁( if ▁not , ▁it ▁is ▁said ▁to ▁have ▁no ▁explan atory ▁power ). ▁The ▁null ▁hypothesis ▁of ▁no ▁explan atory ▁value ▁of ▁the ▁estimated ▁regression ▁is ▁tested ▁using ▁an ▁F - test . ▁If ▁the ▁calculated ▁F - value ▁is ▁found ▁to ▁be ▁large ▁enough ▁to ▁exceed ▁its ▁critical ▁value ▁for ▁the ▁pre - ch osen ▁level ▁of ▁significance , ▁the ▁null ▁hypothesis ▁is ▁rejected ▁and ▁the ▁alternative ▁hypothesis , ▁that ▁the ▁regression ▁has ▁explan atory ▁power , ▁is ▁accepted . ▁Otherwise , ▁the ▁null ▁hypothesis ▁of ▁no ▁explan atory ▁power ▁is ▁accepted . ▁ ▁Second , ▁for ▁each ▁explan atory ▁variable ▁of ▁interest , ▁one ▁wants ▁to