can you please provide a typed solution? it does not need full , just the solution and in type not handwriting. Thank you very much!!!

below is a previous solution to this question if that helps... the typed (not handwritten) solution does not to be as detailed. Just so long as its enough information to present the answer to the question.

Again thank you so much!

UMP - uniformly most powerful

04 = V 1690.10 Let, X1 1 X2 - - 1 xn be ild random sample from N(OIL ). UMP Test of Jize x, Ho: 07, 00 W. HI: 05 00 1 / 2 ( x4012 soln . Likelihood is - 1 ( 01X ) = TTfim; 0)= IT e GeTingz e 2121 - y. Ry ( xi - 0 ) 2 e 2 121 and MLE of 1 0 = X = thi "Me- Sample Mean = x ] . UMP fest is obtained by likelihood ratio fest statistic - Max 1 10121) 13 ( xi - 00 ) 2 e 121 Max (lol ) ( x X/ 2 7 1 1 - 2 - 1/21, 2(xi - 012 - 3 ( xi -x / ] And . * Reject Ho ; if 156 : 0 -12 [ { (x- 012 - E (xi- x ) 27 Taking logarithm both sides-3. Let X1, . .. , Xn be iid random sample from N(0, 1). Find the UMP test of size a, Ho : 0 2 00; H1 : 0