1. Explain what is meant by the most powerful test at level a between two simple hypotheses? /5/ 2 The strength Y of a proxluct has the Weibull distribution with pdf; 10) = fumy"-te(-y, for y> 0, u> 0 0, for y > 0, elsewhere Tfu is unknown and r is known. The strength of a random sample of a products are, Y, Y, ., Yp: Find the form of the most powerful test of the a) null hypothesis u = 0.5 against the altemative hypothesis a = 1.0. Show that Y" has an exponential distribution. /15/ b) /5/ Usc x? tables to find the critical region of the most powerful test at the 10% level when n=20. [Assume the result that if X1, X2, ..X, are independent, cach with an exponential distribution, mean 2, then 22 EX; has the Xn distribution.) c) /5/