The strength Y of a product has the Weibull distribution with pdf; ?(?) = {????−1?(−???), ...

The strength Y of a product has the Weibull distribution with pdf;

?(?) = {????−1?(−???), ??? ? > 0, ? > 0 0, ?????ℎ???

If ? is unknown and ? is known. The strength of a random sample of n products are, ?1,?2,…,??.

a) Find the form of the most powerful test of the null hypothesis ? = 0.5 against the alternative hypothesis ? = 1.0.

b) Show that ? ? ? has an exponential distribution.

c) Use ?2 tables to find the critical region of the most powerful test at the 10% level when n=20. [Assume the result that if ?1,?2,…,?? are independent, each with an exponential distribution, mean ?, then 2?∑?? has the ?2? 2 distribution.]

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