### Question details:

1. An experiment was conducted to assess the accuracy and variability of a machine whose mean is believed to be 1000 and whose standard deviation is believed to be 40. Data on 100 samples was collected and gave the sample mean 990 and sample variance 2450.

(a) Find the rejection region to test a hypothesis that the mean is lower than 1000 at significance level 0.025, and decide whether the data supports this alternative hypothesis.

(b) Find the rejection region at significance level 0.005 to test the hypothesis that the standard deviation is lower than 40, and decide whether the data supports this alternative hypothesis.

2 An experiment was devised to test whether the parameter λ of a sample from the density f(y) = yeλy, y _ 0 is equal to a believed value λ0 = 50.

(a) Derive the most powerful test for the null hypothesis H0 : {λ = λ0} vs alternative hypothesis Ha : {λ = λa} for λa = 40.

(b) Discuss whether this test is uniformly most powerful to test against a composite alternative Ha : {λ _ λ0}.

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