answer all questions with detailed explanation

Suppose that a current industry-standard engineering model predicts that a certain type of concrete block will have a 40% chance of failure when subject to a two-tonne load. A civil engineer wants to test the accuracy of this claim by obtaining 50 independent samples of such blocks, subjecting each to a two-tonne load and then recording the total number X of blocks that broke. She decides that she will reject the claim Ho : p = 0.4 in favour of H1 : p # 0.4 if X 25. (a) (1 mark) What is the significance level o of this test? (b) (1 mark) Graph the power of this test as a function of p. The civil engineer proposes an alternative model for predicting the chance of failure of concrete blocks. Under the alternative model, the probability of failure for this type of concrete block is predicted to be 55%. She wants to test whether her new model is more accurate than the current industry standard. (c) (3 marks) Construct a likelihood ratio test (LRT) of Ho : p = 0.4 versus H1 : p = 0.55 based on the number of failures X from 50 trials. (d) (1 mark) Show that the LRT is equivalent to the test with rejection region {X 2 c} for some cutoff c. (e) (1 mark) Changes in industry standards should not be taken lightly, so a significance level of 1% is often used in engineering instead of 5%. Find the cutoff value c for carrying out this test at the 1% significance level. (f) (1 mark) If the alternative model is actually correct, what is the power of this test? (g) (2 marks) What sample size is needed to increase the power of this test to 90%, while keeping the level of the test at 1%