You wish to test whether a coin lands "heads" and "tails" with the same probability. Use p to denote the probability that the coin lands "heads".

(a) State the relevant null and alternate hypotheses.

(b) Suppose you choose to reject the null hypothesis when in 10 tosses you get more than 7 or less than 3 "heads".

i. What is the power of your test when the true probability of "heads" is 0.6?

ii. What is the probability that your test results in a Type II error when the true probability of "heads" is 0.3?

iii. Write an expression for the power function ?(p) of the test.

iv. What is the size of the test?

Question 1 [14 marks total] (a) Let X = {1, 3, 5, 42}, Y = {0, 1, 5, 7, 12}, and Z = {4, 7}. Use the basic set operations to find the following: [8 marks) (i) Xny, (ii) XUY. (iii) (Xny) u(Ynz), (iv) P(Z), the power set of Z. (b) Let A, B and C be arbitrary sets contained in some universal set U. Use set identities (attached) to prove: (please number each identity used) 6 marks) (i) (AUB) - (C - A) = (AUB)n (C'UA), (ii) (AUB) - (C - A) = AU(B -C).5. [12%] Give the time complexity for the following algorithm using two different approaches. You need to identify the basic operation first. F(int n) { If (n > 1) Return 2*F (n-1) ; Else Return 1; 1) [2%] What is the basic operation? 2) [2%]Set up the recurrence. Let T(n) be the # of basic operations. T(n) = 3) [4%] Use backward substitution to solve the recurrence and indicate the time complexity. 4) [4%] Write an iterative algorithm to solve the same problem.11. Which of the following are subspaces of the vector space of real n xn matrices? (a) Upper triangular matrices. (b) Diagonal matrices. (c) Invertible matrices. (d) Symmetric matrices. (e) Skew-symmetric matrices. (f) Singular matrices