Regression analysis is used to analyze data that has two or more variables in order to investigate interdependence of the variables. Regression analysis is used in business to predict trends and analyze a variety of data types for business decision-making.

Describe the assumptions necessary to conduct statistical tests involving the hypothesized-regression model. Give examples for the practical application of regression analysis in business. Explain how the regression analysis is used to predict future performance of a business.

Fundamental rules of probability: 1. P(0) = 0, P(0) = 1 2. For any event A, P(A) 2 0. 3. For any countable list of disjoint events A1, A2, A3, . .. (i.e. AinA, = 0) for any i # j), P(UP, A;) = En, P(A;). Special case: for any finite list of disjoint events A1, A2, A3, . .. , An (i.e. Ain Aj = 0) for any i # j), P(U_, Ai) = Et_, P(Ai). Use only the fundamental rules of probability (and basic set operations) to prove Problems 1 and 2. Problem 2 (2 points) Prove that for any events A, B, P(A) = P(An B) + P(An Bc).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).(a) Prove or give a counter example: If An B - An C, then B = C. [b) Simplify AU(BU ((AU B) nC))" using basic set operations. Do not use Venn diagrams (except, perhaps, to check your result)! (c) Let A, B. C, and D be subsets of S. Using set operations (C, D.n, U,"), express the situation that exactly three of (A, B. C, D) occur. You can use Venn diagrams to help visualize the problem, but your answer must be expressed using set operations (e.g., An B).the patient is in the group treated with interferon alfa, and let B denote the event that the patient has a complete response. Determine the following probabilities. (a) P(A) (b) P(B) (c) P(AnB) (d) P(AUB) (e) P(A'UB) 2-81. A computer system uses passwords that contain exactly eight characters, and each character is one of 26 low- ercase letters (a-z) or 26 uppercase letters (A-Z) or 10 inte- gers (0-9). Let 2 denote the set of all possible passwords, and let A and B denote the events that consist of passwords with only letters or only integers, respectively. Suppose that all passwords in 2 are equally likely. Determine the probabil- ity of each of the following: (a) A (b) B (c) A password contains at least 1 integer. (d) A password contains exactly 2 integers. lying basic set operations to individual events. Unions of events, vents, such as AnB; and complements of events, such as A'-are bility of a joint event can often be determined from the probabili- it comprises. Basic set operations are also sometimes helpful in joint event. In this section, the focus is on unions of events. 2-1 lists the history of 940 wafers in a semiconductor manu- wafer is selected at random. Let / denote the event that the