ANSWER ALL QUESTIONS

Question 1 [10 marks]

a) Your instructor gives the class a surprise four-question with 3-choice quiz. You have not studied the material being quizzed and therefore decide to answer the four questions by randomly guessing the answers without reading the questions or the answers. i) what is the probability that you will have more than half of the answers correct? [3 marks] ii) What do you think the class “average” number of correct answers will be, if an entire class answers the quiz by guessing ? [2 marks] b) The incomes of senior executive in a large corporation are normally distributed with a standard deviation of RM 1200. A cutback is pending, at which time those who earn less RM 28,000 will be discharged. If such a cut represents 5% of the junior executives, what is the current mean salary of the group of senior executives? [5 marks]

Question 2 [30 marks]

a) In a test of Shettles method to increase the probability of conceiving a boy. Among 574 women using that method, 525 had baby boys. Assuming that the method has no effect so that boys and girls are equally likely, find the probability of getting at least 525 boys among 574 babies. Justify the types of distribution used in this experiment. Does the result suggest that the Shettles method is effective? Explain. [10 marks] b) If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a price is 1/100, i) what is the probability that you will win at least prize ? [2 marks] (ii) what is the probability that you will win exactly one prize ? [2 marks] iii) what is the least number of lotteries that you should go so that the probability of win at least one is at least 0.5 ? [6 marks] c) In a large class, suppose your instructor tells you that you need to obtain a score in the top 10% of your class to get an A on a particular exam. From past experience she is able to estimate the mean and standard deviation on this exam will be 72 and 13, respectively. Assume that the grades will be approximately normal distributed, i) what will be the minimum score needed to obtain an A? [4 marks] ii) find the 33rd quartile for exam scores. [3 marks] iii) what is the probability that a person picked at random has an exam score of 60? [3 marks]

Question 3 [10 marks]

Many experts predicted that party B would gain control of the executive branch of the federal government in the year 2013 election. At one time, the general consensus among political scientists was that if party B fielded their best presidential candidate, they would have a 65% chance of winning the 2013 election. In addition, a poll of several thousand voters from across the nation indicated that 54% intended to vote party B.

Revise the experts’ opinion in light of the information from the poll using a Bayesian tabular analysis. [10 marks]

Question 4 [30 marks]
(a) Determine the value of 3s+t when A=[pic] is a symmetric matrix. [4 marks]
(b) Compute the determinant of matrix B=[pic]and obtain the inverse of matrix B. [8 marks]
(c) Consider a system of equations below, x1 + 3x2 + 5x3 = 14 2x1 – x2 – 3x3 = 8 4x1 + 5x2 – x3 = 6 Apply Gauss Elimination method to determine the values of x1, x2 and x3. [9 marks]

(d) Use Cramer’s rule to solve the following simultaneous equations. rx + 2sy + 3tz = 17 3rx + 2sy + tz = 11 4rx – 5sy + tz = -5 Express the values of x, y and z in terms of r, s and t. [9 marks]

Question 5 [20 marks] a) Is [pic]diagonalisable? [5 marks] b) A is a matrix [pic]. i) Find the eigenvalues. [5 marks] ii) State the matrix P. Then find P-1 and B such that P-1AP = B where B is a diagonal matrix. [10 marks]