QUESTION 3 (20 Marks) a) Discuss with the example three conceptual approaches to probability. (6 Marks) b) A traffic management company wants to conduct a study on proportion of trucks that use a single roadway between 8am and 9am. A traffic camera records the details of every vehicle and produces a list of data. There are 1000 vehicles are recorded within the hour on one day. The company uses a systematics sampling technique to select a random sample of 5% of the data for their research. Determine which vehicles will be in the sample. (8 Marks) c) The number of items returned to a leading retailer by its customers was recorded for the last 25 days. 21 8 17 22 19 18 19 14 17 11 6 21 25 19 9 12 16 16 10 29 24 6 21 20 25 i. Construct a frequency distribution table for above data. Use five class intervals with the lower boundary of the first class being five days. (3 Marks) ii. Draw a relative frequency histogram for the above data and explain the relationship between the area under the histogram and the relative frequencies of the observation. (3 Marks)c) A traveler from Malaysia wants to visit each of the cities shown on the map exactly once, starting and ending in Brussels. The distance (in km) between each pair of cities is given in the table. Find a Hamiltonian circuit that minimizes the total distance traveled. . Berlin . Dusseldorf Brussels . Luxembourg . Paris . Munich Berlin Brussels Dusseldorf Luxembourg Munich Brussels 783 Dusseldorf 564 223 Luxembourg 764 219 224 Munich 585 771 613 517 Paris 1057 308 497 375 832 (4 Marks)QUESTION 2 (20 marks) a) For each positive integer n, let An = {x E R| - 2 >) i. Find A1 U A2 U A3 (2 Marks) ii. Find A, n A2 n A3 (2 Marks) ii. Find Un=1 An and On=1 An (4 Marks) b) Let P = {0,1,2,3} and define relations S on P as follows: S = {(0,0), (0,2), (0,3), (2,3)} i. Is relation S reflexive, symmetric and transitive? Justify your answer. (6 Marks) ii. Draw the directed graph for the relation S. (2 Marks)QUESTION 4120 Marks) a) SS Sdn. Bhd. produces two types of radios. 60% are X radio and 40% are Y radio. A radio is randomly selected from a population line to check if it is malfunction. From the past inspection, it is known that 5% of X radio and 3% of Y radio are malfunction. i. Draw a tree diagram for the above situation. (3 Marks) ii. Find the probability of getting a malfunction radio. (2 Marks) b) The gross performance of two groups of Country X managed funds with a rating of 3 and 4 stars, over a 1-year investment period for a sample of ve trusts are shown below. 4-star rated 3-start rated 7.31 10.44 9.82 8.38 10.15 9.68 12.42 10.62 9.65 11.25 Which type of 1-year investment (4-star rated or 3-star rated managed funds) appears to have higher level of risks? (15 Marks) QUESTION 5 (20 Marks) a) Explain with the example the difference between point estimator and interval estimator. (2 Marks) b) A survey was conducted to find out the opinion of the general public on whether the individual smoker or society should bear the medical cost of smoking-related illness. In a random sample of 200 people, 128 were of the view that society should bear the cost of medical treatment. Estimate with 95% confidence, the true proportion of people who were of the opinion that society should bear the medical cost of smoking related illness. (8 Marks) c) A pharmaceutical company conducts an experiment to test the effect of a new cholesterol medication. The company selects 15 subjects randomly from a larger population. Each subject is randomly assigned to one of three treatment groups. Within each treatment group, subjects receive a different dose of the new medication. In Group 1, subjects receive 0 mg/day; in Group 2, 50 mg/day; and in Group 3, 100 mg/day. At a=0.05 does dosage level have a significant effect on cholesterol level? Group 1 (0 mg): 210, 240, 270, 270,300 Group 2 (50mg): 210, 240, 240, 270,270 Group 3 (100mg): 180, 210, 210, 210,240 (10 Marks)SECTION A Structured questions 100 marks Answer all questions. QUESTION 1 (20 Marks] a) Let R be the domain of the predicate variable x. Determine which of the following are true and which are false. Give justication for the true statement and counterexample for the false statement. i. x>2>x>1 (2 Marks) II x > 2 ) x2 > 4 (2 Marks) Ill 3:2 > 4 > x > 2 (2 Marks) Iv x2>4>|x|>2 (2 Marks) b) Use Logical Equivalence Theorem to verify the logical equivalences below. ((pq)A~q)-'~pET (12 Marks)