MCA-102 Statistical Mathematics Past Paper December 2020 PDF

Total No. of Questions : 8] [Total No. of Printed Pages : 4 Roll No.................................. MCA-102 M.C.A. I Semester (Two Year Course) Examination, December 2020 Statistical Mathe...

Total No. of Questions : 8] [Total No. of Printed Pages : 4 Roll No.................................. MCA-102 M.C.A. I Semester (Two Year Course) Examination, December 2020 Statistical Mathematics Time : Three Hours Maximum Marks : 70 Note: i) Attempt any five questions. ii) All questions carry equal marks. 1. a) Explain Kolmogrov’s axioms of probability. 7 b) An Urn contains 10 white and 3 black balls, whether an urn contains 10 white and 3 black balls while another contains 3 white and 5 black balls. Two balls are drawn from first urn and put into the second urn and then a ball is drawn from the latter. Find the probability that 7 i) It is a white ball ii) It is a black ball ⎡1 2 1 ⎤ 2. a) Determine the row rank of A = ⎢ 2 3 1 ⎥ 7 ⎢ ⎥ ⎢⎣1 1 2 ⎥⎦ b) What is the Cayley–Hamilton Theorem? 7 3. a) Use the Intermediate Value Theorem to show that 25 – 8x2 – x3 = 0, 25 – 8x2 – x3 = 0 has at least one root in the interval [–2, 4] [–2, 4]. Note that you are NOT asked to find the solution only show that at least one must exist in the indicated interval. 7 MCA-102 PTO  b) Determine if the given function is continuous or discontinuous at the indicated points. 7 4x + 5 f ( x) = at x = −1 9 − 3x 4. a) Average heart rate for Americans is 72 beats/minute. A group of 25 individuals participated in an aerobics fitness program to lower their heart rate. After six months the group was evaluated to identify is the program had significantly slowed their heart. The mean heart rate for the group was 69 beats/minute with a standard deviation of 6.5. Was the aerobics program effective in lowering heart rate? 10 b) Define Sampling distribution. 4 5. A randomized trial is designed to evaluate the effectiveness of a newly developed pain reliever designed to reduce pain in patients following joint replacement surgery. The trial compares the new pain reliever to the pain reliever currently in use (called the standard of care). A total of 100 patients undergoing joint replacement surgery agreed to participate in the trial. Patients were randomly assigned to receive either the new pain reliever or the standard pain reliever following surgery and were blind to the treatment assignment. Before receiving the assigned treatment, patients were asked to rate their pain on a scale of 0-10 with higher scores indicative of more pain. Each patient was then given the assigned treatment and after 30 minutes was again asked to rate their pain on the same scale. The primary outcome was a reduction in pain of 3 or more scale points (defined by clinicians as a clinically meaningful reduction). The following data were observed in the trial. 14 MCA-102 Contd... PTO  Treatment n Number of Proportion with Group Reduction of Reduction of 3+ points 3+ Points New Pain 50 23 0.46 Reliever Standard pain 50 11 0.22 Reliever Test whether there was a significant difference in the proportions of patients reporting a meaningful reduction (i.e., a reduction of 3 or more scale points) using a Z statistic. 6. a) Define the following terms using examples and venn diagrams: 8 i) Sets ii) Subsets iii) Power set iv) Union of two sets. b) Prove that x ∈ A ⇔ { x} ⊆ Ax ∈ A ⇔ { x} ⊆ A , for any element x ∈ U. 6. 7. a) What is a Queue Data Structure? Describe basic features and application of Queue? What is the implementation of Queue data structure? 7 b) Prove that the following equality P(n) holds, for every positive integer n: n ( n + 1) ∑ k =1 k = n by induction. 7 2 MCA-102 PTO  8. Employers want to know which days of the week employees are absent in a five-day work week. Most employers would like to believe that employees are absent equally during the week. Suppose a random sample of 60 managers were asked on which day of the week they had the highest number of employee absences. The results were distributed as in the table below. For the population of employees, do the days for the highest number of absences occur with equal frequencies during a five-day work week? Test at a 5% significance level. Day of the Week Employees were Most Absent 14 Monday Tuesday Wednesday Thursday Friday Number of Absences 15 12 9 9 15 ****** MCA-102 PTO