GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER-V (OLD) EXAMINATION - WINTER 2018
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Subject Code:151601 Date: 30/11/2018
Subject Name: Computer Oriented Statistical Methods
Time: 10:30 AM TO 01:00 PM Total Marks: 70
Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
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Q.1 (a) Define absolute error, relative error, truncation error and round off error. Evaluate the sum s = v3 + v5 + v7 to four significant digits and find its absolute and relative error.
(b) Using Newton’s divided difference interpolation, compute the value of f(6) from the table given below:
| X | 1 | 2 | 7 | 8 |
|---|---|---|---|---|
| f(x) | 1 | 5 | 5 | 4 |
Q.2 (a) Perform three steps of False Position method to find a real root of f(x) = x3-2x - 5.
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(b) Show that the Newton-Raphson method is 2nd order convergent.
OR
(b) State Budan’s theorem. Solve x4- 4x3 + 3x2 - 10x + 8 = 0 by Budan method, in the interval [-1, 0] and [0, 1].
Q.3 (a) Solve the following system of equations by Gauss Seidel method : x+y+54z = 110, 27x+6y-z = 85, 6x+15y+2z = 72.
(b) Use Lagrange’s interpolation to find the value of y when x = 10 if the following values of x and y are given:
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| X | 5 | 6 | 9 | 11 |
|---|---|---|---|---|
| y | 12 | 13 | 14 | 16 |
OR
Q.3 (a) Write Gauss Elimination algorithm. Also include pivotal condensation.
(b) Derive the Recurrence relation for Chebyshev polynomials and using it define T2(x),T3(x) and T4(x).
Q.4 (a) Write an algorithm for cubic spline interpolation.
(b) Fit a second degree polynomial to the following data using least squares method.
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| X | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 1 | 1.8 | 1.3 | 2.5 | 6.3 |
OR
Q.4 (a) Use Runge kutta method of order 4 to compute y(0.2) given that y(0) = 1, and dy/dx = x + y2 . Take h= 0.1
(b) Fit a curve of type y = axb for the following data:
| X | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 2.50 | 8.00 | 19.00 | 50.00 |
Q.5 (a) Calculate 5-yearly moving averages for the following data
| Year | Value | Year | Value |
|---|---|---|---|
| 2009 | 464 | 2014 | 540 |
| 2010 | 515 | 2015 | 557 |
| 2011 | 518 | 2016 | 571 |
| 2012 | 467 | 2017 | 586 |
| 2013 | 502 | 2018 | 612 |
(b) Derive trapezoidal rule. Evaluate ?06 (1/(1+x)) dx Taking n = 6, correct to four significant digits by Simpson’s 1/3rd rule.
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OR
Q.5 (a) Calculate seasonal indices by the ratio-to-moving average method from the following data:
| Year | Quarter I | Quarter II | Quarter III | Quarter IV |
|---|---|---|---|---|
| 2016 | 68 | 62 | 61 | 63 |
| 2017 | 65 | 58 | 66 | 61 |
| 2018 | 68 | 63 | 63 | 67 |
(b) The sale of a company rose from Rs.60000 in the month of October to Rs.69000 in the month of November. The seasonal indices for these two months are 105 and 140 respectively. The owner of the company was not at all satisfied with the rise of sales in the month of November by Rs.9000. He expected much more because of the seasonal index for that month. What was his estimate of sales for the month of November?
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