Pune University BE (Civil Engineering) Statistical Analysis Question Papers

B.E. (Civil)

STATISTICAL ANALYSIS & COMPUTATIONAL METHODS IN

CIVIL ENGINEERING

(2008 Pattern) (Elective – IV) (Sem. – II)

Time :3 Hours]                                                                                              [Max. Marks :100

Instructions to the candidates:

1)            Answer any 3 questions from each section.

2)            Answer 3 questions from Section -1 and 3 questions from Section – II.

3)            Answers to the two sections should be written in separate answer books.

4)            Neat diagrams must be drawn wherever necessary.

5)            Pigures to the right indicate full marks.

6)            Use of logarithmic tables, slide rule, Mollier charts, electronic pocket calculator and steam tables is allowed.

7)           Assume suitable data, if necessary.

SECTION – I

QI) a) Explain the role of statistics in engineering applications.                                 [3]

b)        Write an expression for coefficient of skewness and coefficient of Kurtosis and state what these coefficients indicate.                                                                      [5]

c)         The following table gives the S02 concentration in ppm in the ambient air observed at a monitoring station. Determine the mean, median, mode and standard deviation for this data.                                                                                                              [8]

Sr.No

l

2

3

4

5

6 7

8

9

10

SOp conc. in ppm

0-1.5

1.5-3.0

3-4.5

4.5-6.0

6.0-7.5

7.5-9.0 9.0-10.5

10.5-12

12-13.5

13.5-15

No. of observations

l2

l8

6

2

4

3 1

1

2

1

 

OR

Q2) a) Write a short note on methods of sampling.                                                        [4]

b)       The following table gives the BOD in mg/l observed at a sampling station. Determine the Karl Pearson’s coefficient of skewness and coefficient of Kurtosis.      [12]

Sr.No

1

2

3 4 5 6

7

BOD (mg/l)

0-10

10-20

20-30 30-40 40-50

50-60

60-70

No. of observations

8

12

20

10

6

3

1

 

Q3) a) In the first week of August every year, the probability for a rainy day is 0.75. What is the probability that there will be exactly 5 rainy days in that week? What is the probability that there will be atmost 2 rainy days? [4]

b)       The diameter D of wires installed in an Electro Static Precipitator (ESP) has a standard deviation of 0.01 in. What should be the value of mean if the probability of its exceeding 0.21 in is to be 1%. Use the std. normal table given in Q.4b.                      [4]

c)       

 

a = 0.05

3

7.8147

4

9.4877

5

11.07

6

12.59

7

14.067

 

Mechanical engineers while testing a new arc welding technique, classified welds with respect to appearance and X-ray inspection. The results are shown in table below. Use the 0.05 level of significance to test the independence of the criteria of classification. [Use chi-square Table]. [8]

T

<D

O

§

<D

d

d

Bad

Normal Good

X-ray

Inspection i

Bad

20

7 3

Normal

13

51 16

Good

7

12 21

 

OR

Q4) a) The ppm concentration of a toxic substance in a wastewater is known to be normally distributed with mean ^ = 100 and standard deviation a = 2.0. Calculate the probability that the toxic substance concentration C is                                                           [6]

i)              less than 98

ii)           between 98 and 104

iii)         greater than 104

Use the standard normal distribution table given in Q.4b.

b) The following table gives tensile strength of concrete cylinders in lb/in2. Test the goodness of fit for normal distribution at 5% significance level using chi-square test.    [10]

Tensile strength of concrete cylinders

No. of observations

< 325

6

325 – 335

6

335 – 345

11

345 – 355

14

355 – 365

16

365 – 375

15

375 – 385

8

385 – 395

10

395 – 405

8

> 405

6

 

Use the following chi-square distribution table for a = 0.05.

u

3 4 5 6

7

^2

7.8147 9.4877 11.07

12.59

14.067

 

Use the Standard Normal Distribution Table given below.

z

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

area

0.0000

0.0398

0.0793

0.1179

0.1554

0.1915

0.2257

0.2580

0.2881

0.3159

0.3413

 

 

 

z

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

area

0.3643

0.3849

0.4032

0.4192

0.4332

0.4452

0.4554

0.4641

0.4713

0.4772

 

z

2.1

2.2

2.3

2.4

2.5

area

0.4821

0.4861

0.4893

0.4918

0.4938

 

Q5) a) The pressure and volume of a gas are related by the equation

 

pv = a or

Fit this equation

vb =1 a where a and b are constants p

 

[12]

for the following data using the principle of least squares.

P

0.5

1.0

1.5

2.0

2.5

3.0

v

1.62

1.00

0.75

0.62

0.52

0.46

 

b) For the following data find fg) using Newton’s forward interpolation formula. [6]

 

8

10 12 14

16

A*)

1000

1900

3250

5400

8950

 

OR

Q6) a) The amount A of a substance remaining in a reacting system after an interval of time t in a certain chemical experiment is given in the following table. Find the value of A when t = 6. Use Lagrange’s Interpolation formula.                                                        [6]

t

3

7 9

10

A

168

120

72

63

 

b) The average yearly rainfall over a basin and the corresponding yearly runoff, both expressed in cm, for a period of 9 years are given below. Establish the relation between rainfall and runoff of the form Y = ax + b. Also compute the coefficient of correlation between them.           [12]

Year

1

2

3

4

5

6

7

8

9

Rainfall

113

127

108

167

99

152

165

160

149

Runoff

74

96

59

109

57

109

124

134

106

 

SECTION – II

Q 7) a) Solve the following system of equations by Gauss – elimination method. [8]

xx + x2 + 2xQ = 4 2x1 + 5x2 – 2xQ = 3 x1 + 7x2 – 7xQ = 5

b) Use Gauss – Seidel iterative method to solve the following equations. [The percent relative error Es < 5%]                                                                                       [8]

83xj + 11x2 – 4xQ = 95

7xx + 52x2 + 13xQ = 104

3xx + 8x2 + 29xQ = 71

OR

Q8) a) Solve the following equations using Gauss – Jordan method.                          [8]

2xx + x2 + xQ = 10 3xx + 2x2 + 3xQ = 18 xj + 4x2 + 9xQ = 16

b) Use Gauss – Seidel iterative method to solve the following equations. [Relative error E < 5%]                                                                                                                 [8]

5* + x2 + 2xQ = 19 x1 + 4x2 – 2xQ = -2 2x1 + 3x2 + 8xQ = 39

Q9) a) Find the positive real root of                                                                                   [8]

x log10 x = 1.2

Using bisection method in four iteration in the interval (2, 3)

b) Find the real root of xQ – 3x + 1 = 0 lying between 1.5 and 2 upto three decimal places by Newton Raphson method.                                                                          [8]

OR

QIO) a) Using False Position method, find the root of                             [10]

fx) = x2 – loge x – 12 = 0

upto four iteration, in the interval (3, 4)

b) Explain – Newton Raphson method, to find the roots of the nonlinear equation.            [6]

QII) a) A river is 80 m wide. The depthy in meter at a distance x from one bank is given by the following table. Calculate area of cross section of the river using Simpsons’s rule    [9]

x

0

10

20

30

40

50

60

70

80

y

0

4

7

9

12

15

14

8

3

 

t _ 1 dx

b) Evaluate 1 = J                 7 using Gauss Quadrature three point formula. [9]

0     1 + x

OR

QI2) a) The following table gives the velocity (v) of a particle at time (t) Find the distance moved by the particle in 12 seconds.                                                                           [9]

t(sec)

0

2

4

6

8

10

12

v (m/s)

4

6

16

34

60

94

136

 

1      dx

b) Using Simpson’s 3/8th rule. evaluate I = J                    .                                    [9]

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