JE – 939

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I Semester M.E. (Civil Engg.) Degree Examination, July/August 2013 SS-101 : APPLIED STATISTICS (Common to HW/CT/WRE/Env. Engg.) Time : 3 Hours

Max. Marks : 100

Instruction : Answer any five questions. 1. a) Compute the mean and standard deviation, given the following data : Class :

44.5 – 49.5

49.5 – 54.5

54.5 – 59.5

Freq. :

2

8

16

59.5 – 64.5

64.5 – 69.5 69.5 – 74.5

24

19

7

b) Explain skewness and kurtosis. How do you measure them ? c) Sketch a ogive curve for the data in 1 a).

(7+8+5)

2. a) Two balanced dice are rolled. What is the probability of getting a sum of five ? b) What are disjoint and independent events ? Illustrate with examples. c) The following table shows the promotional status of police officers for the past ten years. (M) Men

(W) Women

Promoted (A)

288

36

Not promoted ( A)

672

204

Compute P(A|W) and P(A|M) and hence comment. d) Give an application of Bayes theorem.

(5+5+5+5)

3. a) What is a Poisson distribution ? Illustrate with an example. Derive the mean and variance of Poisson distribution. b) If p varies with n such that np = λ , then show that lim b(x; n, p ) = p( x; λ ) ,

n→∞

where b (x; n, p) and p( x; λ) are binomial and Poisson probability mass functions (p.m.f’s). P.T.O.

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c) If X has an exponential distribution with a mean of 1.4 minutes, where X denotes the time (in mts) between detections of a rare particle at a Geiger 1 counter, compute P( x < ) , P(3 < X < 3.5), P (X > 3) and [E (X)]–1. 2

(8+7+5)

4. a) Explain i) moment estimator, and ii) maximum likelihood estimator. Find these estimators based on a random sample of size n from B(1, p) distribution and examine them for consistency. b) The reaction time of a driver to visual stimulus is normally distributed with a mean of 0.4 second and a standard deviation of 0.05 second. What is the probability that reaction time requires i) more than 0.5 second ? and (ii) between 0.4 and 0.5 second ? What is the reaction time that is exceeded 90% of the time ? c) If X ~ U [ 0, 20 ] , where X is the current measured in a thin copper wire in milli amperes, find P (5 < X < 10), E(X) and standard deviation of X. (10+4+6) 5. Given the following data, where x1 is oil viscosity, x2 is load and y denotes wear of a bearing, fit a multiple linear regression model to this data. y

193

230

172

91

113

125

x1

1.6

15.5

22

43

33

40

x2

851

816

1058

1201

1357

1115

Predict y given x1 = 20, x2 = 1000. 6. a) What is the problem of hypothesis testing ? b) Test the hypothesis that the mean life of a brand of light bulbs is 1000 hrs. against that it is greater than 1000 hrs. based on the following data : n = 100 ; x = 985 and σ 2 = (100 )2 . what would be your testing procedure if the data are : n = 25 ; x = 985 and s = 90.2. Provide 95% confidence interval for μ in both the cases.

20

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c) Given the following data on HDL (high density Lipoprotein), test for equality of means at 5% level of significance. Samples

n

x

s

Elite runners 20

56

12.1

Non-runners

49

10.5

72

(4+8+8)

7. a) Fit a Poisson distribution to the following data and test for goodness of fit. x

0

1

2

3 >3

f

59 27 09 1

0

b) Analyse the following fabric strength data on three chemical types by carrying out one-way ANOVA. Chemical type 1

1.3

1.6

0.5

1.2

2

2.2

2.4

0.4

2.0

3

1.8

1.7

0.6

1.5

(7+13)

8. a) Explain i) simple random sampling, ii) systematic sampling, and iii) stratified random sampling. Give an example in each case. b) Discuss correlation and linear regression. How do you measure correlation ? Illustrate with an example. (10+10) 9. Write short notes on any four of the following : i) Discrete probability distribution ii) Law of total probability iii) Measures of dispersion iv) Sampling distributions v) Negative exponential distribution vi) Tests for proportion. —————————

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