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I Semester M.E. (Civil ) Degree Examination, March 2013 SS-101 : APPLIED STATISTICS (Common to HW/CT/WRE/Env. Engg.) Time : 3 Hours
Max. Marks : 100
Instruction : Answer any five full questions.
1. a) Define dispersion, state its measures and explain any two of them. b) Compute arithmetic mean, mode, standard deviation and coefficient of variation to the following data of percentage of impurities in a certain chemical substance. % of impurities : 0-5
5-10
Frequency :
15
8
10-15 22
15-20 20-25 17
6
25-30 2
(8+12)
2. a) Prove i) If A ⊂ B ⇒ P(A) ≤ P(B) ii) P(A ∪ B) = P (A) + P(B) – P(AB) b) State and prove Bayes theorem. c) The first of the three carpenter shops in a locality has 7 doors of teak wood and 10 jungle wood, the second has 5 teak wood and 12 jungle wood and the third has 17 teak wood doors. The contractor chooses a shop at random and purchases a door from that shop. It is a teak wood door. Find the probability that the door is purchased from i) The first shop ii) The second shop and iii) The third shop
(4+6+10)
3. a) Define i) pdf f(x) ii) pmf p(x) and iii) cdf F(x) P.T.O.
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b) The probability that a house constructed using a particular brand of water proof cement will have leakage is 0.2. If a constructor uses this brand of water proof cement for constructing 10 houses, what is the probability that i) none of them will have leakage ii) only one of them will have leakage iii) at least two of them will have leakage ? c) Define Poisson distribution and find its mean and variance.
(3+10+7)
4. a) Define normal distribution and state its properties. b) Suppose X ~ N (μ, σ 2 ) , μ = 80 , σ 2 = 4 . Find the probability that 76.08 < X< 83.92. c) If X ~ U (10, 20) write the pdf. Find its mean and variance. Compute P{12 ≤ X ≤ 13.5} .
(6+6+8)
5. a) Define correlation coefficient and state its properties. b) Compute correlation coefficient γ xy between sales (X) and purchases (Y) for the following data. Sales (X)
: 91 97 108
Purchases (Y) : 71 75 69
121
67
124 51
73
111 57
97
70
91
61
80
c) Fit regression line of Y on X for the above data.
39
47 (4+8+8)
6. a) Explain : i) Type I and Type II error ii) Level of Significance iii) Most Powerful Test b) An aircraft manufacturer needs to buy aluminium sheets of 0.05 inches in thickness. Thinner sheets would not be appropriate and thicker sheets would be too heavy. The aircraft manufacturer takes a random sample of 100 sheets and finds that their average thickness is 0.048 inches and their standard deviation is equal to 0.001 inches. Should the aircraft manufacturer buy the aluminium sheets from his supplier if he wants to make the decision at 5% level of significance ?
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c) A certain diet newly introduced to each of the 12 pigs resulted in the following increase in body weight 6, 3, 8, –2, 3, 0, –1, 1, 6, 0, 5 and 4. Can you conclude that the diet is effective in increasing the weight of pigs ? (6+6+8) (given, t 0.05 (10 ) = 2 .23, t 0.05 (11) = 2 .20, t 0.05 (12 ) = 2 .18 )
7. a) Fit Poisson distribution to the following data and test goodness of fit. x
:
0
1
2
3
4
f
:
123
59
14
3
1
b) A trucking company wishes to test the average life of each of the four brands of tyres. The company uses all brands on randomly selected trucks. The records showing the lives (thousands of miles) of tyres are as given below. Brand 1
Brand 2
Brand 3
Brand 4
20
19
21
15
23
15
19
17
18
17
20
16
17
20
17
18
16
16
Test the hypothesis that the average life for each brand of tyres is the same at (10+10) 1% level of significance (given F(3,14), 0.01=5.56). 8. a) Explain i) Simple random sampling and ii) Stratified random sampling. How do you estimate the population mean by these methods ? Give the variance of the estimators. b) The stress of two brands of steel rods are as follows : Brand A : 23.8,
18.6, 20.4,
17.3,
Brand B : 19.4,
21.2, 22.7,
19.9,
26.1,
21.5
20.1
Test for any significant difference in the mean stress for the two brands.
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c) Given the following data, carry out the Chi-square test. Not infectual
Infectual
Vaccinated
28
17
Not vaccinated
16
35
9. Write short notes on any four of the following. i) Ogives ii) Exponential probability distribution iii) Rank Correlation iv) Two way analysis of variance v) Method of maximum likelihood vi) Confidence interval.
______________
(8+7+5) (5×4=20)
