PED – 001
*PED001*
I Semester M.E. (Civil Engineering) Degree Examination, January 2015 SS-101 : Applied Statistics (Common to HW/CT/WRE/Env. Eng.) Time : 3 Hours
Max. Marks : 100
Instruction : Answer any five full questions. 1. a) What are histogram and cumulative frequency curve ? How are they useful ? b) Define mean and median. Mention their merits and demerits. c) For the following frequency distribution of wire bond pull strength : Class Interval : Frequency
:
11 − 15 15 − 19 4
15
19 − 23
23 − 27
27 − 31
24
18
9
i) Draw histogram and find the value of the mode from this graph. ii) Draw a cumulative frequency curve and find median value from this curve. (5+5+10) 2. a) Define : i) Coefficient of variation. How is it useful ? b) Define skewness and give a measure of it. c) For the frequency distribution given in question no. 1(c) compute coefficient of variation and coefficient of skewness. (4+4+12) 3. a) State and prove addition rule of probability. Given P( A ) = P( A ∪ B ) =
1 , P(B ) = 1 , 2 4
5 , find P(A ∩ B ) and P( A | B ) . Are A and B independent ? 8
b) State and prove Bayes’ theorem. c) Of the mangoes supplied to a cooperative, 35%, 45% and 20% come from the orchards A, B and C respectively. Rotten mangoes account for 2%, 2.5% and 1.5% of the A, B and C orchards respectively. A randomly selected mango is found to be rotten. What is the probability that it is from orchard A ?
(8+5+7) P.T.O.
PED – 001
*PED001*
-2-
4. a) Find k such that the function ⎧k x (1 − x) 0 < x < 1 f( x ) = ⎨ 0 otherwise ⎩ 1 is a p.d.f. Find i) E(x) and ii) P⎛⎜ x > ⎞⎟ 4⎠ ⎝
b) Define Binomial and Poisson distributions. Find their mean and variance. c) The number of failures (x) of machines per day in a certain firm has Poisson distribution with mean λ = 2. Find P(x = 0) and P( x ≥ 2) .
(6+10+4)
5. a) State normal distribution and its properties. b) What is a sampling distribution ? State the sampling distribution of the sample mean and sample variance, when the sample is drawn from a normal population. c) The contents (x) of a certain beverage in a case follows normal distribution with mean μ = 250 gms and standard deviation σ = 5 gms. Find P(255 < x < 265). Suppose x denotes the sample mean based on n = 9, drawn from this (6+6+8) population, find P( x > 253) . 6. a) Define correlation coefficient and state its properties. b) What are the objectives of regression analysis ? c) Following table gives purity of oxygen (y) in a chemical distillation process and percentage of hydrocarbons (x) present in the main condenser : x
:
0.99
1.02
1.29
1.46
y
:
90.01
89.05
93.74 96.70
1.23
1.15
1.01
91.77
92.52 89.54
i) Fit a linear regression of y on x. ii) Predict the value of y when x = 1.3. iii) Find correlation coefficient and interpret its value.
(4+4+12)
*PED001*
PED – 001
-3-
7. a) Define : i) type I error ii) type II error and iii) P-value. b) Nine samples of the autoclaved aerated concrete material were tested and the interior temperature (x) (°C) reported as follows : 23.01, 22.22, 22.04, 22.62, 22.59, 23.12, 24.11, 23.82, 24.22. Assuming
X ~ N(μ, σ2 ) , Find i) an unbiased estimator of μ and its standard error. ii) construct 95% confidence interval for μ . iii) test the hypothesis H0 : μ = 22 .5 vs H1 : μ ≠ 22 .5 . c) The deflection temperature under load for two different types of plastic pipe is being investigated and data are given below : Type 1 :
97,
87,
96,
89,
101,
81
Type 2 :
81,
94,
87,
88,
80,
82,
86, 92
Do the data support the claim that the mean deflection temperature under load for type 1 pipe exceeds that of type 2. Test at α = 0.05 level of significance. (4+8+8) 8. a) Explain briefly basic principles of experimental design. b) The following table gives tensile strength of paper (kN/m2) for three hard woods concentrations in the pulp : Concentrations (%) C1
C2
C3
7
12
14
8
17
18
15
13
19
11
18
17
Carry out analysis of variance to test the equality of mean tensile strengths for the three concentrations. Take α = 0.05.
PED – 001
-4-
*PED001*
c) A study is being made of the failures of an electronic component. There are four types of failures possible and two mounting positions. The following data are obtained : Failure type Mounting position
A
B
C
D
1
20
48
20
7
2
4
17
6
12
Test at α = 0.05 level of significance, for the association between the two factors. (4+8+8) 9. Write short notes on any four of the following : a) Stratified sampling b) Chi-square distribution c) Test for proportion d) Goodness of fit test e) Randomized block design f) Kurtosis. ———————
20
