Question 1. Below are ordered data relating to the permeability K (in appropriate units) of core samples from an oil reservoir.
|
0.63
|
2.64
|
2.98
|
6.08
|
6.26
|
6.37
|
6.45
|
6.53
|
6.78
|
6.86
|
6.94
|
6.97
|
7.64
|
7.74
|
|
7.80
|
8.07
|
8.29
|
8.35
|
8.40
|
8.40
|
8.45
|
8.57
|
8.62
|
8.67
|
8.70
|
8.72
|
8.72
|
8.81
|
|
8.88
|
8.91
|
8.93
|
8.96
|
9.01
|
9.03
|
9.05
|
9.06
|
9.09
|
9.14
|
9.17
|
9.20
|
9.22
|
9.25
|
|
9.28
|
9.30
|
9.32
|
9.34
|
9.35
|
9.40
|
9.42
|
9.45
|
9.45
|
9.45
|
9.46
|
9.47
|
9.47
|
9.48
|
|
9.50
|
9.51
|
9.54
|
9.62
|
9.70
|
9.70
|
9.71
|
9.74
|
9.75
|
9.76
|
9.77
|
9.77
|
9.82
|
9.90
|
(a) Calculate (to 3 decimal places) the median, lower and upper quartiles of the data.
(b) It is proposed to make the following transformation of the data K:
y = ln(10 - k)
where ln denotes the natural logarithm (i.e. logarithm base e).
Calculate (to 3 decimal places) the median, lower and upper quartiles of the transformed data.
(c) Using the median, lower and upper quartiles calculated in parts (a) and (b), discuss the shape of the two distributions and hence describe the effect of the transformation on the shape of the distribution.
Question 2. The tensile strength S of cement as a function of curing time t is predicted to be of the form
S = R exp [-α/t], t>0,
where R is the final (residual) strength and α a curing parameter. In a series of experiments the following data were obtained (in appropriate units)
|
t
|
0.50
|
0.60
|
0.70
|
0.80
|
0.90
|
1.00
|
1.10
|
1.20
|
1.30
|
1.40
|
|
S
|
30.09
|
33.43
|
34.55
|
34.56
|
40.15
|
39.06
|
40.58
|
39.88
|
39.64
|
40.73
|
|
t
|
1.50
|
1.60
|
1.70
|
1.80
|
1.90
|
2.00
|
|
S
|
42.611
|
45.32
|
43.12
|
44.43
|
46.93
|
43.24
|
(a) Provide a scatter diagram of the data.
(b) Use the transformation y = ln S and x = 1/t and draw another scatter diagram for the transformed data, and comment.
(c) Compute the sample correlation coefficient between x and y and comment why a regression analysis is plausible.
(d) Use the least squares linear regression of y on x to estimate values for R and α (to 3 decimal places).
(e) Predict (to 3 decimal places) the tensile strength at time t = 2.6 and t = 1.45 and comment on the reliability of your results.
Question 3. In a comprehensive investigation of mobile phone mast, faults could have been found to have developed in three possible sections: the antenna (fault A), the receiver/transmitter electronics (fault B) and line connections (fault C), with the following probabilities
P (A ) = 0.3 P (B) = 0.08 P (C) = 0.15
P (A ∩ B) = 0.026 P (B ∩ C) = 0.012 P (A ∩ B ∩ C)=0.003
It has also been found that the occurrence of fault A is independent from the occurrence of fault C
(a) What is P (A ∪ B), P (B ∪ C), P (A ∪ C)?
(b) Is A independent from B ? Is C independent from B?
(c) Given that it is known that there were antenna faults, what is the probability that there are receiver/transmitter electronics faults.
(d) Given that it is known that there were receiver/transmitter faults, what is the probability that there are faults due to line connections.
(e) What is the probability that a randomly chosen mast has at least one of the three faults.
(f) What is the probability that a batch of 8 masts has no mast with faults, assuming masts are independent?
Question 4. The distribution of liquid drop sizes carried in a gas/liquid flow is given by
0 x < 0
p(x) = (6a/b2).x2(1-x) x ∈ [0, 1]
a/2(x -1) exp(-b(x-1)) x > 1,
where a and b are constants and x is the drop diameter (microns). Assume b > 0.
(a) Write down the requirements for p(x) to correspond to a probability density function (pdf) and thus obtain a relationship between a and b.
(b) Derive an expression for the mean drop diameter µ (this expression may depend on b and/or a).
(c) For a particular flow situation it is known that µ = 9/5 microns.
(ci) Compute the corresponding cumulative distribution function (cdf) F (x)
(cii) Calculate the probability that a drop exceeds 6 microns in diameter (to 3 decimal places).
(ciii) Calculate the percentage of drop diameters between 3 and 9 microns (to 3 decimal places).
(civ) Given that a baffle is used that removes all drops of diameter greater than 9 microns, obtain the proportion of drops which remain which will have diameter less than 6 microns (to 3 decimal places).