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Seat No.: Enrolment
GUJARAT TECHNOLOGICAL UNIVERSITY
M.E. SEMESTER I-EXAMINATION -SUMMER 2018
Subject code: 2714601 Date: 08/05/2018
Subject Name: Statistics for Engineers
Time: 2:30 p.m.-5:00 p.m. Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
4. Statistical Tables are required.
Q.1

i A stamping machine produces can tops whose diameters are
normally distributed with a standard deviation of 0.01 inch. At
what "nominal" (mean) diameter should the machine be set so
that no more than of the can tops produced have diameters
exceeding 3 inches?
ii Human error is given as the reason for 75% of all accidents in a
plant. Use the formula for the binomial distribution to find the
probability that human error will be given as the reason for two of
the next four accidents.
04
03

As part of the investigation of the collapse of the roof of a building, a testing laboratory is given all the available bolts that connected the steel structure at 3 different positions on the roof. The forces required to shear each of these bolts (coded values) are as follows:
Position 1
90
82
79
98
83
91
Position 2
105
89
93
104
89
95
86
Position 3
83
89
80
94
Perform an analysis of variance to test at the 0.05 level of significance whether the differences among the sample means at the 3 positions are significant.
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Q.2

i The specifications for a certain kind of ribbon call for a mean
breaking strength of 180 pounds. If five pieces of the ribbon
(randomly selected from different rolls) have a mean breaking
strength of 169.5 pounds with a standard deviation of 5.7 pounds,
test the null hypothesis pounds against the alternative
hypothesis pounds at the 0.01 level of significance.
Assume that the population distribution is normal.
ii The dean of a college wants to use the mean of a random sample
to estimate the average amount of time students take to get from
one class to the next, and she wants to be able to assert with 99%
confidence that the error is at most 0.25 minute. If it can be
presumed from experience that minutes, how large a
sample will she have to take?
04
03

i Find the joint probability distribution function of two random
variables with probability density given by
ii Find mean and variance of the uniform probability density
function
04
03
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OR

An experiment was designed to study the performance of 4 different detergents for cleaning fuel injectors. The following "cleanness" readings were obtained with specially designed equipment for 12 tanks of gas distributed over 3 different models of engines:
Engine 1
Engine 2
Engine 3
Detergent A
45
43
51
Detergent B
47
46
52
Detergent C
48
50
55
Detergent D
42
37
49
Considering detergents as treatments and the engines as blocks, obtain the appropriate analysis of variance table and test at the 0.01 level of significance whether there are differences in the detergents or in the engines.
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Q.3

i At an electronics plant, it is known from past experience that the
probability is 0.86 that a new worker who has attended the
company's training program will meet the production quota and
that the corresponding probability is 0.35 for a new worker who has
not attended the company's training program. If 80% of all new
workers attend the training program, what is the probability that a
new worker will meet the production quota? Also, find the
probability that a new worker who meets the production quota
will have attended the company's training program?
ii The probability that a new airport will get an award for its design is
0.16, the probability that it will get an award for the efficient use of
materials is 0.24, and the probability that it will get both awards is
0.11.Find the probability that the airport will get the design award
given that it did not get the award for the efficient use of materials.
04
03

i The probability that an integrated circuit chip will have defective
etching is 0.12, the probability that it will have a crack defect is
0.29, and the probability that it has both defects is 0.07.
a What is the probability that a newly manufactured chip will
have either an etching or a crack defect?
b What is the probability that a newly manufactured chip will
have neither defect?
ii A quality-control engineer inspects a random sample of 3 batteries
from each lot of 24 car batteries ready to be shipped. If such a lot
contains 6 batteries with slight defects, what is the probability that
the inspector's sample will contain at least two of the batteries
with defects?
04
03
OR
Q.3

i Write the axioms of probability. Prove that, for any two events
and
ii Define independent events. Check whether the events and are
independent if and
04
03

i The number of gamma rays emitted per second by a certain
radioactive substance is a random variable having the Poisson
distribution with If a recording instrument becomes
inoperative when there are more than 12 rays per second, what is
the probability that this instrument becomes inoperative during
any given second?
ii If has mean 4 and variance 9 while has mean and variance
and the two are independent, find a and
b
04
03
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Q.4

i A safety engineer feels that 30 of all industrial accidents in a
plant are caused by failure of employees to follow instructions. If
this figure is correct, find approximately the probability that among
84 industrialized accidents in this plant anywhere from 20 to 30
(inclusive) will be due to failure of employees to follow instructions.
ii If the annual proportion of erroneous income tax returns filed with
the IRS can be looked upon as a random variable having a beta
distribution with and what is the probability that in
any given year there will be fewer than 10% erroneous returns?
04
03

i The amount of time that a surveillance camera will run without
having to be reset is a random variable having the exponential
distribution with days. Find the probabilities that such a
camera will
a have to be reset in less than 20 days;
b not have to be reset in at least 60 days.
ii If the distribution of the weights of all men travelling by air between
Dallas and EI Paso has a mean of 163 pounds and a standard
deviation of 18 pounds, what is the probability that the combined
gross weight of 36 men travelling on a plane between these two
cities is more than 6000 pounds?
04
03
OR
Q.4

i The arrival of trucks at a receiving dock is a Poisson process with a
mean arrival rate of 2 per hour.
a Find the probability that exactly 5 trucks arrive in a two-hour
period.
b Find the probability that exactly 2 trucks arrive in a one-hour
period and exactly 3 trucks arrive in the next one-hour period.
ii If the probability density of a random variable is given by
find the probabilities that the random variable will take on a value
a between 0.2 and 0.8 b between 0.6 and 1.2.
04
03

i The following are the times between 6 calls for an ambulance in a
certain city) and the patient's arrival at the hospital: 27, 15, 20, 32,
18 and 26 minutes. Judge the reasonableness of the ambulance
service's claim that it takes on the average of 20 minutes between
the call for an ambulance and the patient's arrival at the hospital.
ii Find if the probability that a random variable having the standard
normal distribution will take on a value less than is 0.0217.
04
03
Q.5

i The following show the gain in reading speed of 8 students in a
speed-reading program, and the number of weeks they have been in
the program:
No. of weeks
3
5
2
8
6
9
3
4
Speed gain
86
118
49
193
164
232
73
109
Fit a straight line by the method of least squares.
ii The coefficient of correlation between two variables X and Y is
0.48. The covariance is 36. The variance of X is 16. Find the
standard deviation of Y.
05
02

Find Karl Pearson's coefficient of correlation between sales units) and expenses Rs.) of the following ten firms:
Firms
1
2
3
4
5
6
7
8
9
10
Sales
50
50
55
60
65
65
65
60
60
50
Expense
11
13
14
16
16
15
15
14
13
13
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OR
Q.5

i Ten competitors in a beauty contest are ranked by three judges
B and C in the following order:
A
1
6
5
10
3
2
4
9
7
8
B
3
5
8
4
7
10
2
1
6
9
C
6
4
9
8
1
2
3
10
5
7
Use the rank correlation method to determine which pair of judges
has the nearest approach to common tastes in beauty.
ii The coefficient of rank correlation of the marks obtained by 10
students in two subjects was found to be 0.5. It was later
discovered that the difference in ranks in two subjects obtained by
one of the students was wrongly taken as 3 instead of 7. What
should be the correct value of the coefficient of rank correlation?
05
02

The following data shows sales units) of a firm from year 2001 to 2011. Calculate three yearly moving averages:
Year











Sales
242
250
252
249
253
255
251
257
260
265
262
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