CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Examinations 2007/08
Module Title: Bridging Mathematics
Module Code:
School: Mechanical & Process Engineering
Programme Title:
Bachelor of Science (Honours) in Process Plant Technology - Award
Bachelor of Science (Honours) in Advanced Manufacturing Technology - Award
Programme Code: EPPTE_8_Y4
EAMTN_8_Y4
Internal Examiner(s): Mr. C. O’Conaill
Dr.T. Creedon
Instructions: Answer THREE questions. Statistical tables are provided
Duration: 1.5 HOURS
Sitting: Spring 2008
Requirements for this examination:
Note to Candidates: Please check the Programme Title and the Module Title to ensure that you have received the
correct examination paper.
If in doubt please contact an Invigilator.
1. (a) Machines A and B make components. Of those made by machine A, 95 % are reliable; of
those made by machine B, 92 % are reliable. Machine A makes 70 % of the components
with machine B making the rest. Calculate the probability that a component picked at
random is
(i) made by machine A and is reliable;
(ii) made by machine B and is unreliable;
(iii) reliable. (7 marks)
(b) Let A and B be events such that P(A) = 0.6, P(B) = 0.4 and P(A or B) = 0.9.
(i) Evaluate P(A and B).
(ii) Evaluate P(B|A).
(iii) Are A and B mutually exclusive events?
(iv) Are A and B independent events? (6 marks)
(c) A manufacturer sets up the following sampling scheme for accepting or
rejecting large crates of identical items of raw material received. He takes a random
sample of 20 items from the crate. If he finds more than two defective items in the sample,
he rejects the crate, otherwise he accepts the crate. It is known that 5 % of these type of
items are defective. Calculate the proportion of crates that will be rejected.
(7 marks)
2. (a) The number of breakdowns of a machine is a Poisson random variable, with on average, 2
breakdowns per month.
(i) What is the probability of at least one breakdown in any particular month?
(ii) What is the probability of 3 breakdowns in a two-month period?
(5 marks)
(b) Certain shipments of insulators were subject to inspection by a high-voltage laboratory.
Each shipment consists of a batch of 1000 insulators. One of the important tests was a
destructive one. The procedure adopted was as follows:
Select 6 insulators at random from the batch and test these. If all 6 pass the test, accept the
batch. If 2 or more fail, reject the batch. If only 1 insulator fails, take a second sample of 6
insulators. If all 6 insulators in the second sample pass the test, accept the batch; otherwise
reject it. What is the probability of accepting batches in which 2% of insulators are non-
conforming?
(8 marks)
(c) The lifetime of a certain kind of automobile battery is normally distributed with a mean of
4 years and a standard deviation of 1 year.
(i) What percentage of the batteries last less than 3 years?
(ii) The manufacturer wishes to print a minimum lifetime for the battery and no more
than 1% of the batteries are to have a lifetime less than this printed time. What
value should this printed time have?
(7 marks)
3. (a) A machine that automatically packs cement into bags is known to operate so that the
contents of bags are normally distributed with mean 20 kg and standard deviation 1 kg.
What is the probability that 5 such bags have a mean weight in excess of 20.8 kg?
(4 marks)
(b) The quality control manager at a light bulb factory needs to estimate the average lifetime
of a batch of 2,000 light bulbs. A sample of 50 light bulbs yielded a sample mean lifetime
of 350 hours and a sample standard deviation of 100 hours.
(i) Find a 95 % confidence interval for the average lifetime of the light bulbs in the
entire batch.
(ii) What sample size would be necessary to estimate this population mean to within
20 hours with 95 % confidence?
(8 marks)
(c) A large shipment of air filters is received by an auto supply company. A sample of 100 air
filters is taken and reveals that 15 of the air filters sampled are unusable.
(i) Find a 95 % confidence interval for the proportion of air filters in the shipment
that are unusable.
(ii) What sample size would be necessary to estimate this population proportion to
within 0.04 with 99 % confidence?
(8 marks)
4. (a) The manufacturer of a brand of emergency flare claims that the average burning time for
the brand is 12.6 minutes. A random sample of 8 flares was tested and yielded the
following burning times (in minutes):
11.4 10.4 7.2 13.4 12.7 11.1 11.8 13.6
(i) State the null and alternative hypotheses.
(ii) Test the manufacturer’s claim at the 0.05 level of significance.
(iii) State your conclusions clearly.
(10 marks)
(b) Management of a steel company wishes to determine if there is any difference in
performance between the day-shift workers and the evening-shift workers. A sample of 25
day-shift workers reveals an average output of 74.3 parts per hour with a sample standard
deviation of 16 parts per hour. A sample of 15 evening-shift workers reveals an average
output of 69.7 parts per hour with a sample standard deviation of 18 parts per hour.
(i) State the null and alternative hypotheses.
(ii) Test the null hypothesis at the 0.1 level of significance.
(iii) State your conclusions clearly.
(10 marks)
5. (a) The repair times for computers have an exponential distribution with a mean of
24 minutes.
(i) Find the probability that a repair time will be less than 12 minutes.
(ii) Estimate the percentage of repair times that are between 20 and 28 minutes.
(iii) How much time should be allowed for each repair so that the chance of any
one repair time exceeding this allowed time is only 0.15?
(8 marks)
(b) Solve the following differential equations:
dx
(i) + 4 x = e −2t , x(0) = 0
dt
d 2x dx
(ii) 2
− 3 + 2 x = e−t , x(0) = 1, x ' (0) = 0
dt dt
(12 marks)
AMT/PPT Formulae and Tables 2008
Addition Law
P ( A or B ) = P( A) + P( B) − P( A and B)
Multiplication Law
P ( A and B) = P( B | A) P( A)
Binomial Distribution
P ( X = r ) = nCr p r (1 − p ) n − r
Poisson Distribution
e− m mr
P( X = r ) =
r!
Hypergeometric Distribution
M
Cr N − M C n − r
P( X = r ) = N
Cn
Exponential
f ( x) = λ e− λ x , for x ≥ 0
P ( X ≤ x) = F ( x) = 1 − e − λ x
Sample mean
x=
∑x
n
Sample standard deviation
(∑ x)
2
∑ (x − x ) 2 ∑x 2
−
n
s= =
n −1 n −1
Normal Distribution Theory
x−µ
z= , where X is N ( µ , σ )
σ
x −µ
z= , where X is N ( µ , σ )
σ
n
Sampling Theory
Means Proportions
σ π (1 − π )
E(x ) = µ SD( x ) = E ( p) = π SE ( p ) =
n n
s p(1 − p)
x±Z p±Z
n n
s N −n p(1 − p) N − n
x±Z p±Z
n N −1 n N −1
2 2
Z s Z 2 p(1 − p )
n= 2 n=
E E2
n n
nf = nf =
n n
1+ 1+
N N
Hypothesis Testing
One Sample t test Two sample t test
d.f. = n-1 s12 s22
2
+
d.f.= n1 n2
2 2
s12 s22
n1 + n2
n1 − 1 n2 − 1
t=
X −µ
t=
(X 1 − X 2 ) − ( µ1 − µ 2 )
s
s12 s22
n +
n1 n2
Laplace Transforms
For a function f (t ) the Laplace Transform of f (t ) is a function in s defined by
∞
F ( s ) = ∫ e − st f (t )dt , for s > 0.
0
f (t ) F ( s)
A = constant A
s
tN N!
s N +1
e at 1
s−a
sin ωt ω
s + ω2
2
cos ωt s
s + ω2
2
f ' (t ) sF ( s ) − f (0)
f ' ' (t ) s 2 F ( s ) − sf (0) − f ' (0)
First order linear D.E.s:
1
[
• y = Ρ ∫ e Ρ g ( x)dx + c
e
]
Second order linear D.E.s with constant coefficients:
• y = c1e r1 x + c 2 e r2 x
• y = c1e r1 x + c 2 xe r1x
• y = c1e αx cos βx + c 2 e αx sin βx
