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Cork Institute of Technology
Bachelor of Science (Honours) in Advanced Manufacturing Technology ā Award
Bachelor of Science (Honours) in Process Plant Technology ā Award
(NFQ ā Level 8)
January 2007
Bridging Mathematics
(Time: 1.5 hours)
Answer three questions. Examiners: Mr. K. J. Kelly
All questions carry equal marks. Ms. K. Kelleher
Statistical tables are provided.
- (a) Five percent of items produced by a production process are defective. The items are subjected to a manual inspection procedure. This inspection procedure is not foolproof - in fact 10% of defective items are accepted and 4% of good items are rejected.
(i) An item has been rejected by the inspection procedure - what is the probability that it is defective?
(ii) An item has been accepted by the inspection procedure - what is the probability that it is defective? (5 marks)
(b) An incoming batch of 20 items is to be inspected. Acceptance procedure is to sample and test five items (without replacement) and accept the batch if the sample contains at most one defective. Find the probability the incoming batch will be accepted if eight out of the twenty in the batch are defective. (5 marks)
(c) The lives of component C3PO is normally distributed with mean 2500 hours and standard deviation 500 hours. What percentage of these components will meet a mission requirement life of 1850 hours? (5 marks)
(d) Ropes produced by a process have breaking strength normally distributed with mean Ī¼ kg and standard deviation 10 kg. What is the minimum value Ī¼ can have if the probability of a rope having a breaking strength less than 1000 kg is to be 0.01 or smaller? (5 marks)
- (a) A box contains SIX components three of which TWO are known to be defective. The items are to be tested in turn (without replacement) until the TWO defectives are found. Find the probability that exactly THREE tests are required to identify all three defectives. (4 marks)
(b) Ten percent of items supplied by manufacturer X fail during the guarantee period. If twenty items are purchased find the probability that during the guarantee period
(i) exactly three of the items fail (ii) less than two of them fail. (7 marks)
(c) A particular type filling machine jams on average 1.2 times per hour, the pattern of such occurrences following a Poisson distribution. For a particular machine find the probability of
(i) Exactly two jams during a one hour period. (ii) At least one jam in a two hour period.
If the company has three of these machine running for a period of one hour what is the probability that no jam will occur during that period? (9 marks)
- (a) A sample of 8 alloy filaments was selected at random from the output of a particular line. The melting points of the filaments ( o^ C) were determined as follows:
350 370 330 315 325 335 316 340
(i) Calculate the sample mean and the sample standard deviation. (ii) Find a 95% confidence interval for the mean melting point (iii) Find also a 99% lower confidence limit on this mean learning time. (12 marks)
(b) The average assembly time for the R2D2 product was over time established to be 26.5 minutes. A modification was made to the assembly method and a sample of 18 assembly times using the new method yielded a sample mean of 24.3 minutes with a sample standard deviation of 5. minutes. Has the new method successfully reduced the average assembly time or does the previous average stand? Perform a formal test of hypothesis and report your findings clearly Use a 5% level of significance (Ī±=0.05). (8 marks)
AMT/PPT Bridging Statistical Formulae and Tables 200 6
Addition Law
P ( A or B) = P ( A ) + P B ( ) ā P ( A and B)
Multiplication Law:
P ( A and B) = P ( A P B A ) ( )
Discrete Distributions :
Pr ( )
x n p qx^ n^ x
x
= ļ£«^ ļ£¶^ ā
Pr ( )
m e^ x^ m
x
x
ā
Pr ( )
D N D
x n x
x
N
n
(Negative) Exponential Distribution
at
f t ae
ā
at
F t e
ā
at
R t e
ā
E x
a
Normal distribution Theory
, where ( , )
x z X N
, where , x z X N n n
Estimation
n 1
s x t n
Hypothesis Testing
n
s
x t (^) n 1 0
ā Ī¼ ā =
( ) ( ) 1 2
2 2 1 2 1 2 2 1 1 2 2 2 1 2 1 2
where
n n p p
x x n^ s^ n^ s
t s
n n
s
n n
ā ā ā ā^ +^ ā
Regression
( )
n
x S (^) xx x
2 2 (^ )
2 2 yy
y S y n
( )( )
n
x y S (^) xy xy
xy xx
S
b S
= a = y ā bx
xy xx yy
S
r S S
(^2 )
or equivalently
n s s
r n
t t r
r r
n
ā
NB. You may consult Murdoch & Barnes tables for other formule.
