Download Probability and Statistics Exam Questions from Cork Institute of Technology and more Exams Mathematics in PDF only on Docsity!
Cork Institute of Technology
Bachelor of Science (Honours) in Advanced Manufacturing Technology – Award
Bachelor of Science (Honours) in Process Plant Technology –
Award
(EPPTE_8_Y4)(EAMTE_8_Y4)
Summer 2008
Bridging Mathematics
(Time: 1.5 Hours)
Instructions Answer any THREE questions. All questions carry equal marks. Statistical tables are provided.
Examiners: Dr. T. Creedon Mr. C. O’Conaill Mr. N. Kingston Mr. J. Phelan
Q1. (a) In a factory, machines A, B and C produce, respectively, 20, 30 and 50 percent of the total output. Of their respective outputs 5, 3 and 2 percent are defective. An item is chosen at random. (i) What is the probability that it is defective? (ii) If the item is found to be defective, what is the probability that it was produced by machine A? (7 marks)
(b) Given that P(A or B) = 0.7 and P(B) = 0.2, find P(A) and P(A and B) if (i) A, B are mutually exclusive (ii) A, B are independent (6 marks)
(c) A company purchases large lots of a certain type of electronic device. A method is used that rejects a lot if 2 or more defectives are found in a random sample of 100 units. (i) What is the probability of accepting a lot that is 1% defective? (ii) What is the probability of rejecting a lot that is 5% defective?
(7 marks)
Q2. (a) The hourly number of failures of a testing instrument is a Poisson random variable with, on average, 1 failure in 5 hours. (i) What is the probability of at least one failure in an hour? (ii) What is the probability of 2 failures in a 10-hour interval? (5 marks)
(b) Specifications require that a product should have certain quality characteristics, which can only be determined by a destructive test. The product is made in batches of 1,000. The current inspection scheme is to select four items from each batch. If all four meet the quality specification, we accept the batch. If two or more fail, we reject the batch. Otherwise, we take a further sample of two items. If both pass the inspection, we accept the batch, otherwise we reject the batch. What is the probability that we accept a batch that is 5% defective? (8 marks)
(c) Cement is packed into bags. The weights of the bags are normally distributed with a mean weight of 30 kg and a standard deviation of 0.09 kg. (i) What percentage of bags weigh between 29.9 kg and 30.6 kg? (ii) Under EU law a minimum weight must be printed on each bag and not more than 1 % of the bags are to weigh less than this printed weight. What value should this printed weight have? (7 marks)
Q3. (a) A machine that automatically packs sugar into bags is known to operate so that the contents of bags are normally distributed with mean 1 kg and standard deviation 0.1 kg. What is the probability that 5 such bags have a mean weight in excess of 1.05 kg? (5 marks)
(b) A random sample of 12 metal rods is selected from the output of a machine and the diameters are measured, with the following results (in mm):
8.23 8.21 8.20 8.28 8.24 8. 8.25 8.22 8.21 8.24 8.26 8.
Find a 99 % confidence interval for the mean rod diameter. (7 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)
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 ) = n^ C pr r^ (1 − p ) n^ − r
Poisson Distribution
e m^ mr P X r r
− = =
Hypergeometric Distribution
M N M r n r N n
C C
P X r C
− = = −
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
( )
2 2 2 ( ) 1 1
x x x x s n n n
−^ −
∑ ∑ ∑
Normal Distribution Theory
, where is ( , )
x z X N
, where is ( , )
x z X N
n
Sampling Theory Means Proportions
E x ( ) SD x ( ) n
= μ = E p ( ) SE p ( ) (1^ )
n
s x Z n
± (^) p Z p (1^^ p ) n
s N n x Z n N
p p N n p Z n N
2 2 2
Z s n E
2 2
Z p (1 p ) n E
f
n n n N
f
n n n N
Hypothesis Testing
One Sample t test Two sample t test d.f. = n- 1
d.f .=
2 2 2 1 2 1 2 2 2 2 2 1 2 1 2 1 1 2 1
s s n n s s n n n n
X
t s n
− μ = (^1 2 )^ (^1 2 ) 2 2 1 2 1 2
X X
t s s n n
