Average Burning Time - Bridging Mathematicsy- Exam, Exams of Mathematics

Download Average Burning Time - Bridging Mathematicsy- Exam and more Exams Mathematics in PDF only on Docsity! 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) AMT/PPT Formulae and Tables 2008 Addition Law ( or ) ( ) ( ) ( and )P A B P A P B P A B= + − Multiplication Law ( and ) ( | ) ( )P A B P B A P A= Binomial Distribution ( ) (1 )n r n rrP X r C p p −= = − Poisson Distribution ( ) ! m re mP X r r − = = Hypergeometric Distribution ( ) M N M r n r N n C CP X r C − −= = Exponential ( ) , for 0xf x e xλλ −= ≥ ( ) ( ) 1 xP X x F x e λ−≤ = = − Sample mean x x n = ∑ Sample standard deviation ( )22 2( ) 1 1 x xx x ns n n −− = = − − ∑∑∑ Normal Distribution Theory , where is ( , )xz X Nµ µ σ σ − = , where is ( , )xz X N n µ µ σσ − = Sampling Theory Means Proportions ( ) ( )E x SD x n σµ= = (1 )( ) ( )E p SE p n π ππ −= = sx Z n ± (1 )p pp Z n − ± 1 s N nx Z Nn − ± − (1 ) 1 p p N np Z n N − − ± − 2 2 2 Z sn E = 2 2 (1 )Z p pn E − = 1 f nn n N = + 1 f nn n N = + Hypothesis Testing One Sample t test Two sample t test d.f. = n-1 d.f.= 22 2 1 2 1 2 2 22 2 1 2 1 2 1 21 1 s s n n s s n n n n   +               + − − Xt s n µ− = ( ) ( )1 2 1 2 2 2 1 2 1 2 X X t s s n n µ µ− − − = + Laplace Transforms For a function ( )f t the Laplace Transform of ( )f t is a function in s defined by 0 ( ) ( ) , for 0.stF s e f t dt s ∞ −= >∫ ( )f t ( )F s A = constant A s Nt 1 ! N N s + ate 1 s a− sin tω 2 2s ω ω+ cos tω 2 2 s s ω+ ' ( )f t ( ) (0)sF s f− ' ' ( )f t 2 '( ) (0) (0)s F s sf f− − First order linear D.E.s: • [ ]∫ += ΡΡ cdxxgeey )( 1 Second order linear D.E.s with constant coefficients: • xrxr ececy 21 21 += • xrxr xececy 11 21 += • xecxecy xx ββ αα sincos 21 +=