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6 Pages

### EG243 Control Systems 08 paper

Course: ENGINEERIN EG 243, Spring 2010
School: Swansea UK
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Word Count: 1008

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ABERTAWE PRIFYSGOL SWANSEA UNIVERSITY School of Engineering SEMESTER 1 EXAMINATIONS JANUARY 2008 EG-243 CONTROL SYSTEMS LEVEL 2 UNIVERSITY CALCULATORS ONLY Translation dictionaries are not permitted, but an English dictionary may be borrowed from the invigilator on request. Time allowed: 2 hours Answer Question 1 and TWO other questions Formulae and Transform Tables included TURN OVER PAGE 1 OF 6 1. Attempt...

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ABERTAWE PRIFYSGOL SWANSEA UNIVERSITY School of Engineering SEMESTER 1 EXAMINATIONS JANUARY 2008 EG-243 CONTROL SYSTEMS LEVEL 2 UNIVERSITY CALCULATORS ONLY Translation dictionaries are not permitted, but an English dictionary may be borrowed from the invigilator on request. Time allowed: 2 hours Answer Question 1 and TWO other questions Formulae and Transform Tables included TURN OVER PAGE 1 OF 6 1. Attempt all parts For Question 1 parts (a), (b), (c) and (d) assume a control system with forward path G(s) and negative feedback path, H(s), given by: G( s) = K ( s + 1)( s + 2) and H ( s) = ( s + 5) ( s + 10) (a) Derive the closed loop steady state error (ie output - input) for (i) a unit step input (ii) unit ramp input. Assume K = 1. [5 marks] (b) Derive the system closed loop frequency response as 0 . Assume K=1. [5 marks] (c) Give an expression for the closed loop system impulse response, when K is just above zero. Numerical values for the expression coefficients are not required. [5 marks] (d) Sketch the Nyquist diagram giving values at (i) 0 and (ii) . [5 marks] (e) For the control system in Figure Q1e derive the close loop transfer function, C(s)/R(s). [5 marks] J(s) R(s) E(s) C(s) G(s) R(s) E(s) + C(s) G(s) + H(s) + H(s) + Figure Q1abcd Figure Q1e TURN OVER PAGE 2 OF 6 2. (a) Define the characteristic equation of a feedback control system in its canonical form. Describe (i) the importance of this equation in control systems and (ii) the fundamental principle used to solve the equation when using the root locus technique. NB a list of root-locus plotting rules is not required. [9 marks] (b) The open loop transfer function of a negative feedback control system is given by: K ( s + 4)( s + 1) 2 GH ( s ) = . s 3 ( s + 10) 2 Sketch the root locus for the system indicating (i) values of any asymptotes (ii) points on the loci when K is zero. Comment on the nature of the system output (oscillatory, asymptotic, stable) as K increases from zero. [11 marks] (c) Estimate the minimum value of K to minimise the number of oscillatory components in the system closed loop response. [5 marks] 3. (a) Explain the stability criterion of a negative feedback control system using a Nyquist diagram. Illustrate your answer with the example of a transfer function: GH ( s ) = K . ( s + a )( s + b)( s + c) [7 marks] (b) Derive an expression for Ks, the value of K at the stability limit, in terms of a, b and c. Determine Ks when a = b = c = 1. [14 marks] (c) Comment on the case when: GH ( s ) = K ( s + 1) n lim n [4 marks] TURN OVER PAGE 3 OF 6 4. (a) Describe typical performance criteria in feedback control system. Use an example such as a passenger lift (elevator) in a building to illustrate answer. your [5 marks] (b) Figure Q4 shows a feedback control system where GP represents the mechanical and drive systems and GC represents a compensation unit. Describe the three terms in a conventional compensation unit and directly link these terms to your response to part (a) above. [5 marks] (c) If the initial representations are: GP = n2 1 , H = 1 and GC = K , 2 ( s + 1) ( s 2 + 2n s + n ) explain the nature of the system output as , n and K vary. [6 marks] (d) Determine the system steady state error for a unity step input using the values for GP , H , and GC given in (c) above. Comment on the response as K is increased. [6 marks] (e) Propose a transfer function for Gc that would help reduce the steady state error of the system. [3 marks] R(s) + _ GC GP C(s) H Figure Q4 TURN OVER PAGE 4 OF 6 EG-243: Control Systems Formulae and Transform Tables for Examination Use Steady-state Performance For a system with open-loop transfer function G(s) and unity-gain feedback the steady-state error [sse] is related to system type according to the following table: SYSTEM TYPE Type 1 Type 0 Type of input sse Type 2 position error velocity error acceleration error Step Ramp Parabola 1 1+ K p 1 1+ K p 0 0 0 1 Kv 1 Ka 1 Kv 1 Ka Kp = G(s) at s = 0; Kv = s G(s) at s = 0; Ka = s2 G(s) at s = 0. The error constants are: position error constant, velocity error constant, acceleration error constant, Second order Transient Response Second order Transfer Function 2 n Y (s ) = 2 2 X (s ) s + 2 n s + n 2 2 d + d = . (s + d )2 + d2 or X(s) n2 2 s 2 + 2 n s + n Y(s) n is the undamped natural frequency; is the damping ratio; d = n(1 2) is the damped natural frequency; d = n is the exponential damping frequency; frequencies are in rad s-1. TURN OVER PAGE 5 OF 6 Laplace and Z-Transforms Transform Pairs f(t) F(s) 1 F(z) 1 (t ) (t ) t t2 1 s 1 s2 2 s3 z z -1 t z ( z - 1) 2 t 2 z ( z + 1) e - at te - at sin bt cos bt sin(bt + ) cos(bt + ) e - at sin bt e - at cos bt 1 - e - at at - 1 + e - at ( z - 1) 3 1 s+a 1 (s + a)2 b s + b2 s 2 s + b2 s sin + b cos s 2 + b2 s cos - b sin s2 + b2 b (s + a) 2 + b2 s+a (s + a) 2 + b2 a s( s + a ) 2 z z - e - t t ze- at ( z - e - a t ) 2 z sin bt 2 z - 2 z cos bt + 1 z ( z - cos bt ) 2 z - 2 z cos bt + 1 z [ z sin + sin(bt - ) ] z 2 - 2 z cos bt + 1 z [ z cos + cos(bt - ) ] z 2 - 2 z cos bt + 1 ze - at sin bt z 2 - 2 ze - at cos bt + e-2 at z ( z - e - at sin bt ) z 2 - 2 ze - at cos bt + e-2 at z (1 - e- at ) ( z - 1)( z - e - at ) a2 s 2 (s + a) e- smt e- smt s 1 - e- st s z at ( z - e - at ) - ( z - 1)(1 - e - at ) ( z - 1) 2 ( z - e- at ) z -m z - ( m-1) z -1 1 (t - mt ) (t - mt ) (t ) - (t - t ) END OF PAPER PAGE 6 OF 6
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Swansea UK - ENGINEERIN - EG 243
Page:1 of 5SCHOOL OF ENGINEERINGEXAMINATION MODEL ANSWER SHEETPaper Title: Paper No: EG-243 Question No 1. (a) MAY/JUNE 2009 Examiner(s): JSM MarksC K (s ) = R ( s + a)(s + b)(s + c) + K[2 marks](b)C ( ) 0 = K R abc + KType 0 system so: Thus sse f
Swansea UK - ENGINEERIN - EG 243
Page:! of5SCHOOL OF ENGINEERINGEXAMINATION MODEL ANSWER SHEETPaper Title: Paper No:Question No1.EG-243MAY lJUNE 2009Examiner(s): ISMMarks(a)c (s)= R(s+ a)(s + b)(s + c) + KK[2 marks(b)[2 marks(c)Type 0 system so: Thus sse for:KP=-K a
Swansea UK - ENGINEERIN - EG 243
Page:30f5SCHOOL OF ENGINEERINGEXAMINATION MODEL ANSWER SHEETPaper Title: Paper No:Question NoEG-243MAY/JUNE 2009Examiner(s): JSMMarksnormal The root locus plots systems, from to the upwards), complex s plane as set on loci, thegain K vari~ loc~ &quot;
Swansea UK - ENGINEERIN - EG 243
PRIFYSGOL ABERTAWE SWANSEA UNIVERSITY School of EngineeringSEMESTER 2 EXAMINATIONS MAY/JUNE 2009EG-243 CONTROL SYSTEMS LEVEL 2UNIVERSITY CALCULATORS ONLYTranslation dictionaries are not permitted, but an English dictionary may be borrowed from the inv
Swansea UK - ENGINEERIN - EG 243
Figure 1.1 Simplified description of a control systemControl Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley &amp; Sons. All rights reserved.Rates of change: Newton's law: velocity, acceleration of a mass Force = mass x ac
Swansea UK - ENGINEERIN - EG 243
Swansea University SCHOOL OF ENGINEERING EG 243: Control SystemsCharacteristic EquationThe CE, 1+GH=0, determines the nature of the closed loop system response. To determine values related to the stability limits, then the 2 conditions, magnitude and ph
Swansea UK - ENGINEERIN - EG 243
Figure 4.1a. System showing input and output; b. pole-zero plot of the system; c. evolution of a system response. Follow blue arrows to see the evolution of the response component generated by the pole or zero.Control Systems Engineering, Fourth Edition
Swansea UK - ENGINEERIN - EG 243
Figure 6.1 Closed-loop poles and response: a. stable system; b. unstable systemControl Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley &amp; Sons. All rights reserved.Figure 6.2Common cause of problems in finding closed-l
Swansea UK - ENGINEERIN - EG 243
Figure 5.2 Components of a block diagram for a linear, time-invariant systemControl Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley &amp; Sons. All rights reserved.Figure 5.3 a. Cascaded subsystems; b. equivalent transfer
Swansea UK - ENGINEERIN - EG 243
Figure 8.1 a. Closedloop system; b. equivalent transfer function 1+GH=0 is the characteristic equation (CE) CE determines the nature of the output c(t) H is the feedback path, often arranged to be unity - UNFControl Systems Engineering, Fourth Edition by
Swansea UK - ENGINEERIN - EG 243
Figure 9.1a. Sample root locus, showing possible design point via gain adjustment (A) and desired design point that cannot be met via simple gain adjustment (B); b. responses from poles at A and BControl Systems Engineering, Fourth Edition by Norman S.
Swansea UK - ENGINEERIN - EG 243
Figure 10.3 System with sinusoidal inputControl Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley &amp; Sons. All rights reserved.Figure 10.4 plots for G(s) = 1/(s + Systems Engineering, Fourth Edition by Norman S. Nise 2):
Swansea UK - ENGINEERIN - EG 243
University of Wales Swansea SCHOOL OF ENGINEERING EG 243: Control SystemsRoot Locus1 Plot and calibrate the root locus diagrams of the following systems as a function of the open-loop gain K:a)2K ; s (s + 1)b)K (s + 2 ) ; s (s + 1)c)K (s + 2 ) s(
Swansea UK - ENGINEERIN - EG 243
Root Locus TechniquesWilliam J. Murphy Washington University in St. Louis Reference: Nise (2004): Chapter 8TOPICSThe Root Locus Method Closed-loop poles Plotting the root locus of a transfer function Choosing a value of K from root locus Closed-loop re
Swansea UK - ENGINEERIN - EG 243
Tutor: Dr J.S.D. MASONJrmie BARRERoot locus in Control SystemAnalysis and DesignUniversity of Swansea11/03/2005SISO Design ToolThis presentation is 4 examples to learn to use a part of SISO Design Tool.Example 1+-K s ( s 2 + 2)Example 1 1sts
Swansea UK - ENGINEERIN - EG 243
The root locusTo make the presentation of the SISO Design Tool I took again the training method of the book which was very practical because very visual and very concrete. This method consisted in showing what one wants to learn and use thanks to the use
Swansea UK - ENGINEERIN - EG 243
Swansea UK - ENGINEERIN - EG 260
A.K. SloneEG-260 Dynamics (1)EG-260 DYNAMICS I DampingEG-260 DYNAMICS I Damping .1 1. Introduction .2 1.1. The idealised dashpot.3 2. Underdamped motion .7 3. Overdamped motion.9 4. Critically damped .11 5. Summary .12a.k.slone 20081 of 14A.K. Slone
Swansea UK - ENGINEERIN - EG 260
A.K. SloneEG-260 Dynamics (1)EG-260 DYNAMICS I General Forced Response1.Introduction .21.1. Impulse Response. .3 1.2. Under-damped system .6 1.3. Undamped system .7 1.4. Response to unit impulse .7 1.5. Example.7 2. 3. Response to arbitrary input .10
Swansea UK - ENGINEERIN - EG 260
A.K. SloneEG-260 Dynamics (1)EG-260 DYNAMICS I Harmonic motionIntroduction . 2 1. Revision - Motion in a Circle . 2 2. Circular Motion and the Sine Function . 4 3. Simple Harmonic Motion (SHM) . 5 3.1. 3.2. 3.3. 3.4. 4. Solution of the SHM equation . 6
Swansea UK - ENGINEERIN - EG 260
A.K. SloneEG-260 Dynamics (1)EG-260 DYNAMICS I Lagrange's Equation 1. 2. 3. 4. 5. 6. 7. 8. Introduction .2 An overview of the procedure. .3 Useful energy expressions.4 A mass-spring example .4 A trolley mounted pendulum example .7 Two pendulum example.1
Swansea UK - ENGINEERIN - EG 260
A.K. SloneEG-260 Dynamics (1)EG-260 DYNAMICS I MeasurementIntroduction.2 1. Measurement .21.1. The logarithmic decrement method. .3 1.1.1. Measurement over more than one cycle.5Example 1 .5 1.2. The added mass method. .6 Example 2 .6a.k.slone 20091
Swansea UK - ENGINEERIN - EG 260
A.K. SloneEG-260 Dynamics (1)EG-260 DYNAMICS I Modelling and Energy Methods 1. Kinetic Energy. 3 2. Potential Energy . 4 2.1. Potential Energy in Elastic springs. 4 2.2. Gravitational PE . 5 3. Relating Potential and Kinetic Energy. 5 4. A vertical spri
Swansea UK - ENGINEERIN - EG 260
A.K. SloneEG-260 Dynamics (1)EG-260 DYNAMICS I Multiple Degrees of FreedomEG-260 DYNAMICS I Multiple Degrees of Freedom . 1 1. 2. 2.1. 2.2. 2.3. 2.3.1. 3. 3.1. 4. 5. 5.1. 5.1.1. 5.1.2. 6. Introduction . 2 Two degrees of freedom. 2 Finding eigenvalues a
Swansea UK - ENGINEERIN - EG 260
EG-260 Dynamics 1 Pre-course Revision Questions 1. a. State what is meant by angular velocity. b. A stone is tied to one end of a cord and rotated in a horizontal circle about a point C with the cord horizontal, as shown in Figure 1. The stone has speed v
Swansea UK - ENGINEERIN - EG 260
Swansea UK - ENGINEERIN - EG 260
A.K. SloneEG-260 Dynamics (1)EG-260 DYNAMICS I Stiffness1. 2. 3.Introduction .2 Material Properties.2 Examples of spring constants.23.1. Longitudinal vibration. .3 3.2. Transverse vibration. .5 3.3. Torsional vibration.6 3.4. Summary of spring consta
Swansea UK - ENGINEERIN - EG 260
A.K. SloneEG-260 Dynamics (1)EG-260 DYNAMICS I Harmonic Excitation 1EG-260 DYNAMICS I Harmonic Excitation 1.1 Introduction to harmonic excitation .2 1. Equation of motion .2 2. Harmonic excitation of an undamped system.4 2.1. Example.8 2.2. The phenome
Swansea UK - ENGINEERIN - EG 260
Swansea UK - ENGINEERIN - EG 260
EG-260 DYNAMICS COURSE REVISIONSTANDARD DIFFERENTIALS AND INTEGRALS. 2 Some Series . 3 1. The exponential series. 3 1.1. Sine and Cosine. 3 2. Trigonometrical formulae . 3 Euler Relationship .
Swansea UK - ENGINEERIN - EG 260
1st Printing Errata If you have the second printing, the following corrections have been made. If you have a first printing you will need to print and refer the following corrections. To tell which printing you have: Look on the page, 4 in from the front
Swansea UK - ENGINEERIN - EG 260
A.K. SloneEG-260 Dynamics (1)EG-260 DYNAMICS I GlossarySymbol k m c ccr e z fn T wn wd w rc c crEquationDescription stiffness coefficient mass damping coefficient critical damping coefficient eccentricity damping ratio natural frequency period of os
Swansea UK - ENGINEERIN - EG 260
EG-260EGD260 Dynamics - Example Sheet 1 Single degree of freedom systems1.Formulate the equation of motion for a simple pendulum of length l. If a grandfather clock requires the pendulum to have a period of 2 seconds, what is the required length? Compu
Swansea UK - ENGINEERIN - EG 260
EG-260EG -260 Dynamics I - Example Sheet 5 Multi degree of freedom systems1. Write down the equations of motions for the torsional system of two circular plates with moments of inertia I1 and I2, and torsional stiffness k 1 and k 2. Use variables 1 and
Swansea UK - ENGINEERIN - EG 260
EG-260EG260 Dynamics - Example Sheet 2 Single degree of freedom systems1. Develop the equations of motion and hence obtain the natural angular frequencies of the following systems: i) A mass-spring-bar system (shown below left), assuming that the bar is
Swansea UK - ENGINEERIN - EG 260
EG-260EG-260 Dynamics I - Example Sheet 3 Damped single degree of freedom systems1. A weight of 2 N is attached to a damped spring having a stiffness of 400 N/m and a damping constant of 9.03 kg/s. Determine a) b) c) critical damping constant undamped n
Swansea UK - ENGINEERIN - EG 260
EG-260EG-260 Dynamics I - Example Sheet 4 Forced single degree of freedom systems1. A spring-damper-mass system is excited by an harmonically varying force. At resonance, the amplitude was measured as 0.58 mm, and at 0.8 of the resonant frequency, the m
Swansea UK - ENGINEERIN - EG 260
EG-260EG -260 Dynamics I - Example Sheet 5 -Answers Multiple degree of freedom systems1. Free Body Diagram1(t)kt11 kt2(2 - 1)First disk2(t)&amp; I 1&amp;1 = -k t1 1 + k t 2 ( 2 - 1 ) &amp; I &amp; + (k + k ) - k = 01 1 t1 t2 1 t2 2Second disk&amp; I 2&amp;2 = - k t 2 (
Swansea UK - ENGINEERIN - EG 260
EG-260 Examples Sheet 1 1. This is a rotational problem, so we express Newton's Law in the form&amp; C = I&amp;where C is the applied moment and I is the moment of inertia. In the pendulum caseC = -mgl sin I = ml 2So that&amp; - mgl sin = ml 2&amp;For small angles
Swansea UK - ENGINEERIN - EG 260
EG-260EG260 Dynamics - Example Sheet 2 Single degree of freedom systems Answers 1. i). Moment of inertiaI = m(a + b )2Let be angle of rotation Then moment - k a (a sin( ) = -ka 2 (small angle approximation) So that the equation of motion is or 2 &amp; ka
Swansea UK - ENGINEERIN - EG 260
EG-260 EG-260 Dynamics I - Example Sheet 3 Answers Damped single degree of freedom systems1.k = 400 N / m c = 9.03kg / s mg = 2 N so ccr = 2mn = 2 km = 2 400 * 2 / g = 40 b) n = k 400 g g = = 20 m 2 2 = 44.3rad / s ( 7.05Hz ) c) = c 9.03 1 = = cr 18.06
Swansea UK - ENGINEERIN - EG 260
EG-260EG-260 Dynamics I - Example Sheet 4 Answers Forced single degree of freedom systems1. EOM ism&amp; + cx + kx = F0 cos t x &amp;The steady state solution is of the form x p = X cos(t - )then the magnitude of the steady state response X is given by: f0 X
University of Arkansas Community College at Morrilton - MATH - 246
NAME: Project 1 KEY Super Bowl Point Spreads The following data represent the number of points by which the winning team won Super Bowls I to XXXIX: 25 10 10 1 15 19 4 29 13 7 9 18 22 35 27 16 17 36 17 3 3 419 23 27 21 12 32 10 3 7 17 4 14 3 17 5 45 7Yo
University of Arkansas Community College at Morrilton - MATH - 246
NAME: Project 2 KEY You will be graded on three basic levels: ability to use MINITAB, statistical written explanation, and proper use of English. A researcher believes that as age increases the grip strength (in pounds per square inch) of an individual's
University of Arkansas Community College at Morrilton - MATH - 246
Name _KEY_ ID#_ Project Three _ Chapter FiveUse complete sentences when appropriate; use complete mathematical sentences when appropriate, for example: either P(gold medal) = value or the probability of a gold medal is value . Failure to do so may result
University of Arkansas Community College at Morrilton - MATH - 246
Name _KEY_ ID# _ Project4 _ Chapter 6 I have included a very detailed KEY that is probably at the level of your solutions but not necessarily at the level of your communications. Highlighted in yellow are the details.Use complete sentences when appropria
University of Arkansas Community College at Morrilton - MATH - 246
Math 246KEYName_Project 5 chapter 7Use complete sentences when appropriate; use complete mathematical sentences when appropriate. Failure to do so may result in a lower score. Show all your work. Indicate clearly the methods you use, because you will
University of Arkansas Community College at Morrilton - MATH - 246
Math 246 Project 6KEYName_ ID# _Chapter 8: Do not send a separate MINITAB file, but do include in your report a copy of the MINITAB output. Submit copies of the graphical summary for questions #1 and #2.Use complete sentences when appropriate; use com
University of Arkansas Community College at Morrilton - MATH - 246
Project 7 Four questions; 20 pointsMath 246 Project 7 Chapter NineName_ ID# _Use complete sentences when appropriate; use complete mathematical sentences when appropriate. Failure to do so may result in a lower score. Show all your work. Indicate clear
University of Arkansas Community College at Morrilton - MATH - 246
Project 8 chapter 10 section 1 KEYName _ ID# _ Project 8 Chapter 10KEYUse complete sentences when appropriate; use complete mathematical sentences when appropriate. Failure to do so may result in a lower score. Show all your work. Indicate clearly the
University of Arkansas Community College at Morrilton - MATH - 246
Project 9/Power total points 20 Name _ UWEC ID# _ Project 9 /Power_Use complete sentences when appropriate; use complete mathematical sentences when appropriate. Failure to do so may result in a lower score. Show all your work. Indicate clearly the metho
University of Arkansas Community College at Morrilton - MATH - 246
Project 10 (10 points)Project 10KEYChapter Ten: Hypothesis Testing A. The average monthly bill for cellular phone service is \$68. It is claimed that these costs are now decreasing. A survey yields the following data: 55 48 52 70 72 78 69 65 66 59Test
University of Arkansas Community College at Morrilton - MATH - 246
Project 11 chapter 11 Name_ ID# _20 pointsKEYA.) A study is conducted to investigate the effect of physical exercise on the serum cholesterol level of the individual. It is thought that regular exercise will reduce the level of cholesterol in the blood
University of Arkansas Community College at Morrilton - MATH - 246
Project 12Math 246KEYProject Twelve _/20_Chapter 12READ EVERYTHING CAREFULLY: Show all your work. Indicate clearly the methods you use, because you will be graded on the correctness of your methods as well as on the accuracy of your results and expla
University of Arkansas Community College at Morrilton - MATH - 246
Project 13Project Thi r teen_KEY_Chapter 13/20 pointsREAD EVERYTHING CAREFULLY: Show all your work. Indicate clearly the methods you use, because you will be graded on the correctness of your methods as well as on the accuracy of your results and expl
University of Arkansas Community College at Morrilton - MATH - 246
3002001000100200Project 14 _/20 Chapter 14KEY300400500READ EVERYTHING CAREFULLY: Show all your work. Indicate clearly the methods you use, because you will be graded on the correctness of your methods as well as on the accuracy of your results
University of Toronto - MATH - 246
Project One Sample -KEY: An investigator is interested in examining the effects of alcohol on individual's ability to perform physical activities. Twenty randomly selected individuals consumed a prescribed amount of alcohol, and their time to complete a s
University of Toronto - MATH - 246
Project Two Sample -KEY: You will be graded on three basic levels: ability to use MINITAB, statistical written explanation, and proper use of English. Consider the following data reflecting lengths (explanatory variable) of stay in the hospital (recorded
University of Toronto - MATH - 246
Math 246 Project 3 Sample KEY Use complete sentences when appropriate; use complete mathematical sentences when appropriate. Failure to do so may result in a lower score. Show all your work. Indicate clearly the methods you use, because you will be graded
University of Toronto - MATH - 246
KEYProject Two _ Chapter 6 Use complete sentences when appropriate; use complete mathematical sentences whenappropriate. Failure to do so may result in a lower score. Show all your work. Indicate clearly the methods you use, because you will be graded o
University of Toronto - MATH - 246
KEY P roject F iveP roblem Talk T ime On a Cell Phone: Suppose the talk t ime in digital mode on a MotorolaTimeport P8160 is approximately normally distributed with mean 324 minutes and standard deviation of 24 minutes. A.) What proportion of the t ime