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1 .,L A_u_t_u_m_n_2_0_1_7_- M_a_i_n_E_x_a_m ~ SEAT NUMBER: iuts UNIVERSITY OF TECHNOLOGY SYDNEY STUDENT NUMBER: SURNAME: {FAMILY NAME) OTHER NAMES: This paper and all materials issued must be returned at the end of the examination. They are not to be removed from the exam centre. Examination Conditions: It is your responsibility to fill out and complete your details in the space provided on all the examination material provided to you. Use the time before your examination to do so as you will not be allowed any extra time once the exam has ended. You are not permitted to have on your desk or on your person any unauthorised material. This includes but not limited to: Mobile phones Smart watches and bands Electronic devices Draft paper (unless provided) Textbooks (unless specified) Notes (unless specified) You are not permitted to obtain assistance by improper means or ask for help from or give help to any other person Mathematical Modelling 2 Time Allowed: 2 hours and 10 mi Includes 10 minutes of reading time. Reading time is for reading only. You are n paper in any way during reading time. If you wish to leave and be re-admitted (including to use the toilet), you have to wait until 90 mins has elapsed. All questions are of equal value Answer all questions Answer each question in a separate booklet clearly indicating the question number on the front cover During the exami seek permission ( from a supervisor befo Leaving early Using the toilet Accessing your bag The necessary statistical tables and formula sheet are at the end of the paper. Disciplinary action will be taken against you if you infringe university rules. Do not open your exam paper until instructed.. Page 1 of 18

2 Mathematical Modelling 2 - Main Exam Rough work space Do not write your answers on this page. P'~9~ 2-~t ; a

3 33230 Mathematical Modelling 2 - Main Exam ********ANSWER IN A SEPARATE BOOKLET******** Question 1. {4 marks+ 3 marks+ 8 marks+ 5 marks= 20 marks) a) (i) Find the inverse of the matrix A=G i ). -2 {ii) Hence or otherwise find the solution to the system of linear equations 4x + 2y + z = 3 4x + y + 2z = 4 4x- y- 2z = 12. b) Find and classify the critical points of the function c) Find the volume of the solid bounded on i er above by the paraboloid and below by the plane... Page 3 of 18

4 Mathematical Modelling 2 - Main Exam ******** ANSWER IN A SEPARATE BOOKLET******** Question 2. (5 marks + 8 marks + 7 marks = 20 marks) ( -13 a) LetA= (i) Show that it 1 = 3 is an eigenvalue of the matrix A., (ii) Find the eigenvectors u 1 associated with A- 1. b) Use the method of Lagrange multipliers to find the maximum value ofth f(x,y) = 3x 2 + 2xy + 3 on the circle c) Use spherical coordinates to evaluat each point is proportional to the z coordinate. P~9~ 4 ~t ;-;;... _.... :

5 33230 Mathematical Modelling Main Exam ******** ANSWER IN A SEPARATE BOOKLET******** Question 3 ( ) a) The following time series plot shows the change in the number oflong term visitors entering Australia over time. Identify any trends or cycles in this series. Time Series Plot of Number of Long Term Visitors to Australia ~ T 0 Q) c Q)..c E :: z V (Continues on next page)... ~.... Page 5 of 18

6 P'~9~-6~f ; a ~ -~~~~-~-~~-~~-~~~~i-~-~!.~~~~!~i-~-~-~.-:..~.~~-~--1!-.~~-~--- b) Using the following graphs and summary statistics, corresponding to the same set of measurements, describe the distribution of the measurements. In your answer, you should only refer to appropriate measures for centre and spread. You should also mention the symmetry/skewness of the distribution and identify any outliers (or state that there are none). Descriptive Statistics: Variable Mean StDev C Minimum Ql Median Q Maximum IQR Histogram of Measurements Normal 25 > '-> a: 10 il) * * ~ ~../Over

7 ... c) Consider the circuit below Mathematical Modelling 2 - Main Exam (i) Suppose that the lifetime of an individual component is exponentially distributed with a mean of 6 hours. What is the probability that an individual component lasts longer than 4 hours? (ii) Assume that all components have lifetimes that take the distribution in part (i), and are independent of the other components. What is the probability that the entire system is working after 4 hours? d) The reaction time of a driver to visual stimulus is normally di of0.7 seconds and a standard deviation of0.04 seconds (i) What is the probability that a reaction seconds? (ii) What is the distribution of the mean sampled drivers? (iii) What is the probability that them 0.78 seconds?.../over... ~... ~.... Page 7 of 18

8 Mathematical Modelling 2 - Main Exam ******** ANSWER IN A SEPARATE BOOKLET******** Question 4 ( ) a) Two different analytical tests can be used to determine the impurity level in steel alloys. Eight specimens are tested using both procedures, and the results are shown below. Use the Minitab output provided to answer the following questions. (i) Perform a hypothesis test to determine whether the differences between the observations are normally distributed. State your hypotheses, the test statistic, a p-value, the decision made and your conclusion. (ii) Test whether the mean level of impurity for both of the types of alloy ar same or not. State your hypotheses, a p-value, the decision indicate which output you used to make your decision. ur conclusion. Also Output5.1: Two-Sample T-Test and CI: Test 1, Two-sample T for Test 1 vs Test 2 N Test 1 8 Test 2 8 Mean StDev Difference mu (Test 1) - Estimate for difference: - 95% CI for difference: T-Test of difference = P-Value DF 13 Output 5.2: Probability Plot of Testl-Test2 Normal Mean S!Dev 0.1Sil8 N AD o.40d P-Value /Over f'~9~ a ~r-; a ~

9 ... Output 5.3: Paired T-Test and CI: Test 1, Test 2 Paired T for Test 1 - Test Mathematical Modelling 2 - Main Exam Test 1 Test 2 N 8 8 Mean StDev SE Mean Difference % CI for mean difference: ( , ) T-Test of mean difference = 0 (vs not = 0): T-Value = P-Value = 0.01 b) A group of chemical engineers studies the relationship between the temperature 0 C) and the viscosity (mpa.s) of a certain solvent. Minitab Output 6.2 and Minitab Output 6.3 (on the page) were obtained. (i) What is the equation of the regression line? (ii) Using the regression output, test whether there is a signific of the solvent and its viscosity. State your hypothese made and your conclusion. he temperature decision (iii) Would it be reasonable to use temperature answer using appropriate statistical measur (iv) Use the residual plots in Minitab Out appropriateness of the linear model for this data. Minitab Output 6.2: Regression emperature s SE Coef T % R-Sq(adj) 98.1% p Regression 1 Residual Error 6 Total 7 DF ss MS F p /Over... Page 9 of 18

10 .~~~~.~.~~.~~.~~~~i.~~!.~.c?.~~!!i.~.~.~.::.~.~.i.~.. ~'!.~.~.... Minitab Output 'i a II :/... --/-~..--- Normal Probability Plot 7'' Residual Plots for Viscousity /.. -,.,...-"" !..-/ : _A ~ "li =.. -;; _.?.,.. Ill: Versus Fits o.6 o.a 1.0 Residual Fitted Value c =- 1.0 &i: Hlstogram o.oo Residual ~. "li ' = Ill: p-~9~. 1 "6. ~f "

11 Mathematical Modelling 2 - Main Exam Table of Integrals d -secx = secxtanx xn+1 dx xndx= --+K, n+1 sinxdx =- cosx + K sinhxdx = coshx + K ---;::.::::;;::::::: FORMULA SHEET n # n-1 1 =~ dx = sin - 1 ~ + K a2- x2 a sin 2 x dx = 2 x - 4 sin 2x + K ~ dx = log lxl + K cosn u du = - cosn-l u sin u + -- n n sinn 'U du = _..!._ sinn-1 u cos Formulas for multiple integrals n. cosxdx = sinx + K coshxdx = sinhx + K 2 2 dx =-tan a +x a cos 2 xdx = 1 n-1 1. Cylindrical coordinates: x = r 2. Spherical coordinates: p 2 sin dpd db psin sinb, z = pcos, dv 3. Mass of a solid: z dv, where p(x, y, z) is the density 4. )dv, y= ~jjj yp(x,y,z)dv, z= ~jjj zp(x,y,z)dv Volume= lf(x,y)da Volume= L dv an of the transformation given by x = x ( u, v) and y = y( u, v) is ]( ) = o(x, y) = u,v o(u,v) ox ox ou ov oy oy ou ov ox fy ox oy ou fv fv ou... Page 11 of 18

12 Mathematical Modelling 2 -Main Exam Statistical Formulae Basic Statistical Results 8 2 = _ 1 (tx; _ n-1 i=l nx 2 ) (Qi) = i(n + 1) 4 x -x Z-score = _z: s Probability Distributions Discrete Distributions P(X::; x) = L P(X =xi) Var(X) = E(X 2 ) - ( E(X)) 2 Continuous Distributions P(X < x) = 1~ (t)dt Var(X) = E(X 2 ) - ( E( -p x = 0, 1,...,n E(X) = np Var(X) = np(1- p) ) n-x Poisson Di ution (Discrete) e-,\,\x P(X = x) = 1 x = 0, 1,... X. Exponential Distribution (Continuous) f(x) =..\e->.x x > 0 E(X) =..\ Var(X) =..\ E(X) = t Var(X) = ]2... Page 12 of 18..

13 ... -~-~~~~- -~~~~.t;.~~~~~-~~. ~.<:>.~~!!~~-~- -~-::. ~-~-i.r:t. -~~~~ Linear Combinations of Random Variables If Y = a1x1 + a2x anxn, then E(Y) = a1e(x1) + a2e(x2) ane(xn), and Var(Y) = aivar(x1) + a;var(x2) a~var(xn) Normal Approximation to the Binomial Distribution P(a <X< b)= p (a np < z < b np) - - np(1- p) - - np(1- p) Inference for a Single Mean Population Variance Known x- Lo a Z = ;;:;; X ± Za/2 ;;:;;n ajyn Y' Population Variance Unknown x-lo _ s T = j ;;:;; X± ta/2 n-1 ;;:;; s yn ' yn Inference for a Single Proportion z = x- npo npo(1- Po) _ (Za/2)2 ~( 1 ~) n- -- p -p E df = ( R - 1) X ( c - 1) Simple Linear Regression T=_b_1_ se(b1 )... ~... ~.... Page 13 of18

14 P"~~~ ;4-~i-1a Mathematical Modelling 2 - Main Exam Cumulative Standard Normal Distribution z ! o.018:m : T s such that P(Z < z) = p p z

15 Mathematical Modelling 2 - Main Exam Cumulative Standard Normal Distribution z : such that P(Z<z)=p p z......, Page 15 of 18

16 Mathematical Modelling 2 - Main Exam Percentage Points of the N(O,l) Distribution Za The tabulated value, Za, is such that Za P~9~ ; 5 ~i ~

17 Mathematical Modelling 2 - Main Exain Cumulative Student t Distribution v Ln a The tabulated value, t a Page 17 of 18

18 Mathematical Modelling 2 - Main Exam Cumulative Chi-Squared Distribution l/ a =a: (\ Xa,v "P'~9~ 18"~i"18... :

~UTS UNIVERSilY OF TECHNOLOGY SYDNEY

Spring 2016 -Main Exam SEAT NUMBER: ~UTS UNIVERSilY OF TECHNOLOGY SYDNEY STUDENT NUMBER: SURNAME: (FAMILY NAME) OTHER NAMES: This paper and all materials issued must be returned at the end of the examination.