The University Of Newcastle School Of Engineering Assignment

The University Of Newcastle School Of Engineeringassignment Cover S

All assignments are the responsibility of the student. The School of Engineering cannot accept responsibility for lost or unsubmitted assignments. You are therefore advised to keep a copy. First Name: ------------------------------------------------- Family Name: ------------------------------------------------- Student Number: ------------------------------------------------- Course Code: MECH2450 Assignment #2 (2016) Date submitted: ------------------------------------------------- Declaration: I have read and understand the University of Newcastle’s Policy for the Prevention and Detection of Plagiarism Main Policy Document, which is located at I declare that, to the best of knowledge and belief, this assignment is my own work, all sources have been properly acknowledged, and the assignment contains no plagiarism. This assignment or any part thereof has not previously been submitted for assessment at this, or any other University. Student Signature: ------------------------------------------------- Date: ------------------------------ MECH2450 Engineering Computations 2 Assignment #) Singapore Due Date: 9:30 pm, Thursday 31st March 2016. Show all working clearly. Q.1 (25 marks) The size y (in millimetres) of a crack in a Pontiac Trans Am’s front sub-frame weld is described by a random variable X with the following PDF:  ïƒ ïƒ¬    elsewhere y yy yf X ,3/,16/ (a) Sketch the PDF and CDF. (5 marks) (b) Determine the mean crack size. (5 marks) (c) What is the probability that a crack will be smaller than 3 mm? (5 marks) (d) Determine the median crack size. (5 marks) (e) Suppose there are four cracks in the weld. What is the probability that only one of the four cracks is larger than 3 mm? (5 marks) Q.2 (25 marks) The average speed of vehicles on a freeway is being studied. Assume that the standard deviation of vehicle speed is known to be 8 km/h. (a) Suppose observations on 120 vehicles yielded a sample mean of 105 km/h. Determine two-sided 99.5% confidence intervals of the mean speed. (Assume a normal distribution). (6 marks) (b) In part (a), how many additional vehicles’ speed should be observed such that the mean speed can be estimated to within 1 km/h with 99.5% confidence? (6 marks) (c) Suppose Jason and Britney are assigned to collect data on the speed of vehicles on this highway. After each person has separately observed 60 vehicles, what is the probability that Jason’s sample mean will be equal to Britney’s sample mean 0.5 km/h? (7 marks) (d) Repeat part (c) if each person has separately observed 120 vehicles instead. (6 marks) Q.3 (25 marks) The fuel consumption of a certain make of car may not be exactly that rated by the manufacturer. Suppose ten cars of the same model were tested for combined city and highway fuel consumption, with the following results: Car No. Observed fuel consumption 1 7.1 litres per 100km 2 6.4 litres per 100km 3 6.8 litres per 100km 4 6.2 litres per 100km 5 7.8 litres per 100km 6 6.1 litres per 100km 7 6.6 litres per 100km 8 7.8 litres per 100km 9 6.4 litres per 100km 10 7.4 litres per 100km (a) Estimate the sample mean and sample standard deviation of the actual fuel consumption of this particular model of car. (10 marks) (b) Suppose that the manufacturer’s stated fuel consumption of this particular model of car is 7 litres per 100km; perform a hypothesis test to verify the stated fuel consumption with a significance level of 2%. (15 marks) Q.4 (25 marks) The occurrence of bushfires in the Port Stephens area may be modelled by a Poisson process. The average occurrences of bushfires ï® is assumed to be either 15 (event A1), 20 (event A2) or 25 (event A3) times a year. The probability of A2 is the same as the probability A1 and the probability of A3 is half of A1. (a) Determine the probability that there will be 20 occurrences of bushfire in the next year. (10 marks) (b) If there are exactly 20 bushfires in the next year (event B), what will be P(A1|B)? Determine P(A2|B) and P(A3|B). (15 marks) Q.5 (Additional Question – No need for submission) Measurements of the velocity in a fully developed flow in a circular pipe whose wall radius is R, gave the following data: r/R 0.0 0.102 0.206 0.412 0.617 0.784 0.846 0.907 0.963 1.0 u/Uc 1.000 0.997 0.988 0.959 0.908 0.847 0.818 0.771 0.690 0.000 where r is the radius at which the velocity is measured, u is the velocity, and Uc is the centerline velocity. For R = 12.35 cm, and Uc = 30.5 m/sec, estimate the average velocity, Uave, in the pipe as defined by: R R ave, where r/R is , and the integral form as specified. (a) Explain why the Trapezoidal Rule is the most appropriate rule to use for the integration and why it is likely to under-estimate the value of Uave. Give your answer to two decimal places. (b) Check your answer by writing a Matlab program to perform the trapezoidal integration. The data is given in the data file pipeflow.dat. The easiest way to read a data file into a Matlab program is to use the ‘load’ command to fill an array with the data for u/Uc and . The short written responses include reflections on the data, interpretation of the results, and understanding of the numerical methods involved.

Paper For Above instruction

The assignment encompasses a comprehensive analysis across multiple engineering and statistical topics, including probability distributions, statistical inference, hypothesis testing, Poisson processes, and numerical integration methods within the context of real-world engineering problems. This scholarly discussion aims to synthesize these topics, applying theoretical principles to practical scenarios typical in engineering disciplines.

Beginning with question one, the focus is on understanding the probability density function (PDF) of crack sizes in a welded component of a Pontiac Trans Am. The PDF, which describes the likelihood of different crack sizes, necessitates sketching the PDF and its cumulative distribution function (CDF). These visualizations aid in grasping the distribution's shape — likely a form of a bounded, possibly asymmetric distribution given the PDF's piecewise nature. The mean crack size calculation involves integrating the product of crack size and the PDF over its support, revealing the expected size of a crack. The probability that a crack is smaller than 3 mm is obtained by integrating the PDF from the lower bound up to 3 mm, useful in risk assessment of structural integrity. The median crack size is determined by solving for the value of crack size at which the CDF equals 0.5, implying an equal probability of the crack being smaller or larger. For the probability that only one of four cracks exceeds 3 mm, the binomial probability distribution is employed, considering individual crack size conditions.

Question two shifts to statistical inference concerning vehicle speeds on a freeway. The known standard deviation informs the construction of confidence intervals for the population mean using the normal distribution, with the sample mean from 120 vehicles guiding the calculation at a 99.5% confidence level. The subsequent task involves determining the necessary sample size to estimate the mean speed within ±1 km/h at the same confidence level, which requires applying formulas for margin of error and sample size. The comparison of two independent sample means, representative of Jason and Britney’s data collection, utilizes properties of the sampling distribution to calculate the probability that their respective means are within ±0.5 km/h of each other, taking into account sample sizes and variability. Repeating this for larger samples underscores the influence of sample size on the precision of estimates and the likelihood of close agreement between independent observations.

The third question addresses fuel consumption variability. Calculating the sample mean and standard deviation from given data provides insights into the actual consumption characteristics of the vehicle fleet. The hypothesis test examines whether the observed mean significantly deviates from the manufacturer-stated 7 litres per 100 km, using a significance level of 2%, which encapsulates the core of inferential statistics by testing assumptions against observed data.

Question four explores modeling bushfire occurrences with a Poisson process, where the average incident rate is uncertain but hypothesized as one of three possible values with specified prior probability relationships. The problem entails calculating the probability of exactly 20 bushfires in a year using the Poisson distribution. Following this, applying Bayes’ theorem helps update beliefs about which incident rate is most likely given the observed data, effectively combining prior probabilities with the likelihood of the data conditional on each rate. This Bayesian approach pinpoints the posterior probabilities for each scenario, demonstrating the utility of Bayesian inference in environmental risk modeling.

Finally, the additional question, although not submitted, involves estimating the average flow velocity in a pipe using numerical integration. The problem discusses the suitability of the Trapezoidal Rule for this task, emphasizing its approximation attributes and tendency to underestimate the true integral. A MATLAB implementation further showcases the practical application of numerical methods, reinforcing understanding of computational techniques in fluid mechanics. The integration of observed velocity ratios and radius ratios encapsulates key concepts of finite element numerical analysis, enhancing students’ ability to interpret complex experimental data and apply mathematical tools in engineering contexts.

Overall, this assignment encapsulates critical competencies in engineering analysis, from probabilistic modeling to statistical inference and numerical computation, fostering a holistic understanding essential for advanced engineering practice.

References

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