University Of Newcastle School Of Engineering Assignment

The University Of Newcastle School Of Engineeringassignment Cover S

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 advised to keep a copy. The assignment includes several questions related to probability, statistics, and engineering computations, involving data analysis, statistical inference, hypothesis testing, and probability modeling of real-world engineering scenarios. The questions require sketching probability density functions, calculating means and probabilities, establishing confidence intervals, performing hypothesis testing, estimating parameters, and applying statistical models such as Poisson processes and numerical integration methods. Each problem emphasizes demonstrating clear working steps, applying appropriate statistical formulas, and interpreting results within an engineering context. The tasks involve mathematical rigor, data analysis, and simulation, essential skills in engineering computations and reliability assessment.

Paper For Above instruction

Introduction

Engineering reliability and statistical analysis are fundamental to ensuring safety, performance, and durability of mechanical components and systems. In the context of automotive engineering, understanding crack sizes, vehicle speeds, and fuel efficiency requires rigorous statistical analysis, modeling, and hypothesis testing. This paper discusses various engineering problems involving probability distributions, confidence intervals, hypothesis testing, and numerical integration to illustrate how statistical methods underpin decision-making in engineering practice.

Question 1: Crack Size Analysis in Automotive Welds

The first problem involves analyzing the size of a crack in a vehicle's weld, modeled by a probability density function (PDF). The given PDF indicates a specific distribution over a range of values, requiring graphing, calculating the mean, median, and probabilities, and understanding the behavior of the random variable.

Plotting the PDF involves identifying the shape based on the provided expression, which is non-zero over a specific interval. The cumulative distribution function (CDF) can be derived by integrating the PDF over the range. Calculating the mean crack size involves integrating y times the PDF over the interval. The probability that a crack is smaller than 3mm is obtained by evaluating the CDF at y=3. The median crack size is the value y_m where the CDF equals 0.5. For the probability that only one of four cracks exceeds 3 mm, the binomial probability applies, considering the individual probability of a single crack exceeding the threshold.

Question 2: Vehicle Speed Analysis

Second, the focus shifts to analyzing vehicle speeds on a freeway. With known standard deviation and a sample mean of 105 km/h from 120 observations, constructing a 99.5% confidence interval provides bounds within which the true mean speed lies, accounting for sampling variability. Additional observations are required if the estimate must be within ±1 km/h with the same confidence level, which involves calculating the sample size for the desired precision. The comparison of sample means between two observers involves calculating the probability that their means differ by a certain amount, assuming independence and known variance, modeled using normal distribution properties.

Question 3: Fuel Consumption Testing

The third problem involves analyzing data from fuel consumption tests on a fleet of cars. Estimating the sample mean and standard deviation requires calculating the arithmetic mean and standard deviation of the sample data. A hypothesis test then assesses whether the true mean equals the manufacturer's claimed fuel consumption using a t-test at a specified significance level. Calculating the t-statistic and comparing it to critical values determines if there is sufficient evidence to reject the null hypothesis.

Question 4: Bushfire Occurrence Modeling

The fourth problem models bushfire occurrences using a Poisson process, with an average rate that varies among different scenarios. Probabilities associated with the number of occurrences are computed using the Poisson probability mass function. Conditional probabilities, like the probability of a specific scenario given observed events, are derived using Bayesian updating, which involves calculating the prior probabilities, likelihoods, and normalized posterior probabilities.

These problems highlight the importance of probabilistic modeling and statistical inference in engineering applications, facilitating critical decision-making in maintenance, safety assessment, and resource allocation.

Conclusion

Effective engineering analysis relies on understanding and applying statistical principles to real-world data. From analyzing crack sizes to vehicle speeds, fuel consumption, and natural hazards, statisticians and engineers must employ probability distributions, hypothesis testing, confidence intervals, and numerical methods. Such approaches ensure informed decisions, optimized performance, and enhanced safety in engineering systems.

References

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  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability and Statistics for Engineers and Scientists. Pearson.
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