FORMAT: This task involves a combination of pen-and-paper work and MATLAB-based work. However, all solutions should be provided as written/typed solutions (you do not need to submit any .m ﬁle for this task). See the ﬁnal page for the marking scheme. Working must be shown in order to receive marks! Mass Production To save on costs, a certain circuit is mass-produced, and then adjusted before shipping to dif- ferent manufactures to tailor it to their needs. Speciﬁcally, ﬁve different sub-components of the circuit can be switched on or off, according to what the different manufacturers buying the circuit require. Historical production data for these circuits suggests that the truncated Poisson distri- bution is the best model for the probability that manufacturers request a circuit with X of these sub-components switched to on. That is, the probability distribution for X is given by the proba- bility mass function x Pr(X = x) = Aµ , (1) x! with µ = 3 and x ∈ {0, 1, 2, . . . , 5}. A range of similar devices each use one of these circuits, but depending on the number of active sub-components for a given circuit, its expected lifetime changes. Speciﬁcally, the baseline failure rate of the circuit during any given day is 1%, and then this is increased by 0.5% for each sub- component of the circuit that is set to on. This results in a conditional probability for the device failing on day Y, given the number of components that are on, X, Pr(Y = y|X = x) = (1 − 0.01 − 0.005x)y−1(0.01 + 0.005x). (2) PROBLEM SOLVING TASK 1 P Pen-and-Paper Exercises 1. Find the value that A must be so that the probability mass function for X, equation (1), is valid. (Hint: Zero factorial is equal to one, that is, 0! = 1) 2. Find the mean and variance of X using the probability mass function in equation (1). 3. Using the law of total probability and equations (1) and (2), ﬁnd the chance that a given circuit (selected at random) fails on precisely its second day of use, Pr(Y = 2). 4. Using Bayes’ theorem and your result for Q3, ﬁnd the probability that, if a device failed on its second day of use, it had precisely three of its internal circuit components acti- vated, Pr(X = 3|Y = 2). Bridge Pylons In the construction of bridges, there is variability in the manufacture of individual pylons. A certain manufacturer produced a number of these pylons for a few different bridges, advertising a mean lifetime of 20 years. Many years later, we have now obtained data regarding the actual lifetimes (in years) of several of these pylons, [ ] 12.78 16.25 20.35 18.81 17.64 21.08 14.37 21.72 19.92 17.24 21.14 19.27 . To inform a decision on whether the manufacturer’s product is suitable for a bridge that is to be built, we wish to analyse this data, and use it to investigate the company’s original claim. Pen-and-Paper Exercises 1. Use this data to estimate the mean lifetime and its associated variance for the pylons produced by this manufacturer. 2. Based on this data, determine the 90% and 99% conﬁdence intervals for the expected lifetime of the pylons. 3. Using the given data, compute the value of the test statistic that would be appropriate for testing the company’s claimed mean lifetime for the pylons. 4. Find the p-value corresponding to this test statistic, and use it to draw a reasonable con- clusion about whether the manufacturer’s claimed lifetime for these pylons is accurate. 2 Spring Testing To help inform a washing machine design project, the properties of the springs that are to be used to suspend the drum have been directly tested in a simple experiment. Placing different masses on the spring and measuring the resultant extension, a set of data has been collected. This data is available as a .mat ﬁle on the Blackboard page for this task. Also provided is a .m ﬁle that will load this data, then use MATLAB to ﬁt a linear model for this data. This function also plots the residuals. In this question, you will run this function and then interpret the results. Pen-and-Paper Exercises 1. Interpreting the output for the linear model ﬁt, how strong is the evidence (if there is any) of a relationship between the weight placed on the spring and its extension? 2. Explain in 1-2 sentences what the R-squared value speciﬁcally tells you about this model. 3. Calculate a 99% conﬁdence interval for the slope of the linear relationship that has been ﬁtted between these two variables. 4. From the conﬁdence interval in Question 3, and using a value of g = 9.81 m s−2, deter- mine the conﬁdence interval for the spring constant (assuming Hooke’s law holds). (This will require a little bit of modelling. You may disregard the intercept term for this calcula- tion) 5. In 2-3 sentences, comment on the residual plot. Does it indicate any other concerns with the validity of the regression analysis? Does Hooke’s law appear to hold for these springs — that is, is the relationship between mass and extension truly linear or not? 3 Marking Scheme Mass Production (Total: 40%) 10% – Q1 (ﬁnding A) 10% – Q2 (calculate mean and variance) 10% – Q3 (total probability) 10% – Q4 (Bayes’ theorem) Bridge Pylons (Total: 30%) 08% – Q1 (sample mean and variance) 08% – Q2 (conﬁdence intervals) 06% – Q3 (test statistic) 08% – Q4 (p-value and appropriate conclusion) Spring Testing (Total: 30%) 05% – Q1 (evidence of relationship) 04% – Q2 (interpreting R2) 07% – Q3 (conﬁdence interval for slope) 07% – Q4 (conversion to spring constant) 07% – Q5 (interpreting the residual plot) 4

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