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Problem Solving Task Part C
Weight: 1
3 of 50%
Sharing this document or parts of it online, or asking for solutions online, constitutes academic
misconduct which could lead to failing this unit or suspension of enrolment.
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 different 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 distribution 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 probawith 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 subcomponent 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,
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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 activated, 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 conclusion about whether the manufacturer’s claimed lifetime for these pylons is accurate.
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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, determine 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 calcula5. 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?
CRICOS No. 00213J 3
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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)
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