4.8: Chapter 4 Excercises
P4.1 (4 pts)
Many classrooms are equipped with
video projectors that can be connected to portable, laptop
computers for use during class meetings.
a. Based upon your experience with such systems, what is the
probability of projection system failure over the course of a year
of regular classes in such a room?
b. Develop a fault tree of potential causes for a classroom video
projection system.
P4.2 (8 pts)
Appearing below is a series of roof inspection condition summaries, where 1 is excellent and 5 is poor. Note that an inspection 1997.5 occurred in the second six months of 1997, whereas 1997 occurred in the first six months of 1997. The roof was replaced in 1985. Answer the questions below. You might use software aids, such as EXCEL or MATLAB, for this problem.
| DATE | CONDITION |
| 1985 | 1 |
| 1985.5 | 1 |
| 186.5 | 2 |
| 1997 | 2 |
| 1997.5 | 2 |
| 1998 | 2 |
| 1998.5 | 2 |
| 1999 | 3 |
| 1999.5 | 3 |
| 2000 | 4 |
| 2000.5 | 4 |
| 2001 | 4 |
| 2002 | 4 |
| 2002.5 | 4 |
| 2003 | 5 |
a. Estimate an ordinary least squares regression deterioration model of the form: Condition = \(a + b(age)\) where age is the age of the roof in years. Report your parameter estimates, standard errors, t-statistics, and \(R^2\) values. Note that there is a gap in the data from 1985 to 1996!
b. Suppose I have a comparable roof that is 12 years old. What would your regression model in (a) predict for its condition? What would it predict for age 18? At what age is the condition expected to become 5?
c. Plot the data and your regression line.
d. Do you think a non-linear regression model would fit the data better? Try a quadratic model (Condition = \(a + b(age) + c(age^2)\) and an exponential model (Condition = \(a*age^b\)) and discuss your results. Which model has a better-adjusted \(R^2\)? Which model would you use in practice for deterioration prediction?
P4.3 (16 pts)
Formulate a simple Markov process model of roof condition. Assume that transitions occur every six months and can either be a return to current condition or a transition to the next worse condition (except for state 5 which is an absorbing state without an exit in this deterioration model…)
a. Draw your process model as a series of states (in circles) and transition possibilities (as arrows) for five states corresponding to roof conditions 1 to 5.
b. Assume the probability of remaining in state 1 is 0.93 in any one transition. Estimate (from the data above) or calculate (when appropriate) the remaining transition probabilities and mark them on your process diagram.
c. Develop a Markov transition matrix for your process.
d. Suppose you start at time 0 in state 1. Calculate the probability of being in each state for the next twenty years (or 40 transitions) based on your model.
e. Suppose you believe that a roof must be replaced when the roof condition reaches state 5. Starting with a new roof (state 1 in time 0), plot the probability of being in state 5 as a function of time.
f. Calculate the expected service time of the roof based on your data in part e. You can assume that the expected service time is when the probability of entering state 5 reaches 50%.
g. If you ran your model to the limit (infinite time), what is the probability of being in each state?
h. How does your Markov process model compare with your linear regression model in Question 1? In particular, is the expected service time different? Is the Markov model non-linear? Why or why not? Which is preferable and why?
P4.4 (6 pts)
Suppose I have the simple piping
system shown below:
For this simple system, I develop a fault tree for failure analysis
as:
Suppose further that the estimated failure probabilities of the four sub-events are independent and are as follows (left to right in the figure):
| Event | D empty | A broken | B blocked | C blocked |
| Event Probability | 0.15 | 0.05 | 0.1 | 0.1 |
P4.1 (4 pts)
Many classrooms are equipped with
video projectors that can be connected to portable, laptop
computers for use during class meetings.
a. Based upon your experience with such systems, what is the
probability of projection system failure over the course of a year
of regular classes in such a room?
b. Develop a fault tree of potential causes for a classroom video
projection system.
P4.6 (4 pts)
With the growth of internet service providers, a researcher decides to examine whether there is a correlation between cost of internet service per month (rounded to the nearest dollar) and degree of customer satisfaction (on a scale of 1 - 10 with a 1 being not at all satisfied and a 10 being extremely satisfied). The researcher only includes programs with comparable types of services. A sample of the data is provided below.
| Dollars | Satisfaction |
| 11 | 6 |
| 19 | 8 |
| 17 | 10 |
| 15 | 4 |
| 9 | 9 |
| 5 | 6 |
| 12 | 3 |
| 19 | 5 |
| 22 | 2 |
| 25 | 10 |
a. Plot the data. Do you think dollars and satisfaction are related (or correlated)?
b. Estimate a linear regression with Satisfaction = a + b*dollars. Discuss your results.
P4.7 (3 pts)
Which of the following are transition matrices for Markov processes? Explain.
a. \begin{bmatrix}
.4 & .3 & .3 \\
.2 & .4 & .4 \\
.6 & .1 & .3 \\
\end{bmatrix}
b. \begin{bmatrix}
.2 & .3 & .5 \\
.6 & .1 & .2 \\
.7 & .1 & .3 \\
\end{bmatrix}
c. \begin{bmatrix}
.25 & .15 & .3 & .4\\
.5 & 0 & .15 &
.3 \\
.15 & .35 & .4 & .2 \\
.1 & .5 & .2 & .2 \\
\end{bmatrix}