Homework Log
Stat 567:Statistical Reliability
All homework is due on the Monday following the Lecture
date on which it is assigned unless otherwise specified. Make sure to identify
all homework with the Lecture number (ordinal).
To avoid confusion homework from different classes should
be handed in distinct packages. All pages of your homework should have your
name and Lecture number. Pages within a Lecture homework package should
be numbered sequentially.
First Lecture (August 27, 1997)
Second Lecture (September 3, 1997)
- Show that the cumulative hazard function of an increasing
hazard (failure) rate (IFR) distribution is convex (i.e., holds water).
- Show that if T is a positive random variable with an
exponential distribution then:
- Show that if T measured in minutes is exponential with
hazard rate lambda, the T*=T/60 measured in hours is exponential with hazard
rate 60 time lambda.
- Show that the Weibull distribution is:
- IFR if the shape parameter is greater than 1
- DFR if the shape parameter is less than 1
- Exponential if the shape parameter is equal to 1
- Show that alpha is the 63.2 percentile of the Weibull
distribution
- Take a look at Military
Reliability Standards and Handbooks to learn about MIL-STD-217 and
other military reliability standards.
- Check the web site: Weibull
Data for Parts
Third Class (September 8, 1997)
- Send me e-mail on what you found to be the less clear
points of my lectures so far.
- Show that if T is a random variable with Weibull survivor
function
Then X=logT has a Gumbel survivor function
where
Fourth Class (September 10, 1997)
- For the ball bearing data of Example 2.1 on Page 37 of
your textbook. obtain 95% confidence intervals for S(50) and H(50)
.
- For the ball bearing data of Example 2.1 on Page 37 of
your textbook plot the empirical survivor function using the plotting points
derived on Page 40. Then plot the empirical integrated hazard. Do you think
that the underlying distribution of the data is IFR? Why?
Fifth Class (September 15, 1997)
- Take a look at the homepage of the course Graphical
Data Analysis out of the University of Montana
- Use JMP to do Weibull and lognormal plots of the Aircraft
component data of Example 2.2 on Page 43 of your textbook.
You can see how this is done below. You can download
a text file containing these data. (Instructions
for importing a text file into JMP)
- Identify the standard survivor function G that leads
to the Gumbel and normal QQ plot. Find G inverse for the Gumbel plot.
Sixth Class (September 17, 1997)
- Let A1, A2, A3, and A4
be nested events with A1 being the smallest and A4
being the largest. Show that
P(A1)=P(A1|A2)P(A2|A3)P(A3|A4)P(A4)
- Show that in Example 2.2 on Page 43 of your textbook
the Product Limit estimator is the same as the empirical survivor function
- Analyze the rat diet data supplied in class. Write a
professional quality report.
Attach a cover letter directed at management. The report itself should
be directed at the scientists or engineers who are neither management nor
statisticians but are technically oriented. The report should have:
- Summary and Conclusions
- Data Analysis
- Appendix with the raw data and other things that interfere
with the smooth flow of the Data Analysis section of the report.
(Due October 1, 1997)
Go to the bottom of this page to download the data for
this homework
Hints:
- Are the three groups - low fat, saturated fat, unsaturated
fat, significantly different? What test are you using? If two different
tests lead to different significance levels can you explain the reason?
- Avoid vague language such as "they appear to be
different." Instead use language such as: "there is statistical
evidence (alpha=.05) to indicate that the three groups are different."
- Are the mortality (hazard) rates constant, decreasing,
or increasing for each group? Why can you use JMP's exponential plot to
find this information out? Why is this information useful?
- What distribution fits each group best? Why? Why is this
information useful?
- Avoid causal language such as: "Saturated fat causes
tumors." Instead, use language such as: "In this study the x
group had a significantly lower mortality rate than the y group."
- Use JMP's help menu to learn about the properties of
the statistical procedures you are using.
Seventh Class (September 22, 1997)
- In the analysis in this class summary
we determine the effect on reliability of removing Cause 9 failures. Do
a similar analysis with the Cause 6 failures. Notice that you can download
a JMP file with the data by going to the end of this class
summary.
- Write a professional
quality report documenting the data analysis in the class
summary and your analysis in the homework above. As usual, your report
should include a cover letter to management. (Due Wed, October 8, 1997.)
- Write a proposal for the final project. What experiment
are you going to do? Who are you going work with? (Due Monday, October
6, 1997.)
Eighth Class (September 24, 1997)
- Read Chapter 3 of your textbook
- Derive the likelihood function for the following data:
- Left censoring times: 2
- Right censoring times: 7, 7, 11
- Failure times: 3, 5.5
(Homework due Wednesday, October 1)
Recall that you have a Midterm Exam Monday, September
29
Ninth Class (October 1, 1997)
- Redo the questions you missed in the midterm exam. Be
neat! Append you solutions to the exam and turn this package in. You will
receive a homework grade for this work.
Tenth Class (October 6, 1997)
- Use JMP's Nonlinear Fit platform to fit an exponential
distribution to the Aircraft components data.
What is the value that JMP gives you for the MLE of lambda? How does this
value compare with the number of failures divided by the total time on
test? What 95% confidence interval does JMP gives you for lambda? How does
this confidence interval compare with the one that one would obtain using
the standard error of the maximum likelihood estimate found in Example
3.1, Page 56 of your textbook? (This standard error can also be obtained
from the JMP output.)
- Obtain the variance covariance matrix for the Gumbel
in Example 3.2, Page 57 of your textbook given by
from the JMP output of the Gumbel fit of the log of the
ball bearings data. (Given in the class
notes.) Clearly show all formulas that you are using.
- Reproduce the JMP output of the Gumbel fit of the log
of the ball bearings data. (Given in
the class notes.)
- Verify that the values:
in Problem 3.2, Page 57 of your textbook satisfy Equations
(3.8) and (3.9) on Page 55 of your textbook.
- Verify that the variance covariance matrix for the Gumbel
in Example 3.2, Page 57 of your textbook given by
can be obtained by using the three equations on top of
Page 56 of your textbook and the formula:
- Remember that you have to write a proposal for the final
project. What experiment are you going to do? Who are you going work with?
(This homework was due Monday, October 6, 1997.)
- Rewrite the rat diet report for Wednesday, October
15. Read the revised professional
quality report guidelines for suggestions on improving your report.
Recall that you are to attach a cover letter directed at management. Also
recall that the report itself should be directed at the scientists or engineers
who are neither management nor statisticians but are technically oriented.
The report should have the following sections:
- Summary and Conclusions
- Data Analysis
- Appendix with the raw data and other material that interferes
with the smooth flow of the Data Analysis section of the report.
Eleventh Class (October 8, 1997)
- Obtain the variance covariance matrix for the Normal
in Example 3.2, Page 57 of your textbook given by
from the JMP output of the Normal fit of the log of the
ball bearings data. (Given in the class
notes.) Clearly show all formulas that you are using.
- Reproduce the JMP output of the normal fit of the log
of the ball bearings data. (Given in
the class notes.)
- Consider the aircraft components
data of Example 2.2, Page 43. Assume an exponential distribution. Find
b10 and its standard error. Calculate
a 95% confidence interval for b10.
- Consider the two confidence intervals for S(.5) given
in class for the Aircraft components data
of Example 2.2, Page 43.
- What are the assumptions leading to each one?
- Which confidence interval do you prefer? Justify your
answer with an exponential plot of the data.
- Why were we able to use the empirical survivor formula
for the aircraft components data of Example
2.2, Page 43 instead of the more complex Product-Limit estimator formula?
Twelves Class (October 13, 1997)
- Verify the calculations leading to the numerical values
of estimates and standard errors on Table 3.2, Page 58 of your textbook.
- Estimate the lower 5% percentile and its standard error
for the data used in Table 3.2, Page 58. Assume a Weibull distribution
- Assuming a Weibull model for the data of Example 2.1
and 3.2 (and of Table 3.2) find the MLE of S(60) and its standard
error. Compare your results with those you get without the Weibull assumption
using the empirical survivor function.
- For each of the distributions listed below what functions
of the parameters are used to calculate the standard errors of quantiles
and survival probabilities via the Delta method:
- Weibull
- Gumbel
- Exponential
- Lognormal
- Normal
- Rewrite the Appliance data report for Wednesday, October
22. Read the revised professional
quality report guidelines for suggestions on improving your report.
Recall that you are to attach a cover letter directed at management. Also
recall that the report itself should be directed at the engineers who are
neither management nor statisticians but are technically oriented. The
report should have the following sections:
- Summary and Conclusions
- Data Analysis
- Appendix with the raw data and other material that interferes
with the smooth flow of the Data Analysis section of the report.
Thirteenth Class (October 15, 1997)
Homework Assignments:
- Reproduce the JMP output for the exponential fit of the
aircraft component data. (Given in today's
class notes.)
- Reproduce the JMP output of the Gumbel fit of the log
of the ball bearings data. (Given in
the class notes.)
- Reproduce the JMP output of the Gumbel fit of the log
of the ball bearings data with sigma
locked at the value one. (Given in the class notes.)
- Read the material in the notes. The likelihood ratio
method will open a whole new world of statistical application outside the
traditional linear model and normal distribution framework. You need to
understand the logic behind the likelihood ratio method to be able to use
this method correctly.
Fourteenth Class (October 20, 1996)
- Use JMP's Nonlinear Fit platform to find a likelihood
ratio confidence interval for the b10 of the ball bearing data. Use
the following screen shots as a guide. As you can see I have reparametrized
the Gumbel distribution of log data in terms of b10
and sigma.
Fifteenth Class (October 22, 1997)
- Use JMP's Nonlinear Fit platform to find a likelihood
ratio confidence interval for the mediam of the ball
bearing data.
- Use the Weibull versus Lognormal tests with ball bearing data.
- Use JMP's Nonlinear Fit platform to fit a log normal
to the cord data
- Use the SAS JMP software to obtain the PP and SP plots
for the cord data of Example 2.3 on Page 46 of your
textbook. Hand these plots in. Use the JMP screen shots at the bottom
of the class summary as a guide to obtaining
these plots.
- At the end of Chapter 3 in you textbook the confidence
interval for S(53) of the cord
data obtained by way of the PL estimator (0.54, 0.83) is compared to
the confidence interval for S(53) obtained using a Weibull fit and
the Delta method (0.58, 0.76). Show the calculations leading to these confidence
intervals. Which interval do you prefer and why?
- For extra credit: Can you obtain a LR confidence interval
for S(53) of the cord data
assuming a Weibull fit?
(October 27, 1997)
Homework Review
- No additional homework.
- Homework that was due today had its due date postponed
to Wednesday, October 29.
Sixteenth Class (October 29, 1997)
Because of the exam on Monday, November 3 this homework
assignments are due Monday, November 10)
- Read the class notes and the web summary for this class.
In particular, note that there is a great deal of material in the web summary
(and its links) that complements the class lecture. Make sure that you
understand how we used JMP to obtain the values of maximized log-likelihoods.
- Reproduce the last page of the Nonlinear Fit JMP platform
analysis of the insulating fluid data
presented in class. Calculate the variance-covariance data.
- Calculate an approximate 95% confidence interval for
b10 at 26 kilovolts for the insulating fluid data. (Assume the model
used in class.)
- What happens if you try to estimate the mediam life at
20 kilovolts?
- Calculate an approximate 95% confidence interval for
the mediam at 30 kilovolts for the insulating
fluid data. (Assume the model used in class.)
- Do a likelihood ratio test for equal sigma versus unequal
sigmas similar to the one done is Fit
a Regression Model but do not exclude the data at 32 Kilovolts
- Extra credit: Can you improve on the confidence intervals
above by getting likelihood ratio confidence intervals?
Seventeenth Class (November 5, 1997)
Eighteenth Class (November 10, 1997)
- Use JMP's Nonlinear Fit platform to fit the model:
to the following data:
Nineteenth Class (November 12, 1997)
Computer Demonstration
Nineteenth Class (November 17, 1997)
- Reproduce the last JMP output of the links from this
class summary
Twentieth Class (November 19, 1997)
- Reproduce the last JMP output of the links from this
class summary
Twenty First Class (November 24, 1996)
Twenty Second Class (December 1, 1997)
Twenty Third Class (December 3, 1997)
- Take a look at Military
Reliability Standards and Handbooks to learn about MIL-STD-217 and
other military reliability standards.
- A system is to be designed with an overall reliability
of .999 using components having individual reliabilities of .7. What is
the minimum number of componets that must be connected in parallel?
Twenty Seventh Class (December 2, 1996)
Do you have something
to tell me?
