Department of Urban Studies and Planning
Massachusetts Institute of Technology
11.220 Quantitative Reasoning and Statistical Methods for Planning I
Spring 1998
Midterm Exam
Date:
Wednesday, April 22, 1998.
Format:
Open book, calculators allowed.
Question 1
Question 2
Question 3
Question 4
Question 5
Tips:
Total Possible
12 points
13 points
12 points
8 points
12 points
Total
57 points
EXTRA CREDIT
12 points
Total Possible
69 points
Your Score
(1) Please be sure to show all your work. We will give partial credit.
(2) Don’t forget to draw pictures when they are appropriate or helpful. For many of
these questions how you set up the problem is just as important as whether or not
you ultimately get the right answer.
(3) If you have any questions about the wording of the questions, please ask.
(4) Question 3 requires more reading time than the others, so plan accordingly.
(5) Please note that the last three parts of Question 5 are for extra credit. The exam
will be graded on the basis of 57 points. Thus, the extra 12 points can help pull
your course average up.
Your Name:
_________________________________________________________________
11.220: Quantitative Reasoning and Statistical Methods for Planning I
Midterm Exam
Page 2
Question 1
In order to proceed with a proposed development in the town of Middletown, a developer
needs to obtain a zoning variance. Historical data indicate that in Middletown an average
of 70% of all such applications are approved by the town. Because there are costs
involved in submitting an application for a zoning variance, the developer wants to avoid
the expense that would be involved in submitting an application that will not be
approved.
The developer is considering hiring a consultant who analyzes zoning variance
applications and predicts their success. This consultant has made a specialty of studying
the various factors that tend to increase or decrease the probability that an application for
a variance will be approved, factors that the developer has not studied. The consultant’s
previous experience indicates that when a variance was approved he had actually
predicted that it would be approved 9 times out of 10. But when a variance was not
approved, he had predicted that it would not be approved only 6 times out of 10.
(Note that in this utopian example hiring the consultant does not change the probability of
approval; it merely increases the developer’s information about the relative likelihood of
the outcomes.)
[6]
(a)
Draw a probability tree to represent this problem. Clearly identify each of the
nodes, branches and outcomes and place the appropriate probabilities on the tree.
11.220: Quantitative Reasoning and Statistical Methods for Planning I
Midterm Exam
(b)
Page 3
The developer would like to know something about the consultant’s reliability.
[3]
•
What is the probability that the variance will be approved if the consultant
says it will be approved?
[3]
•
What is the probability that the variance will not be approved if the
consultant says it will not be approved?
Question 2
The primary job of building inspectors is to detect violations of the building code, but
building inspectors sometimes miss violations that are actually there. A particular
building inspector detects an average of 90% of all the building code violations that
actually exist in the buildings that she inspects. This inspector never “discovers” code
violations when they in fact do not exist.
[3]
(a)
In a particular building the inspector has detected 15 code violations. Calculate a
point estimate of the true number of code violations in this building. Explain your
work.
[6]
(b)
Assume that the inspector is equally likely to detect each potential code violation
and that all potential code violations are independent of one another. In a building
that actually has 10 code violations, what is the probability that she will detect
eight or more of these code violations?
[4]
(c)
In part (b) you made two assumptions. Is each of those assumptions reasonable?
Why or why not?
11.220: Quantitative Reasoning and Statistical Methods for Planning I
Midterm Exam
Page 4
Question 3
On January 21, 1998 Atlantic Marketing Research presented to the Cambridge
Community Development Department its Cambridge Rental Housing Study: Impacts of
the Termination of Rent Control on Population, Housing Costs, & Housing Stock. Rent
controls were eliminated in Cambridge on January 1, 1995, and this report had been
commissioned to test what the implications had been for renters in Cambridge. The
central questions, of course, were the degree to which rents had risen and for whom, but a
number of other questions were addressed as well.
Atlantic Marketing took two basic samples. The first sample was a straightforward
simple random sample taken from a list of all renter-occupied housing units in
Cambridge. But this sample would not have included anyone who had lived in a rent
controlled unit in Cambridge prior to January 1, 1995 and had moved out of Cambridge
or had bought a unit in Cambridge.
The second sample was an explicit attempt to identify and sample tenants who had lived
in rent controlled housing and had moved either to other Cambridge addresses or to
addresses outside of Cambridge. Using various lists compiled by the City of Cambridge,
Atlantic Marketing constructed a complete list of the approximately 600 apartments that
had been formerly subject to rent control and from which individuals had moved between
1994 and 1997. Letters and mail survey questionnaires were sent to all of these tenants at
their former, rent-controlled Cambridge addresses with the hope that the mail would be
forwarded to their current addresses. Anyone who responded to this survey who had
lived in a rental unit in Cambridge at the time of the survey was eliminated from this
second sample because they already had the appropriate probability of being included in
the first sample.
(a)
[6]
With respect to the second sample, the report states, “Significant difficulty was
expected and was experienced in attempting to locate such households. While
this latter effort falls outside truly random surveying techniques, it was believed to
be the best way to reach relocated tenants, particularly those who moved outside
Cambridge.”
Identify two ways in which this second sample falls “outside truly random
surveying techniques” and can introduce bias into the survey results.
What can you say, if anything, about the likely direction of these biases?
11.220: Quantitative Reasoning and Statistical Methods for Planning I
Midterm Exam
Page 5
Eventually, the researchers combined the two samples for purposes of analysis. This
combined sample included various groups of tenants, each of which would be particularly
interesting to study on its own. The accuracy with which one can make estimates about
each of these groups varies. Recognizing this, the analysts prepared the table below. (I
have changed the descriptions of the various groups to make them more explicit, but
otherwise the table remains the same.)
In the words of the final report, this table is intended to give a guide as to how “survey
results can be interpreted at a 95% confidence interval.”
[3]
(b)
Pick one of the groups that is identified in this table and show how the “accuracy”
was calculated for that group.
Group
Tenants who remained in the same unit they had
occupied under rent control.
Number in Sample
293
Accuracy
± 5.7%
Tenants who had resided in a rent controlled unit but
who had moved out of that unit.
97
± 10.0%
Tenants who moved into a decontrolled unit after the
elimination of rent control but had not lived in a rent
controlled unit.
179
± 7.3%
Tenants of market rate units (units that had not been
subject to rent control when it was eliminated).
432
± 4.7%
All tenants who lived in decontrolled units at the time of
the survey.
474
± 4.5%
All tenants who lived in Cambridge market rate units at
the time of the survey.
470
± 4.5%
All current Cambridge renters.
940
± 3.2%
1000
± 3.1%
All tenants in combined sample.
11.220: Quantitative Reasoning and Statistical Methods for Planning I
Midterm Exam
[3]
(c)
Page 6
Accuracy obviously refers to the process of estimation. What type of estimation
are the accuracy levels calculated in this table useful for?
11.220: Quantitative Reasoning and Statistical Methods for Planning I
Midterm Exam
Page 7
Question 4
Based on careful and complete collection of the relevant historical data you have
established that the time that it takes you to get from your apartment or dorm room to the
QR classroom is distributed normally with a mean,  , equal to 20.0 minutes and a
standard deviation,  , equal to 3.9 minutes. This morning you wanted to study until the
last possible minute before heading off to the midterm exam.
[8]
(a)
You carefully calculated the latest time at which you could leave for the midterm
exam and still be 90% certain of arriving on time (at 9:30 a.m.). What was that
time? (You may ignore any adjustments that may have been necessary for the fact
that we changed the room and you may have gotten lost.)
11.220: Quantitative Reasoning and Statistical Methods for Planning I
Midterm Exam
Page 8
Question 5
A simple random sample was taken to estimate the mean number of sinks in single family
houses in Middletown. A random sample of 36 single family houses was selected.
Suppose that, unknown to the person taking the sample, the true value of  is 2.8 sinks
per house (including kitchen, bathroom, and basement sinks) and the standard deviation
of the number of sinks,  , is 0.4.
[3]
(a)
Calculate the expected value of the sample mean.
[3]
(b)
Calculate the standard error of the sample mean.
[6]
(c)
Calculate the probability that the sample mean will be within 0.1 sinks of the
expected value of the sample mean.
11.220: Quantitative Reasoning and Statistical Methods for Planning I
Midterm Exam
Page 9
The last three parts of this question are for EXTRA CREDIT. They involve concepts that
we did not cover directly in class, but based on our class discussions and the information
given below, you should be able to extend your understanding of the material to answer
these questions.
Let the notation Md indicate the sample median and suppose that you have decided to do
estimation of central tendency using medians rather than means. Like sample means,
sample medians are distributed normally.
[3]
(d)
Calculate the expected value of the sample median.
(e)
The standard error of the sample median is not the same as the standard error of
the sample mean, however. The standard error of the sample median is given by
the following formula:
[6]
[3]
Calculate the probability that the median number of sinks will be within 0.1 sinks
of the expected value of the sample median.
(f)
On the basis of your answers to parts (c) and (e) above, what conclusion can you
draw about the relative advantages of using the sample mean or the sample
median to estimate  ?