11.220: Quantitative Reasoning and Statistical Methods for Planning IPage 1
Midterm Exam
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.
Total Possible / Your ScoreQuestion 1 / 12 points
Question 2 / 13 points
Question 3 / 12 points
Question 4 / 8 points
Question 5 / 12 points
Total / 57 points
EXTRA CREDIT / 12 points
Total Possible / 69 points
Tips:(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:______
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.
(b)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?
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)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.”
[6]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?
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 / Number in Sample / AccuracyTenants who remained in the same unit they had occupied under rent control. / 293 / ± 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%
All tenants in combined sample. / 1000 / ± 3.1%
[3](c)Accuracy obviously refers to the process of estimation. What type of estimation are the accuracy levels calculated in this table useful for?
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.)
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.
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]Calculate the probability that the median number of sinks will be within 0.1 sinks of the expected value of the sample median.
[3](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 ?