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MSBA7003 Quantitative Analysis Methods

Assignment 4 (Due October 29 at 9 a.m.; Please submit your solutions with the template)

Q1.

American Copiers sells and services copy machines to customers in 11 cities throughout the

United States. The company wants to set up service centers in three of these cities. After

American Copiers chooses the location of the service centers, it must assign customers in

each city to one of the service centers. For example, if it decides to locate a service center in

New York and then assigns its Boston customers to the New York service center, a service

representative from New York will travel to Boston when services are required there. The

distances (in miles) between the cities are listed in Table 1. The estimated annual numbers

of trips to the various customers are listed in Table 2. The goal of American Copiers is to

minimize the total annual distance traveled by its service representatives.

Table 1: Distance between Different Cities

Boston Chicago Dallas Denver

Los

Angeles Miami

New

York Phoenix Pittsburg

San

Francisco Seattle

Boston 0 983 1815 1991 3036 1539 213 2664 792 2385 2612

Chicago 983 0 1205 1050 2112 1390 840 1729 457 2212 2052

Dallas 1815 1205 0 801 1425 1332 1604 1027 1237 1765 2404

Denver 1991 1050 801 0 1174 2041 1780 836 1411 1765 1373

Los

Angeles

3036 2112 1425 1174 0 2757 2825 398 2456 403 1909

Miami 1539 1390 1332 2041 2757 0 1258 2359 1250 3097 3389

New

York

213 840 1604 1780 2825 1258 0 2442 386 3036 2900

Phoenix 2664 1729 1027 836 398 2359 2442 0 2073 800 1482

Pittsburg 792 457 1237 1411 2456 1250 386 2073 0 2653 2517

San

Francisco

2385 2212 1765 1765 403 3097 3036 800 2653 0 817

Seattle 2612 2052 2404 1373 1909 3389 2900 1482 2517 817 0

Table 2: Estimated Numbers of Annual Trips to Customers

Boston Chicago Dallas Denver

Los

Angeles Miami

New

York Phoenix Pittsburg

San

Francisco Seattle

885 760 1124 708 1224 1152 1560 1222 856 1443 612

Please develop a Python model with the PuLP or DoCplex package to find out which of the

following statement(s) is(are) true.

A) In the optimal solution, Dallas, San Francisco, and Boston should have service centers.

B) In the optimal solution, Dallas, San Francisco, and New York should have service centers.

C) In the optimal solution, Denver customers should be assigned to the service center in

Dallas.

D) In the optimal solution, Phoenix customers should be assigned to the service center in

Dallas.

E) None of the above.

Q2.

Jenny Wilson Realty is a real estate firm in Alabama. Jenny, the manager, wants to develop a

model to determine a suggested listing price based on the size, age, and the condition

(either good or excellent) of the house. A sample of historical data include selling price (Y),

the square footage (X1), the age (X2), and the condition (X3). Jenny runs a regression of Y

against X1, X2, and X3. Suppose Jenny would like to find out how much can renovating a

house and changing the condition from “good” to “excellent” affect the selling price. Which

of the following statement(s) is(are) true?

A) The estimated coefficient for X3 is not the true effect of interest.

B) Jenny should at least include the reputation of the property developer in the regression.

C) If the estimated coefficient for X3 is 250,000, it means that the renovating a house can

increase the selling price by $250,000.

D) We can use the number of bathrooms should be included in the regression in order to

estimate how X3 influence Y.

E) None of the above.

Q3.

You are a factory manager and originally the workers are paid a fixed salary according to

their skill levels. You want to introduce a productivity-based salary in a hope to increase

worker productivity. To begin with, you randomly selected some male and female workers,

respectively, according to the numbers given in following table. The selected the workers

adopted the new salary scheme in the next month.

Gender Selected Not Selected Total Number

Male 15 35 50

Female 65 85 150

Their average productivities in the next month are shown in the following table.

Male 7 (selected) 6.0 (not selected)

Female 9 (selected) 7.5 (not selected)

Which of the following statement(s) is(are) true?

A) If you implement the new salary scheme for the whole factory, you can expect to

increase workers’ monthly productivity by 1.25 on average.

B) The naïve estimator of (7*15/80 + 9*65/80 – 6*35/120 – 7.5*85/120) is not biased.

C) Let D = 1 for selected workers and 0 otherwise. Let G = 1 for male workers and 0

otherwise. Suppose we run the regression model: Y = a + b*D + c*G + d*D*G + e, where

(a,b,c,d) are parameters and e is the error term. The estimated b should be about 1.5.

D) Among those not selected, their average productivity is expected to be increased by

1.375 if they also adopt the new policy.

E) None of the above.

Q4.

Consider the following causal graph.

Which of the following statement(s) is(are) true?

A) If we estimate the influence of X on Y through matching, we can condition on (B, D, W).

B) Using the regression model 𝑌 = 𝛼 + 𝛽𝑊 + 𝛾𝐴 + 𝜃𝐵 + 𝛿𝑋 + 𝜖, we can correctly estimate

the influence of X on Y given that the causal influences are all linear.

C) If the causal influences are all linear in this system, we can give a causal interpretation to

the estimated 𝛿 in this regression model: 𝑌 = 𝛼 + 𝛽𝐶 + 𝛿𝑋 + 𝜖.

D) If we condition on C, then D and Y are independent.

E) None of the above.

Q5.

You are running an online shopping website that focuses on fashion apparel. Your company

normally purchases a product from a supplier before the selling season starts. During the

season, customers that purchase the product can give a rating of the product on the website.

When the selling season ends, any leftovers will be shipped back to the supplier and partial

refund will be provided. You want to use the historical data to estimate how customer

rating of a product influences the sales of the product. You have prepared data for the

following variables:

Y: the total sales of a product

X: the average customer rating

P: the price of a product

S: the set of dummies that indicate the category of a product (e.g., male versus

female, season, and type)

W: the beginning inventory level

Which of the following statement(s) is(are) true?

A) Given the dataset, you can correctly estimate the influence of X on Y.

B) We can use P as an instrumental variable to estimate the influence of X on Y.

C) We can use S as an instrumental variable to estimate the influence of X on Y.

D) We can use W as an instrumental variable to estimate the influence of X on Y.

E) None of the above.