BUSI2105-E1
A LEVEL 2 MODULE, AUTUMN SEMESTER 2021-2022
QUANTITATIVE METHODS 2A
1. Suppose that you want to test whether the performances of China’s foreign direct investment (FDI) differ across locations. You randomly select some FDI firms and record their return on equity (ROE) in the following table:
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Locations of FDI
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America
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Europe
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Asia
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Africa
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ROE
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0.2
0.15 0.21
0.13 0.17
0.11
0.08
0.15
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0.11
0.14
0.14
0.18
0.16
0.08
0.15
0.16
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0.21
0.26
0.15
0.09
0.13
0.15
0.15
0.14
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0.08
0.07
0.13
0.1
0.12
0.11
0.16
0.11
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(a) What test would you use to study whether the mean ROE of these FDI differ across locations? Explain the intuition why such a test is appropriate for this research objective. (3 Marks)
(b) Based on the above data, test whether the mean ROE is the same across different locations at 5% significance level. (8 Marks)
2. A researcher obtains a sample with number of observations n = 100 , and population standard deviation σ = 3. He uses this sample to formulate the following hypothesis test: H0: μ = 1, and Ha : μ ≠ 1 . He chooses the significance level α = 0.05
(a) What is the probability of making a type I error? (2 Marks)
(b) What is the power of the test if the true population mean μT = 1.5? (6 Marks)
(c) How large a sample size n would be required in (b) so as to obtain a power of the test equal to 90%? (4 Marks)
3. Consider the following contingency table of 600 students. Test whether subject preference is independent of gender at the 1% significance level.
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Subject
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Gender
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Economics
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Finance
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Accounting
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Management
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Female
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100
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120
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50
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80
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Male
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70
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100
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30
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50
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(10 Marks)
4. Suppose that you want to study whether the average time per day spent on homework is the same between male and female high school students. You randomly and independently collect
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two samples whose summary statistics are given below:
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Sample 1 (male)
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Sample 2 (female)
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Sample size: nx = 15
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ny = 12
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Sample mean (hours per day) : ̅(x) = 2.5
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̅(y) = 3.5
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Bessel-corrected sample standard deviation: sx = 1.5
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sy = 0.8
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At the 5% significance level, test whether the average time per day spent on homework is the same between males and females. (12 Marks)
5. Consider the following linear simultaneous equation system in x, y, Z: x + 2Z = 3
y − 2Z = −1 2x + y = 3
(a) Express this system in matrix form. AX = b, with vector X = [x y Z]T . Find the inverse of the coefficient matrix A using its determinant and adjoint matrix. (8 Marks)
(b) Solve the system using Cramer’s Rule. (4 Marks)
(c) Find the Eigenvalues of the coefficient matrix A and their corresponding Eigenvectors. (9 Marks)
6. Henry spends all his income on purchasing food. The price for food is p > 0 per unit. He earns income from working for a factory. His hourly wage is w > 0. Henry decides how much time he works and how much food he buys and consumes. His utility from consuming q units of food and working for k hours is u(q, k) = ln q − k 2 . Henry wants to maximize his utility. He has no savings, so he cannot spend more than his income on purchasing food.
(a) Write out Henry’s optimization problem. (2 Marks)
(b) Use the Lagrange function approach to solve Henry’s optimal consumption of food and time spent on working. (4 Marks)
(c) Use the Bordered Hessian to verify that your solution maximizes Henry’s utility. (4 Marks)
7. Compute the following integration problems:
(a) Indefinite integration:
(5 Marks)
(b) Definite integration: a mobile phone company wants to know how large a town is in order to decide the number of base stations it needs to build in this area. This town can be described by the shaded region between two curves y 2 = x + 1 and y = x − 1 . Calculate the area of this town:
(7 Marks)
8. Solve the following difference equations:
(a) xt = 2xt−1 + 9 with initial conditions x0 = 6. (4 Marks)
(b) yt = −yt−1 + 4yt−2 + 4yt−3 − 12 with initial conditions: y0 = 2 , y1 = 4 , and y2 = 5 . What is the dynamic trajectory of yt? (8 Marks)