BUSI2105-E1
A LEVEL 2 MODULE, AUTUMN SEMESTER 2020-2021
QUANTITATIVE METHODS 2A
1. To stimulate the economy after COVID-19 pandemic, many cities in China introduced the consumer coupon. A researcher wants to study whether such consumer coupon has effectively increased the number of households’ transactions within a given period of time, so he collects a sample of cities that have introduced the consumer coupon and a sample of cities that have not. The summary statistics are listed below (assume that the two populations are normally distributed) :
(a) At α = 0.01, test whether the consumer coupon has effectively increased the number of transactions (assume equal population variance). (10 Marks)
(b) In your opinion, what are the potential problems of the above test of difference in means
using independent samples? Explain why it could be better to use matched samples in this case. (6 Marks)
2. Suppose that you want to study the performances in QM2A of students from different majors. You randomly select some students and record their marks of final exam in the following table, categorizing these students according to the majors they belong to.
|
|
Majors to which students belong
|
|
|
FAM
|
IBE
|
Others
|
|
Marks
Of
QM2A
|
80
72
66
73
85
90
88
72
67
|
79
71
67
70
85
89
84
73
66
|
78
75
69
70
77
80
76
75
75
|
(a) If you want to investigate whether the mean mark of students is the same across the above three majors, what test would you use? Explain the intuition of how such test can achieve this research objective. (4 Marks)
(b) Based on the above data, test whether the mean mark is the same across different majors at the 5% significance level. (12 Marks)
(c) At the 10% significance level, test whether the variance of marks is the same between FAM students and IBE students. (8 Marks)
3. Consider the following linear simultaneous equation system in x, y, z.
x − y + z = 1 (1)
2x + y + z = 4 (2)
5y + 2z = 7 (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. (10 Marks)
(b) Solve the system using Cramer’s Rule. (6 Marks)
4. A household has the utility function U = ln(q1) + ln(q2), where q1 and q2 are the quantities of consumption of two types of goods. The budget constraint is given by p1 q1 + p2 q2 = 200, where p1 and p2 are prices of q1 and q2 respectively. The household is a price taker.
(a) Using the Lagrange function approach, determine the optimal quantities of consumption q1 and q2 that maximize the household’s utility (taking the prices as given). (6 Marks)
(b) Based on the Bordered Hessian verify that your solution indeed constitutes a maximum of utility. (6 Marks)
(c) If the price for goods 1 increases from p1 = 5 to p1(′) = 10, other things equal, how large is the loss of the household’s consumer surplus? (6 Marks)
5. The inventory of a firm Qt adjusts as follows:
Qt+1 = PQ t + τIt
where It is the investment that adds new inventory. P captures the depreciation of inventory such that each period, 1 − P share of the inventory is lost, and 0 < P < 1 . τ measures the efficiency of transforming investment into inventory, and 0 < τ < 1 .
(a) If investment It is a constant It = I(̅), express Qt as a function of t (assume that the initial inventory is Q0 ) (3 Marks)
(b) If It = I(̅) + βQt , where β captures the reaction of investment to the current inventory level, express Qt as a function of t (assume that Q0 is known). (5 Marks)
(c) Discuss the dynamic trajectories of Qt that you obtained in (a) and (b) respectively. (6 Marks)
6. A researcher wants to investigate whether the investment preference is independent of gender. He randomly selects 1000 people and makes the following contingency table:
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Most Favorite Investment
|
|
Gender
|
Stock
|
Time Deposit
|
P2P
|
Trust Fund
|
Real Estate
|
|
Male
|
90
|
36
|
50
|
120
|
310
|
|
Female
|
50
|
63
|
30
|
80
|
171
|
At α = 0.05, can the researcher conclude that the preference is associated with gender (12 Marks)