讲解 BUSI2105 QUANTITATIVE METHODS 2A AUTUMN SEMESTER 2020-2021辅导 Processing

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.

(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:

At α = 0.05, can the researcher conclude that the preference is associated with gender    (12 Marks)