BUSI2105-E1
A LEVEL 2 MODULE, AUTUMN SEMESTER 2022-2023
QUANTITATIVE METHODS 2A
1. Suppose that one student wanted to study the following research question: whether investing abroad can help improve a firm’s productivity? He randomly selected 12 firms, and recorded their productivities before and after they started investing abroad in the following table (assume that firm productivity follows normal distribution) :
|
Firm
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
|
Before
|
1.5
|
1.2
|
1.7
|
1.5
|
2.2
|
2.3
|
2.1
|
1.3
|
1.8
|
1.9
|
2.8
|
2.5
|
|
After
|
1.8
|
1.6
|
1.9
|
1.8
|
2.7
|
2.3
|
2.6
|
1.4
|
2.3
|
2.2
|
3.2
|
2.6
|
(a) At α = 0.05, test whether productivities after firms invest abroad are higher than before. (7 Marks)
(b) Does the approach adopted by this student perfectly answer his research question : does investing abroad help improve a firm’s productivity? Provide some arguments that may challenge his approach and result. (4 Marks)
2. A researcher obtains a sample with number of observations n = 100 , and population standard deviation σ = 1 . 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. 1? (5 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. Suppose that you want to investigate whether movie preference is associated with age. You randomly surveyed 1000 people and obtained the following contingency table.
|
|
Movie
|
|
Age
|
Drama
|
Action
|
Comedy
|
Others
|
|
<20
|
30
|
100
|
80
|
30
|
|
20~40
|
40
|
100
|
130
|
160
|
|
>40
|
140
|
40
|
70
|
80
|
At the 1% significance level, test whether movie preference is independent of age. (10 Marks)
4. Suppose that you want to compare innovation behaviour of firms across different ownership in a given industry. You randomly selected some firms in this industry, and recorded the number of patents they have applied within the same period of time. Assume that populations are normally distributed.
|
|
Firm Type
|
|
|
Private-owned
|
State-owned
|
Foreign-owned
|
|
Number of patents
|
1
5
12
2
2
1
7
1
4
4
|
3
5
1
8
1
10
1
1
1
2
|
3
2
7
4
5
4
5
7
4
4
|
(a) At the 5% significance level, test whether the number of patents is the same between Private-owned firms and State-owned firms. (8 Marks)
(b) At the 5% significance level, test whether the number of patents is the same across all types of firms. (9 Marks)
5. (a) Does the matrix have an inverse? If your answer is yes, use Gaussian
elimination to find the inverse of this matrix. If your answer is no, explain why. (7 Marks)
(b) Suppose f(x, y, z) = x 2 + y 2 + z 2 + xy + yz + x + z. Find the first order conditions and use Cramer’s rule to solve the stationary point(s). Determine whether each stationary point is a local minimum or maximum, or saddle. (9 Marks)
6. Bob’s utility function is given by ln x + lny − k 2 , where x and y are consumptions of two goods, and k is the number of hours spent working.
(a) Optimize this utility function subject to budget constraint px + qy = wk, where p and q are prices of x and y respectively and w is the hourly wage rate. Use the Lagrangian approach to find the stationary point(s) of this optimization problem. (4 Marks)
(b) Verify whether these stationary point(s) are indeed local maximum. (9 Marks)
7. Integration
(a) Compute the indefinite integration
(5 Marks)
(b) Determine the area to the left of g(y) = 3– y2 and to the right of x = −1 .
(6 Marks)
8. Difference Equations.
(a) Solve 2xt + xt−1 = 6 for x0 = 1. (4 Marks)
(b) Solve 6xt − 5xt−1 + xt−2 = 2 for x0 = 1 and x1 = 2 . (7 Marks)