Cardinality of Set - Quantitative Methods in Economics - Exam, Exams of Quantitative Techniques

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Ollscoil na hÉireann, Gaillimh

GX_____

National University of Ireland, Galway

Semester 1 Examinations 2008/

Exam Code(s) 2BA1, 2BA6, 1EK2, 1EM1, 1EK

Exam(s) 2

nd

B.A., 2

nd

B.A. (PSP), H.Dip.Econ.Sc., Erasmus,

Occasional

Module Code(s) EC

Module(s) Quantitative Methods in Economics

Paper No.

Repeat Paper 1

External Examiner(s) Professor Robert Wright

Internal Examiner(s) Professor Eamon O’Shea

Dr. Srinivas Raghavendra

Instructions: Answer ten questions in total by choosing five questions in each

section. Each question carries 20 marks.

Duration 3 hours

No. of Pages 1+

Department(s) Economics

Course Co-ordinator(s) Dr. Srinivas Raghavendra

Requirements :

MCQ

Handout

Statistical Tables Yes

Graph Paper Yes

Log Graph Paper Yes

Other Material

EC 229: Quantitative Methods in Economics

Duration: 3hours Max Marks: 100

Answer ten questions in total by choosing five questions in each section. Each question

carries 20 marks.

Section A: Mathematics

1. Define the following concepts using suitable examples:

(i) Proper subset

(ii) Cardinality of a set

(iii) What does a Bijection mapping mean?

(iv) If S I V = S and S U V=V , then using Venn diagram show that S! V.

(v) Power set.

2. Find the equilibrium price and quantity of the following demand and supply

functions:

14 22

2

= + +

s s

P Q Q 10 150

2

=!! + d d

P Q Q

3. The demand and supply functions of a good are:

P = - 5Q

D

+ 80

P = 2Q

S

+ 10

Where P, QD and QS denote price, quantity demanded and quantity supplied

respectively

i) Find the equilibrium price and quantity graphically and algebraically. [8]

ii) If the government imposes a tax of 15% on the market price of the good,

determine the new equilibrium price and quantity. [12]

4 If the average cost function of a good is

Q

AC Q

13

= 2 + 6 +

Find an expression for MC. If the current output is 15, estimate the effect on TC

of a three unit decrease in Q.

5 Maximize profits! of a firm whose total revenue is R = 4000 Q! 33 Q

2

and total

cost C = 2 Q

3

! 3 Q

2

• 400 Q + 5000 , assuming Q > 0. [Hint:! =R-C]

6 Simplify the following using the rules of logarithms and indices:

(i) ( )

3

2

5

x x + y

(ii)

2

2

!

x y

x y

(iii) log (xy) – log x – log y

(iv)

4

2

1

log( x. y )

(v) 5 5

( 2 )

=

x!

7 If the production function is

Q = 300 L! 4 L

Where Q denotes output and L denotes the number of labor, calculate the value of

MP

L

when L=1, L=9, L=100 and L=2500.

8 Define the following concepts:

a. Critical point of a function

b. What does the second derivative of a function signify?

c. Diminishing Marginal Returns in production

d. What is the break-even point for a firm?

e. Marginal revenue of a firm

9. Sketch, on the same diagram, graphs of the following TR and TC functions:

TR = - 2Q

2

• 14Q

TC = 2Q + 10

i) Use you graphs to estimate the values of Q for which the firm breaks even and

maximises profit. [8]

ii) Confirm your answer to part (i) using algebra. [12]

10. Write down expressions for the savings function, given that the consumption

function is:

i) C = 0.9Y + 72 [10]

ii) C = (Y

2

• 500. / (Y + 10) [10]

Section B: Statistics

1. There are 12 students undertaking a particular course as part of their Masters degree.

The final results for these students were as follows:

60 40 81 51 42 49

70 45 68 84 75 55

(i) Explain why the results are a population.

(ii) Find the mean and median of the results.

(iii) Find the range and the standard deviation of the results.

2. The following are the earnings per share for a sample of 15 software companies for the

year 2005. The earnings per share are arranged from smallest to largest.

€0.09 €0.13 €0.41 €0.51 €1.12 €1.2 €1.49 € 3.

€3.5 €6.36 €7.83 €8.92 €10.13 €12.99 €16.

Compute the mean, median and standard deviation. Find the coefficient of

skewness using Pearson's estimate. What is your conclusion regarding the shape of

the distribution?

3. Southwest Airlines is studying the weight of luggage for each passenger. For a large

group of domestic passengers, the mean is 47 pounds with a standard deviation of 10

pounds. For a large group of overseas passengers, the mean is 78 pounds and the standard

deviation is 15 pounds. Compute the relative dispersion of each group. Comment on the

difference in relative dispersion.

4. The sales manager of a book company has a staff of 40 who send her a weekly report.

This lists the number of prospective buyers called to by the sales people that week.

Listed below, ordered from smallest to largest, are the number of visits last week:

(a) Calculate the median number of calls.

(b) Write out the formula for the location of a percentile. Determine the 25

th

and

75

th

percentiles.

(c)Determine the inter-quartile range for this distribution.

5. The following data give the weekly amounts (€) spent on groceries for a sample of

households:

(a) Define a frequency distribution. How many classes for a frequency distribution

would you recommend?

(b) What class intervals would you suggest?

(c) What would you recommend as the lower limit of the first class?

(d) Organize the data into a frequency distribution.

6. The income of a group of sample observations is €30,000; the standard deviation is

€10,000. If income is normally distributed, according to Chebyshev's theorem, at least

what percent of the incomes will lie between €10.000 and €40,000?

7. Mobile phone customers select their own four-digit personal identification (PIN)

number to protect their phones.

(i) Think of this as an experiment and list four possible outcomes.

(ii) What is the probability Mr. Smith and Mrs. Smith select the same PIN?

(iii) Which concept of probability did you use to answer (ii)?

8. (a) Explain the difference between the special and general rules of

addition in probability analysis

(b) A survey of undergraduates in the School of Social Sciences at

Western University revealed the following distribution regarding the

gender and majors (assume they take one specialized subject) of

students:

(i) What is the probability of selecting a female student?

(ii) What is the probability of selecting a political science

or economics major?

(iii) What is the probability of selecting a female or an

economics major?

(iv) What is the probability of selecting an economics

major given that the person selected is male?

9. A study of 200 grocery chains revealed these incomes after taxes:

What is the probability a particular chain has under $1 million in income after

taxes?

What is the probability of grocery chain selected at random has an income of

greater than $20 million? What rule of probability was applied?

10. Refer the following table.

First Event

Second Event Al A2 A3 Total

B 1 8 6 4 18

B

2

2 3 7 12

Total 10 9 11 30

(i) Determine P (A 1 ).

(ii) Determine P (B 2 |A 2 ).

(iii) Determine P (B 2 and A 3

).

Income after Taxes Number of Firms

Under $10 million 102

$20 million to $50 million 61

$50 million or more 37