ECON1151：Quantitative Methods

Section A

1. For parts a, b and c refer to a good for which the supply and demand equations are given by:

𝑃 = 𝑄𝑠 2 + 25 𝑃 = −20𝑄𝐷 + 550

where QD, Qs and P represent, respectively, the quantity demanded (in 1000s of units per week), the quantity supplied (in 1000s of units per week), and the good’s price in £.

Which of the following statements are true, false or uncertain and fully explain your reasons, drawing a relevant diagram for each part:

a) When P = £425 there is excess demand;

b) When Q = 15 units per week the market is in equilibrium.

c) If wages now change and the supply curve shifts to 𝑃 = 𝑄𝑠 2 + 74, the equilibrium price and quantity will both fall.

Parts d and e refer to a firm that has the following production function:

Q = √KL 2

Where, K and L are the units of capital and labour used in the production process respectively.

a) Calculate the marginal products of labour and capital.

b) What is the value of marginal rate of technical substitution when 4 units of capital and 3 units of labour are used?

Section B

2. A random sample was collected from 15 customers which contained information on the time (measured in minutes) each customer took to complete an online form.

a) Calculate the mean and standard deviation

b) Calculate the inter-quartile range (IQR) and coefficient of variation

c) The skewness value from the data, as produced by EXCEL software, is 0.121758. What does it mean?

d) Using the empirical rule, what interval of the completion times contains approximately 95% of the data?

e) Is it accurate for the firm to use the empirical rule in (d) as a decision making guide? Explain your answer.

Section C

3. An industry of price-taking profit-maximising firms in perfect competition all produce using the same technology, and each has a total cost function given by:

TC = 2q³ − 4q² + 3q

Where TC is measured in thousands of pounds per week, and q is measured in thousands of units per week. Then:

a) If the market price of output is p = £4.5 what is each firm’s output and profit (or loss)? [25 marks]

b) Show that each firm’s long-run output is 1000 units per week (i.e. 1 thousands of units per week). [25 marks]

c) What is the price of output when the industry is in equilibrium? [25 marks]

d) if the market demand is given by P = 11 − 0.01Q, where Q is the industry output in thousands of units per week, what is the industry’s equilibrium output and how many firms are in the industry in equilibrium? [25 marks]

4. A consumer maximizes U = xy subject to the budget constraint M = Pxx + Pyy where Px and Py are respectively the prices of the two goods x and y, and M is the consumer’s income.

a) Find the Lagrange function and the first-order conditions. [25 marks]

b) Find the demand function for goods x and y. [25 marks]

c) What would be the total utility function? Next, differentiate the total utility function and show that it equals the optimal value of the Lagrange multiplier. [25 marks]

d) If it is given that Px = 5, Py = 10 and M = 100, what would be the optimal values of x, y and λ? [25 marks]

Section D

5. The information below lists the calories and sugar contents, in grams, in seven randomly selected breakfast cereals:

a) Compute and interpret the correlation coefficient from the information in table 1. [40 marks]

b) A researcher wants to fit a regression line (Calories = a + b Sugar) to predict the number of calories in other cereals based on the sugar contents.

(i) Using relevant information from question (a), calculate and interpret the value of A (Rsquared).

(ii) Find the values of B and C and write the prediction line and give an interpretation. [15 marks] (iii) Using your equation from b (ii), find the level of calories in a given cereal that has 5.38 grams of sugar. Also predict the grams of sugar needed to produce a calorie level of 93.8. [15 marks] (iv)Find the value of D, and provide an interpretation of this value. [15 marks]

6. a) Assume a survey was conducted in 2009 for customers in a small city who prefers to order different types of meals via a drive-through (not sitting in) from any of the three restaurants in the city. Suppose the results, based on a sample of 100 males and 100 females

If a respondent is selected at random, what is the probability that the respondent:

(i) prefers ordering lunch at the drive-through? [10 marks]

(ii) prefers ordering breakfast or lunch at the drive-through? [10 marks]

(iii) is a male or prefers ordering dinner at the drive-through? [10 marks]

(iv) is a male and prefers ordering dinner at the drive-through? [10 marks]

(v) Given that the person selected is a female, what is the probability that she prefers ordering breakfast at the drive-through? [10 marks]

(vi) Are meal and gender statistically dependent or independent? Explain how you arrived at your answer. [15 marks]

b) The branch manager of a nationwide bookstore chain (located near a UK college campus) wants to study characteristics of her store’s customers. She decides to focus on two variables: the amount of money spent by customers (on items other than textbooks) and whether the customers would consider purchasing educational DVDs. The results from a sample of 70 customers are as follows:

• Amount spent: mean (X̅) = £28.52, and standard deviation(S) = £11.39

• 28 customers stated that they would consider purchasing the educational DVDs

(i) Construct a 95% confidence interval estimate for the population mean amount spent in the bookstore. [10 marks]

(ii) Construct a 90% confidence interval estimate for the population proportion of customers who would consider purchasing educational DVDs. [10 marks]

(iii) Assume that the branch manager of another store in the chain (also located close to a college campus) wants to conduct a similar survey in his store. How many customers need to be selected to have 90% confidence of estimating the population proportion who would consider purchasing the educational DVDs to within ± £0.04? [15 marks]