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SUMMER TERM 2018

ECON1003: STATISTICAL METHODS IN ECONOMICS

SUGGESTED SOLUTIONS

TIME ALLOWANCE: 2 HOURS

Answer ALL questions.

Questions in Part A are multiple choice questions, and they carry 60 percent of the total mark. VERY IMPORTANT: Please record your answers to these questions in the Examination Answer Book that has been provided. Write down the question number, and the letter corresponding to the answer that you think is the correct one. Any answers recorded on the exam paper rather than in the Examination Answer Book that has been provided will not be graded.  

Questions in Part B carry 40 percent of the total mark.

Please note that you are not allowed to remove this question paper from the examination venue.

You may use the blank pages in this question paper for rough work.

For reference, please note that tables for the standard normal CDF, as well as chi-square and student t critical values, are included with the exam paper. Please turn the tables in with your exam paper at the end of the examination period.

PART A

1) A company wishes to evaluate the effectiveness of a marketing campaign. Sixty-five percent of all potential professors were reached in a focused advertising programme. Twenty-one percent of those contacted adopted the book while 8% of those not contacted adopted the book. Define the following events of interest:

A1 = Professor received advertising material

A2 = Professor did not receive advertising material

B1 = Professor adopts the book

B2 = Professor does not adopt the book

What is the probability that a professor who adopts the book received the advertising material?

A) 0.534

B) 0.830

C) 0.897

D) 0.913

2) There are five closed boxes and only one contains a prize. You are asked to choose a box at random and set it aside, without opening it. After you have made your selection, a friend - who knows where the prize is - opens three of the four remaining boxes. He opens on purpose those that do not have the prize. There are now only two closed boxes left: the one you originally chose, and the one that your friend did not open. You are now given the option to revise your original choice of box and switch it with the one that your friend has not opened. To decide whether to switch or not, you decide to calculate the probability that the box that you originally selected contains the prize. This probability is:

A) 1/2

B) 1/3

C) 4/5

D) 1/5

3) Consider the following point estimators, W, X, Y, and Z of μ: ;

; ; and . Suppose that  and  are drawn from an identical distribution, with mean µ and standard deviation σ.

Which of these point estimators is unbiased?

A) W

B) Q

C) Y

D) Z

4) Of the point estimators in questions 3), which one has the smallest variance?

A) W

B) Q

C) Y

D) Z

5) A labour economist studying migration is interested in seeing whether there is a difference between the salaries of natives and immigrants. From a random sample of 20 natives, the mean salary is found to be $41,780 with a standard deviation of $5,426. From a sample of 12 immigrants, the mean salary is found to be $40,136 with a standard deviation of $4,383. Assume that the random sample observations are from normally distributed populations, and that the population variances are assumed to be equal.

What is the upper confidence limit of the 95% confidence interval for the difference between the population mean salary for natives and immigrants?

A) $5,122.835

B) $5,423.276

C) $5,240.833

D) $5,079.791

6) Increasing the probability of a Type I error by one percentage point decreases the probability of accepting a true null Hypothesis by one percentage point. Is this statement true?

A) True

B) False

7) Suppose that  and  are population proportions. The p-value of testing  against  is the smallest significance level that would lead to a rejection of the null.  Is this statement true?

A) True

B) False

8) The following expression gives the coverage probability of a confidence interval.

 

In particular, this is the confidence interval for the mean of a normal distribution when the variance is known. In this expression, which object is random?

A)

B) α

C)

D)

9) Suppose that we are interested in exploring the determinants of student achievement in high-school. One possible measure of achievement is a standardized test. The scores of this standardized test have mean 0 and variance 1 in the population of students. We are interested in verifying if there is an association between family income and student standardised test scores. The following regression line is obtained: "standardized test score = -0.53595 + 0.0244943 log(family income)”, where “log(family income)” is the logarithm of the income of the student’s family. Which of the following statements is true with respect to the given scenario?

A) If family income increases by 0.0244943 pounds, we would expect student test scores to decrease by 0.53595 standard deviations.

B) If student test scores increase by 0.0244943 standard deviations, we would expect family income to approximately increase by 1 pound

C) If family income increases by 1 percent, we would expect student test scores to increase by 2.44943 percent of a standard deviation.

D) If student test scores increase by 0.0244943 standard deviations, we would expect family income to approximately increase by 1 percent.

10) There are 18 professors in the Department of Economics. Eleven of them have received good evaluations from students, while 7 received poor evaluations. Each professor teaches one course. You will take three courses in the Department of Economics next term. What is the probability that at least one of your professors next term has received a poor evaluation?

A) 0.516

B) 0.798

C) 0.258

D) 0.202

11) The following table presents the probability mass function for the number of flights that depart per hour from a small local airport.

# of departing flights	3	4	5	6	7	8
P(x)	0.11	0.16	0.27	0.23	0.13	0.10

What is the average number of flights departing per hour?

A) 5.67

B) 5.81

C) 5.73

D) 5.41

12) In STATA, you summarise a variable called Spanish_score, which contains the students’ score in a Spanish test. You obtain the following output:

 sum spanish_score , detail

                 SIMCE test score - Spanish

-------------------------------------------------------------

      Percentiles      Smallest

 1%       144.17         128.54

 5%       158.66         128.68

10%       168.95         129.59       Obs               9,132

25%       189.01         130.48       Sum of Wgt.       9,132

50%          220                      Mean           223.8808

                        Largest       Std. Dev.      44.89862

75%      253.125          381.4

90%       284.71         387.91       Variance       2015.886

95%       305.53         403.43       Skewness       .4853035

99%       340.27         403.43       Kurtosis       2.865638

What is the median value of Spanish test scores?

A) 223.8808

B) 220

C) 220.03

D) 130.48

13) Imagine that Spanish scores are a continuous random variable. Looking at the same STATA output from question 12, how do you expect the graph of the probability distribution function to look like?

A) A mostly symmetric distribution, with a slightly longer and thinner right tail than left tail

B) A mostly symmetric distribution, with a slightly longer and thinner left tail than right tail

C) A perfectly symmetric distribution

D) A bimodal distribution

14) The number of viewers ordering a particular pay-per-view program is normally distributed. 20% of the time, fewer than 20,000 people order the program. Only ten percent of the time more than 28,000 people order the program. What is the mean μ and standard deviation σ of the number of people ordering the program?

A)   μ= 23,170 , σ= 3,774

B)   μ= 23,170 , σ= 14,243,076

C)   μ= 24,000 , σ= 3,774

D)   μ= 24,000 , σ= 5,640

15) The speed of cars passing through a check point follows a normal distribution with a mean of 44 miles per hour and a standard deviation of 6 miles per hour. What is the probability that the next car passing will not be exceeding 50 miles per hour?

A) 0.1587

B) 0.8413

C) 0.7967

D) 0.9854

16) You are the webmaster for your firm's web-site. From your records, you know that the probability that a visitor will buy something from your firm is 0.44. In one day, the number of visitors is 898. What is the probability that less than 390 of them will buy something from your firm?

A) 0.3557

B) 0.3594

C) 0.9279

D) 0.0721

17) Monthly rates of return on the shares of a particular common stock are independent of one another and normally distributed with a standard deviation of 1.8. A sample of 15 months is taken. Find the probability that the sample standard deviation is more than 2.6.

A) 0.01

B) 0.48

C) 0.74

D) 0.99

PART B

B1) In a certain population, a proportion p of individuals has an autoimmune disease. A new blood test is invented to detect the presence of the disease. The probability that an individual has the disease if the test result is positive is 1/14, while the probability that the test gives a positive result if an individual has the disease is 0.93.

a) Using one of the three postulates of probability, find the upper bound of p. Show all your calculations, and define the probability postulate that you are invoking.

b) What are the other two probability postulates?

c) What happens to the probability that the test gives a positive result if an individual does not have the disease, as p increases? Show all your calculations.

B2) On Sunday evening of week 10 of the academic year, you go to a party. Someone you have never met before comes up to you and tells you that she can read your mind. You are sceptical, and ask her that she proves her claim. She accepts the challenge.

First, she asks you to pick your favourite 3 DVDs out of a set of 6 DVDs, and to order them from your most favourite to your least favourite. While you do this, she goes to another room, so she cannot see which three DVDs you are selecting, and in what order you are placing them on the shelf. While she is still in the other room, she writes down her guess. When she comes back to the room, she shows you her guess of the three DVDs that you selected and of the order from your most to your least preferred. Her guess is correct.

Second, she tells you that she knows that you have not become friends with any Italian student yet. From historical data, you know that, on average, it takes a new student at UCL 19 weeks to meet and become friends with an Italian student.

Third, she asks you to think of a colour: either Red or Blue. You think Red, and she guesses correctly.

a) Imagine that you are performing a scientific test of this woman’s claim that she can read your mind. What is the null hypothesis?

b) Under the null hypothesis, what is the probability that she gets the first guess right?

c) Under the null hypothesis, what is the probability that she gets the second guess right? Indicate which probability distribution you are using.

d) Under the null hypothesis, what is the probability that she gets the third guess right?

e) Under the null hypothesis, what is the probability that she gets all three guesses right?

f) Do you believe her claim? Define the concept of p-value for a general hypothesis test. (Do not set up a specific null and alternative hypothesis to define the p-value!). Explain how your answer is related to the concept of p-value.

g) Now suppose that this person got only the second and third guess correct, but did not guess the first one correctly (the DVDs). What is the probability of this alternative result under the null? Would you still believe her claim? Explain why and explain how your answer is related to the concept of conventional significance levels.