Francis O'toole (Module 1: Statistics) Paul Walsh (Module 2: Mathematics)

EC1030: MATHEMATICS & STATISTICS 2005-06

Francis O'Toole (Module 1: Statistics) [Paul Walsh (Module 2: Mathematics)]

Office: Room 3008 ArtsBuilding (Department of Economics)

Office Hours: 11.00am – 12.00pm Wednesday and 11.45am – 12.45pm Friday.

Teaching Assistants: Adrian Byrne, Clarissa Fernandes, Paola Labrecciosa, Samir Patel, Martin Schmitz, Reetta Suonpera, Denis Tkachenko, Jing Zhang. [Office Hours: To be announced.]

Required Textbook: Basic Statistics for Business & Economics, Lind, Marchal and Wathen, Fifth Edition, McGraw-Hill, International Edition, 2006.

There will be a term test of one and half hours duration in week 10 of Michaelmas Term counting towards 40 per cent of your final grade for the statistics module (i.e. 20 per cent of your final grade for this course). The Statistics portion of the final examination (of three hours total duration) accounts for the remaining 60 per cent of your final grade for this module (i.e. 30 per cent of your final grade for this course). There will be no penalty imposed on students with an excused absence (e.g. medical certificate) from the term test. The term test will consist of multiple-choice questions and short answer type questions. The final examination will consist of more-detailed “homework-type” questions.

Students must attempt and hand in “homeworks” assigned in lectures and reviewed by the teaching assistants in classes (weekly from week 2). These problems will be collected at the start of classes; students are advised to make a copy so as to be able to make notes/corrections during classes. These problems will likely form the basis for some of the final examination questions.

This course provides a thorough introduction to descriptive statistics, elementary probability theory, point and interval estimation and confidence intervals. This material forms the essential basis for future courses in statistical methods in the social sciences. This module will cover material based on the contents of Chapter 1 (What is Statistics?) to at least Chapter 9 (Estimation and Confidence Intervals). Extra lecture notesare available via

Michaelmas Term

Weeks 1 to 3: Chapters 1 to 4 - Descriptive Statistics (approximately 9 lectures)

Weeks 4 & 6: Chapter 5 - Probability and Bayes’ Theorem (approximately 6 lectures)

Weeks 7 & 8: Chapters 6 and 7 - Probability Distributions (approximately 6 lectures)

Week 9: Chapter 8 - Sampling Methods and the Central Limit Theorem (approximately 3 lectures)

Week 10: Term Test - One and a half hours

Hilary Term

Week 1: Chapter 9 - Estimation and Confidence Intervals (approximately 5 lectures)

Week 2: Chapter 10 - One Sample Tests of Hypothesis (approximately 3 lectures)

EC1030 MATHEMATICS & STATISTICS

Michaelmas Term Test

Mansion House

9.30am – 11.00amWednesday 10th December 2003

Lecturer: Francis O'Toole

Teaching Assistants: Vahagn Galstyan, William Hynes, Olivia Mollen and Hampus Willfors

Please fill in the following information

Student Name (please use block capitals):______

Student Signature:______

Student ID Number:______

Tutorial Group (day, time and teaching assistant):______

Please read the following carefully before you answer any questions.

Please ensure that you have addressed the above requests for information before you hand in your script at the end of the examination. (Information with respect to your tutorial group will facilitate in the returning of scripts.) Please attempt all sections of this paper. Please answer all questions in the space provided. There are a total of 100 marks available. Allocate your time carefully. Good luck and enjoy your break.

Section A: Please attempt all multiple-choice questions. Each question has only one correct answer. Please indicate your choice clearly. Each correct answer is worth 3 points. (There is no penalty for an incorrect answer.)

1. In the late 1970s, the Mexico city government “created” a six-lane motorway by repainting a four-lane motorway - there was an 50% increase in the number of lanes. In response to the subsequent increase in fatal accidents, the city government reversed the above procedure and “recreated” a four-lane motorway by repainting the six-lane motorway – there was an approximate 33% decrease in the number of lanes. Overall, what was the percentage change in the number of lanes?

(a)an increase of approximately 17%

(b)no change

(c)a decrease of approximately 17%

2. The variable “Gender” can be regarded as being, in general:

(a)qualitative and ratio level

(b) quantitative

(c)qualitative and nominal level

(d)qualitative and ordinal level

3. The ______is used to determine the average rate of change from one period to another. Which of the following best completes the previous sentence?

(a) arithmetic mean

(b) geometric mean

(d) coefficient of skewness

4. Two events A and B are ______if the occurrence (or non-occurrence) of one event has no effect on the probability of the occurrence (or non-occurrence) of the other event. Which of the following best completes the previous sentence?

(a) statistically independent

(b) mutually exclusive

(d) none of the above

5. Which of the following two statements is correct?

(a)A z-score gives deviations from the mean in terms of standard deviations.

(b)A z-score gives deviations from the standard deviation in terms of means.

6. Which of following is correct?

(a) Var (x) = E(x) – E(x2)

(b) Var (x) = E(x2) – E(x)

(d) Var (x) – [E(x2)]2 – E(x2)

7. About 95 percent of the area under the normal curve is within ______standard deviation(s) of the mean. Which of the following best completes the previous sentence?

(a) one

(b) two

8. The ______represents the “balancing point” of a distribution. Which of the following best completes the previous sentence?

(a) standard deviation

(b) median

9. Which of the following statements is always true?

(a) P(A) + P(Not A) = 1

(b) P(A) – P(Not A) = 0

10. What is the probability of getting exactly two "tails" in four tosses of a fair coin?

(a) 50 per cent

(b) 3/8

Section B. Attempt all questions in the space provided. Each correct answer is worth 5 points.

11. The Bookstall, Inc., is a speciality bookstore concentrating on used books. Paperbacks are €1.00 each, and hardcover books are €3.50 each. Of the 50 books sold last Tuesday morning, 40 were paperback and the rest were hardcover. What was the weighted mean price of a book sold last Tuesday morning?

12. Using a Venn diagram, show that:

P(A or B or C) = P(A) + P(B) + P(C) – P(A and B) – P(A and C) – P(B and C) + P(A and B and C).

13. A study by the National Park Services revealed that 50 percent of vacationers going to the RockyMountain region visit YellowstonePark, 40 percent visit the Tetons and 35 percent visit both.

(a) What is the probability a vacationer will visit at least one of these attractions?

(b) Are the events mutually exclusive? Explain briefly.

14. If you ask four strangers on campus, what is the probability that:

(a) all were born on Sunday?

(b) all were born on different days of the week?

(d) none were born on Sunday?

15. The following table lists the probability distribution for cash prizes in a lottery conducted in Lawson’s Department Store.

Prize (€) / Probability
0 / .45
10 / .30
100 / .20
500 / .05

Calculate the mean, variance and standard deviation of this distribution.

16. A recent study by the American Accounting Association revealed 23 percent of students graduating with a major in accounting select public accounting. Suppose we select a sample of 15 recent graduates.

(a) What is the probability that exactly two select public accounting?

(b) What is the probability that exactly five select public accounting?

17. The amounts dispensed by a cola machine follow the normal distribution with a mean of 7 ounces and a standard deviation of 0.10 ounces per cup. How much cola is dispensed (per cup) in the largest 1 percent of the cups?

18. Assume a binomial distribution with n = 50 and  = .25. Compute the following:

(a) The mean and standard deviation of the random variable.

(b) The probability that X is 15 or more.

Section C. Attempt all questions in the space provided. Each correct answer is worth 15 points.

19. Advertising expenses are a significant component of the cost of products sold. Listed below is a frequency distribution showing the advertising expenditures for 60 manufacturing companies located in the Southwest. Estimate the mean and standard deviation of advertising expense.

Advertising Expenditures (€millions) / Number of Companies
25 up to 35 / 5
35 up to 45 / 10
45 up to 55 / 21
55 up to 65 / 16
65 up to 75 / 8
Total / 60

20. There are five male candidates and three female candidates for three positions. All eight are equally qualified. The three selected are male.

(a) What is the probability of the observed evidence given no discrimination?

(b) The following formula represents Bayes’ Theorem:

P(AB) = P(BA).P(A)____

P(BA).P(A) + P(BNot A).P(Not A)

Given a prior probability of discrimination of 0.9 (and using your answer to the previous question), calculate the posterior probability of discrimination.

Final Exam: May 2004

Section A: Answer three of the following four questions.

Ecommerce.com is studying the lead time (elapsed time between when an order is placed and when it is filled) for a sample of recent orders. The lead times are reported in days.

Lead Time (days) / Frequency
0 up to 5 / 12
5 up to 10 / 14
10 up to 15 / 24
15 up to 20 / 16
20 up to 25 / 14
Total / 80

(a)How many orders were studied?

(b)What is the mid-point of the first class?

(c)What are the co-ordinates of the first class for a frequency polygon?

(d)Sketch histogram.

(e)Sketch a frequency polygon.

(f)How many orders were filled in less than 10 days? In less than 15 days?

(g)Convert the frequency distribution into a cumulative frequency distribution.

(h)Sketch the cumulative frequency polygon.

(i)About 60 per cent of the orders were filled in less than how many days?

SCCoast developed the following frequency distribution on the age of internet users.

Age (years) / Frequency
10 up to 20 / 6
20 up to 30 / 14
30 up to 40 / 36
40 up to 50 / 40
50 up to 60 / 24

Find the mean and standard deviation.

Answer two of the following four questions.

(a) If you meet three strangers, what is the probability that (i) all were born on Tuesday? (ii) all were born on different days of the week? (iii) none were born on Sunday?

(b) A case of 24 cans contains 1 can that is contaminated. Three cans are to be chosen randomly for testing. (i) How many different combinations of three cans could be selected? (ii) What is the probability that the contaminated can is selected for testing?

(c) Suppose 60 per cent of all people prefer Coke to Pepsi. We select 16 people for further study. (i) How many would you expect to prefer Coke? Explain your answer. (ii) What is the probability that (exactly) 10 of those surveyed will prefer Coke? (iii) What is the probability that (exactly) 15 of those surveyed will prefer Coke?

(d) The mean starting salary for College graduates in the spring of 2002 was €31,280. Assume that the distribution of starting salaries follows the normal distribution with a standard deviation of €3,300. What per cent of the graduates have starting salaries: (i) between €30,000 and €35,000? (ii) more than €40,000? (iii) between €35,000 and €40,000?

4.Answer two of the following four questions.

(a) There are five sales representatives at Mid-Motors Ford. The five representatives and the number of cars they sold last week are:

Sales Representative / Cars Sold
Peter / 16
Connie / 12
Ron / 8
Ted / 20
Peggy / 12

(i)How many different samples of size 2 (representatives) are possible?

(ii)List all possible samples of size 2, and compute the mean of each sample.

(iii)Compare the mean of the sampling distribution with the mean of the population.

(iv)Compare (briefly) the dispersion in the sample means with the dispersion in the population.

(b) A population of unknown shape has a mean of 75. You select a sample of size 40. The standard deviation of the sample is 5. compute the probability that the sample mean is: (i) less than 74; (ii) between 74 and 76; (iii) between 76 and 77; and (iv) greater than 77.

(c) A sample of 81 observations is taken from a normal distribution. The sample mean is 40 and the sample standard deviation is 5. Determine the 95 per cent confidence interval for the population mean.

(d) Ms. Maria Wilson is considering running for mayor. She conducts a survey of voters. A sample of 400 voters reveals that 300 would vote for her in the election. (i) Estimate the value of the population proportion. (ii) Compute the standard error of the proportion. (iii) Develop a 99 per cent confidence interval for the population proportion. (iv) Interpret briefly your findings.