Introduction to Statistics

Department of Economics University of Michigan

ECON 405 Fall 2012

Introduction to Statistics Lectures: Instructor:

Mon. and Wed., 4:00pm − 5:30pm, 2306 Mason Hall

Yoonseok Lee (365C Lorch Hall, 615-0177, [email protected])

Office Hours:

Fri. 11:00am − 12:00pm

GSI:

Gail Lucasan ([email protected]) (Office Hours: TBA)

Discussion Sessions: Fri. 9:00am−10:00am, 120 Dennison [DIS 002] Fri. 10:00am−11:00am, 120 Dennison [DIS 003]

Course Description ECON 405 is an accelerated introduction to mathematical statistics that requires complete fluency with advanced calculus including integration and differentiation though no prior knowledge of statistics is assumed. The purpose of the course is to provide students with a theoretical understanding of the principles of statistical inference. Topics include probability theory, experimental and theoretical derivation of sampling distributions, hypothesis testing, and properties of estimators including maximum likelihood and method of moments. Enforced pre-requisite is Calculus II (i.e., MATH 116 or higher) with C- or better. In practice, most economics students enrolling in ECON 405 have experience in advanced calculus beyond the required MATH 116 (e.g., multi-variable calculus), are typically planning to enroll subsequently in ECON 406, and often are considering graduate study in economics. Students looking for a general, less mathematical treatment are urged to consider ECON 404 instead. No credit is granted to those who have completed or are enrolled in IOE 265, STATS 265, 400, or 412. Students with credit for ECON 404 can only elect ECON 405 for 2 credits and must have permission of the instructor. The GSI will hold weekly office hours and discussion sessions. She will go over problem sets and answer questions about materials covered in the class. The class web page is available at http://ctools.umich.edu. Announcements, problem sets and additional course materials will be posted there, so make sure to visit the site frequently. Hard copies of these materials will not be distributed. If you believe you need an accommodation for a disability, please let me know at your earliest convenience. Some aspects of this course may be modified to facilitate your participation and progress. As soon as you make me aware of your needs, we can work with the Office of Services for Students with Disabilities to help us determine appropriate accommodations. I will treat any information you provide as private and confidential.

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Textbooks The required textbook for the course is: Hogg, R.V. and E.A. Tanis (2010). Probability and Statistical Inference, 8th ed., Prentice Hall. Most of the problem set questions are from this textbook. Other references that are helpful, though not required, are: Hogg, R.V., J.W. McKean and A.T. Craig (2004). Introduction to Mathematical Statistics, 6th ed., Prentice-Hall. [Higher level than Hogg and Tanis] Keller, G. (2008). Statistics for Management and Economics, 8th ed., Cengage Learning. [Lower level than Hogg and Tanis] All these books are on reserve at the Shapiro Undergraduate Library.

Organization and Evaluation Grades will be decided based on class attendance, weekly problem sets, and two in-class quizzes. The grading breakdown is as follows: Class Attendance 10%; Problem Sets 30%; Quiz I 30%; Quiz II 30%. Class attendance is to be checked 2-4 times on random dates. The problem sets will be posted on the class CTools cite every Friday afternoon and they are due on the discussion session(s) of the following week. No late submission will be accepted. Students are encouraged to form study groups and collaborate with other students to work on problem sets. You have to, however, write up and submit your own solutions.1 The lowest grade of the problem sets will be dropped to calculate the final grade. The quizzes are scheduled as follows (in the normal classroom space): Quiz I: 4:00pm − 5:30pm, Wednesday, October 17 (in class)

Quiz II: 4:00pm − 5:30pm, Monday, December 10 (in class)

All quizzes are closed-book. No makeup quizzes nor early quizzes will be given for any reason, so please plan your travels smartly. Both Quiz I and II are required to pass this course.

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A word of advice: When you write the solution, provide the major steps of your calculation as you are taking exams. It is a good training for organizing and explaining your idea. When you are taking the quizzes, you will not be able to get the full credit if you simply write down the final answers without providing details.

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Course Outline I. Probability 1. Fundamentals of Probability Theory • Set theory; Probability axioms (Ch.1.1-1.3) • Conditional probability; Independent events; Bayes’ rule (Ch.1.4-1.6) 2. Random Variables and Distribution Theory • Random variables and Random vectors (Ch.2.1, 3.1-3.3, 4.1) • Distribution functions; Density functions; Moment generating functions (Ch.2.1, 2.5, 3.1-3.3, 4.1) • Expectation and inequalities (Ch.2.2-2.3) • Joint probability distributions; Conditional probability; Independence; Conditional expectations (Ch.4.1-4.3) 3. Important Families of Distributions (Ch.2.4, 2.6, 3.4-3.7, 4.4) • Discrete distributions: Bernoulli, (Negative-) Binomial, Poisson (Ch.2.4, 2.6) • Continuous distributions: Uniform, Normal, Gamma, Chi-square, Bivariate normal, (Student’s) ,  (Ch.3.4-3.7, 4.4, 5.2, 5.5) 4. Distributions of Functions of Random Variables • Distribution function technique; Transformation/Change-of-variables technique; Moment generating function technique (Ch.5.1-5.5) II. Statistical Inference 1. Sampling Distribution • Sample theory; Sum of independent random variables; Law of large numbers; Central limit theorem; Approximations for discrete distributions (Ch.5.3, 5.6-5.7, 10.5-10.6) 2. Point Estimation • Properties of estimators: bias, efficiency, mean square error, consistency (Ch.6.1) • Maximum likelihood estimation; Method of moments (Ch.6.1) 3. Interval Estimation • Estimation of confidence intervals for mean, variance, proportions (Ch.6.2-6.6) 4. Hypothesis Testing • Basics of statistical testing; Null and alternative hypothesis; Type I and type II errors; Significance level; Power; -value (Ch.7.1-7.2, 10.2) • Tests of mean, variance, proportions (Ch.7.1-7.7) 3