Math 218, Mathematical Statistics Spring 2016 Professor D Joyce BP 322, 793-7421. Department of Mathematics and Computer Science Clark University

Adrien-Marie Legendre
Adrien-Marie Legendre (1752–1833)     1820 watercolor caricature by     artist Julien-Léopold Boilly (1796–1874)

Please bookmark this page, http://aleph0.clarku.edu/~djoyce/ma218/, so you can readily access it.

General information

• Description of this course: The emphasis of this course is to develop the fundamental statistical concepts of inference and hypothesis testing from a classical perspective using the tools of probability theory. Topics investigated include sampling and sample distributions, graphical data analysis, point and interval estimation, hypothesis testing and an introduction to Bayesian inference.

  Prerequisite: Math 217, or permission.

  See also Clark's Academic Catalog

• Textbook: Statistics and Data Analysis by Ajit C. Tamhane and Dorothy D. Dunlop, Prentice-Hall, 2000. ISBN 0-13-744426-5. Prentice-Hall's web site for the book.

• Course goals.
  • To provide students with a good understanding of the theory of statistics as described in the syllabus.
  • To show how statistical methods depend on the theoretical principles of probability.
  • To illustrate the use of statistics in science, both physical and social science
  • To examine the assumptions of statistical methods.
  • To make connections between statistics and other branches of mathematics, and to see some of the history of statistics.
  
  

Syllabus

• Assignments, quizzes, tests, final: There will be numerous short assignments, mostly from the text, occasional quizzes, two tests during the semester, and a two-hour final exam during finals week.

  Practice problems will be assigned daily from the text to help you master the concepts discussed in class. Periodically, problems will be assigned to be turned in and graded. Although not all practice problems will be submitted for a grade, it is expected that you will keep up to date on the problems. Collected homework is due in class on the assigned day. No late assignments will be accepted.

  Tests are closed notebook, but you may bring one sheet of notes and a calculator. In the event that an emergency arises, you are responsible to contact me before the regularly scheduled exam to make alternative arrangements.

• Course grade. The course grade will be determined as follows:
  2/9 assignments and quizzes,
  2/9 each of the two midterms, and
  1/3 for the final exam.

• General policies.
  Class attendance and class participation are obligatory. During the class meetings the text will be supplemented with more rigorous theory and special topics. Turn off your cell phones during class. Laptops may only be used for class-related purposes—no texting, no browsing, no email.
  
  Clark University is committed to providing students with documented disabilities equal access to all university programs and facilities. If you have or think you have a disability and require academic accommodations, you must register with Student Accessibility Services (SAS), which is located in room 430 on the fourth floor of the Goddard Library. If you have questions about the process, please contact The Director of Student Accessibility Services at (508)798-4368. If you are registered with SAS, and qualify for accommodations that you would like to utilize in this course, please request those accommodations through SAS in a timely manner.

• Time and study. Besides the time for classes, you'll spend time on reading the text, doing the assignments, and studying of for quizzes and tests. That comes to about five to nine hours outside of class on average per week, the actual amount varying from week to week. Here's a summary of a typical semester's 180 hours
  Regular class meetings, 14 weeks, 42 hours
  Two evening midterms and final exam, 6 hours
  Reading the text and preparing for class, 5 hours per week, 70 hours
  Doing weekly homework assignments, 3 hours per week 42 hours
  Meeting with tutors or in study groups, variable 4 to 12 hours
  Reviewing for midterms and finals, 12 hours

• Math 217 class notes from Fall 2014 with information on
  Sets, Combinatorics, Binomial coefficients (combinations), Axioms for probability, Uniform finite probabilities, Conditional probability, Bayes' formula, Independent events, The Bernoulli process, Discrete random variables, Expectation for discrete random variables, Variance for discrete random variables, Continuous probability, Density functions, The Poisson process, Expectation and variance for continuous random variables, Joint distributions, Sums and convolution, Conditional distributions, Covariance and correlation, Order statistics, Moments and the moment generating function, A proof of the central limit theorem

Class notes, quizzes, tests, homework assignments

• Summary of basic probability theory. We'll review some high points of probability theory (Chapter 2). The purpose of this review is just to explain what I assume you already know.
• Collecting data. Begin a discussion of collecting data: types of statistical studies, control groups in a comparative study, sample surveys, prospective and retrospective studies, basic sampling designs, simple and other kinds of random sampling. (Chapter 3) Experimental studies.
• Homework due Friday, Jan 29, from chapter 3: 1, 3, 4, 7, 8, 9, 12, 13, and 15. These are not computational exercises. For most exercises, write a paragraph to answer the question using full sentences. A few of them only ask for short answers. Answers
• Summarizing and exploring data
  Normal probablility plot
• Simpson's paradox.
• Common probability distributions.
  See also the Gallery of distributions at http://www.itl.nist.gov/div898/handbook/eda/section3/eda366.htm.
• Homework due Wednesday, Feb 3, from chapter 4: 2, 5, 10, and select one exercise from 4.29 through 4.46 for presentation in class (if you work in a group, do as many exercises as a group as there are students in your group)
• Distribution of the sample mean, central limit theorem, approximation of the binomial distribution by a normal distribution.
• Sample variance, the chi² distribution, the gamma function.
• Student's t distribution and Snedecor-Fisher's F distribution.
• Homework due Wednesday, Feb 10, from chapter 5: 4, 5, 8 , 14, 16, 17, 23. Answers
• Point estimators, inferences about parameters based on data from a sample.
• Maximum likelihood estimators, both discrete and continuous (Chapter 15)
• Homework due Wednesday, Feb 17, from chapter 15: 1,2,3,5. Answers
• Confidence intervals.
• Hypothesis tests.
• Homework due Wednesday, Feb. 24, from chapter 6: exercises 1, 2ab, 5, 7, 11, 13a, 14, 15ab. Answers
• Bayesian statistics Part I: an example, the basic principle, the Bernoulli process
• First Test, Friday, Feb. 26. Answers.
• Normal sample with known variance, any large sample: z-confidence intervals and hypothesis tests, hypothesis tests for the mean. Small normal sample: t-distribution. Chi²-intervals and tests.
• Bayesian statistics Part II: Bayes pool table, conjugate priors for the Bernoulli process, point estimators, interval estimators
• Homework due Friday, Mar 18, from chapter 6: exercises 17, 18, 19, 20, 22, and from chapter 7: exercises 1, 2, 9, 11, 12. Answers
• Bayesian statistics Part III: the Poisson process & its conjugate priors
• Bayesian statistics Part IV: the normal distribution with known variance
• Bayesian statistics Part V: the normal distribution with unknown variance
• More homework from chapter 6: various exercises presented in class.
• Inferences for two samples, two independent samples and matched pairs.
• Inferences for proportions and count data.
  Inferences for comparing two proportions.
• Homework due Monday, Mar 28, from chapter 7: exercises 17, 18, 19, and from chapter 8: exercises 1, 2, 3, 9. Answers
• Second Test, Friday, April 1. On sections 6.3, 15.2, 15.3, 7.1-7.3, 8.1-8.3. Open book. Answers
  
• Homework from chapter 9: exercises 1-3, 6. Answers
• The method of least squares.
• The model for simple linear regression and its analysis.
• Inferences for simple linear regression.
  Estimating the error variance σ² of the model.
  Confidence and prediction intervals for simple linear regression, regression diagnostics.
• Simple linear regression example.
• Homework from chapter 10: various exercises presented in class.
• Bayesian linear regression.
• Introduction to multiple linear regression
• Introduction to nonparametric statistics, medians and the sign test.
• Presentation on machine learning.
• Introduction to principal component analysis
• Philosophy of statistics from Stanford. Excerpts.

Tests from past years

• First midterm. Answers
• Second midterm. Answers
• Final exam. Answers

Related courses

• Math 120, Calculus I
• Math 121, Calculus II
• Math 130, Linear Algebra
• Math 131, Multivariate Calculus
• Math 217, Probability and Statistics

This page is located on the web at

http://aleph0.clarku.edu/~djoyce/ma218/

David E. Joyce