Math 218, Mathematical Statistics
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.
- Chapter 1: Introduction (1 meeting)
The basic goal of statistics: draw conclusions based on data. There
are various aspects of statistics ranging from formulating the question,
designing experiments to address the question, collecting the data, and
analyzing the data, but we'll be stressing the role of probability
and probability distributions in this process. We'll often begin with
a random sample drawn from a parameterized family of distributions, and
our job is to make conclusions about the parameter.
- Chapter 2: Review of Probability (1 meeting)
We'll quickly review the theory of probability. Sample spaces and events,
Kolmogorov's axioms, principles of combinatorics including permutations
and combinations, conditional probability and independence, Bayes' theorem,
random variables, probability mass functions for discrete random variables,
probability density functions for continuous random variables, cumulative
distribution functions, expected value, mean and variance of a distribution,
selected discrete and continuous distributions.
- Chapter 3: Collecting Data (2 meetings)
Types of statistical studies, observational studies, basic sampling designs,
- Chapter 4: Summarizing and Exploring Data (2 meeting)
Categorial data, numerical data, bivariate data, time-series data
- Chapter 5: Sampling Distributions of Statistics (6 meetings)
- 5.1. Sampling Distribution of the Sample Mean
- 5.2. Sampling Distribution of the Sample Variance
- 5.3. Student's t-distribution
- 5.4. Snedecor-Fisher's F-distribution
- Chapters 6 and 15: Basic Concepts of Inference (9 meetings)
- 6.1. Point Estimation
- 15.1. Maximum Likelihood Estimation
- 6.2. Confidence Interval Estimation
- 6.3. Hypothesis Testing
- 15.2. Likelihood Ratio Tests
- 15.3. Principles of Bayesian statistics. The Bernoulli process.
- Chapter 7: Inferences for Single Samples (4 meetings)
- 7.1. Inferences on Mean (Large Samples)
- 7.2. Inferences on Mean (Small Samples)
- 7.3. Inferences on Variance (if time permits)
- Bayesian inference for samples
- Chapter 8: Inferences for Two Samples (4 meetings)
- 8.1. Independent Samples and Matched Pairs Designs
- 8.2. Graphical methods for comparing two samples
- 8.3. Comparing Means of Two Populations, independent samples and matched pairs
- Chapter 9: Inferences for Proportions and Count Data (3 meetings)
- 9.1. Inferences on Proportion
- 9.2. Inferences on Comparing Two Proportions
- Chapter 10: Simple linear regression and correlation (2 meetings)
- The least squares method
- 10.1. The model for simple linear regression
- 10.2. Fitting a line, goodness of fit
- 10.3. Statistical inference with the simple linear regression model,
prediction and confidence intervals
- 10.4. Regression diagnostics
- Chapter 14: Nonparametric statistics (2 meetings)
- 14.1. Inferences for single samples, sign tests
- 14.2. Inferences for two independent samples
- 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
Binomial coefficients (combinations),
Axioms for probability,
Uniform finite probabilities,
The Bernoulli process,
Discrete random variables,
Expectation for discrete random variables,
Variance for discrete random variables,
The Poisson process,
Expectation and variance for continuous random variables,
Sums and convolution,
Covariance and correlation,
Moments and the moment generating function,
A proof of the central limit theorem
Class notes, quizzes, tests, homework assignments
(This part of the course page will be constantly updated.)
- 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.
- Summarizing and exploring data
Normal probablility plot
- Simpson's paradox.
- Common probability distributions.
See also the
Gallery of distributions at
- 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.
- 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.
- Confidence intervals.
- Hypothesis tests.
- Homework due Wednesday, Feb. 24, from chapter 6: exercises 1, 2ab, 5, 7, 11, 13a, 14, 15ab.
- Bayesian statistics Part I: an example, the basic principle, the Bernoulli process
- First Test, Friday, Feb. 26.
- 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.
- 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.
- Second Test, Friday, April 1. On sections 6.3, 15.2, 15.3, 7.1-7.3, 8.1-8.3. Open book.
- Homework from chapter 9: exercises 1-3, 6.
- 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.
Tests from past years
This page is located on the web at
David E. Joyce