Math 466 - Theory of Statistics

Syllabus for Math 466, Section 003, Spring 2024

Location and Times

Math 466, Section 002, meets in C E Chavez Bldg, Rm 104, TuTh 12:30-12:45pm.

Course Description

Course Prerequisites or Co-requisites

1. MATH 464 - Theory of Probability.
2. or equivalent coursework with instructor permission.

Instructor and Contact Information

Information	Data
Instructor	Professor Marek Rychlik
Office	Mathematics 605
Telephone	1-520-621-6865
Email	rychlik@arizona.edu
Instructor Homepage/Web Server	http://alamos.math.arizona.edu
Course Homepage	http://alamos.math.arizona.edu/math466
Course Homepage (Mirror)	http://marekrychlik.com/math466

Course Format and Teaching Methods

The course format is that of a conventional lecture, with in-class discussion and additional web-delivered content. All lectures will be recorded and available on Zoom and Panopto.

Written homework will be assigned regularly and graded using Gradescope.

In addition, the course incorporates required programming assignments. Numerical experimentation is essential to understanding and using the course subject matter. The assignments will be graded by an autograder implemented in Gradescope.

Course Objectives

This course is an introduction to the theory of stochastics. The student will gain understanding of the following fundamental concepts:

1. Sampling theory
2. Point estimation
3. Limiting distributions
4. Testing Hypotheses
5. Confidence intervals
6. Large sample methods

Learning outcomes

Students who successfully complete this course are expected to be able to:

• In the context of sampling distributions calculate and interpret confidence intervals.
• Define and use consistency, efficiency, and sufficiency for point estimators.
• Apply the method of moments.
• Use method of maximum likelihood to derive optimal statistics.
• Derive and apply t-tests and z-tests.

Generative AI use IS permitted or encouraged

In this course you are welcome and expected to use generative artificial intelligence/large language model tools, e.g. ChatGPT, Dall-e, Bard, Perplexity. Using these tools aligns with the course learning goals such as developing writing and programming skills, and ability to effectively use available information. Be aware that many AI companies collect information; do not enter confidential information as part of a prompt. LLMs may make up or hallucinate information. These tools may reflect misconceptions and biases of the data they were trained on and the human-written prompts used to steer them. You are responsible for checking facts, finding reliable sources for, and making a careful, critical examination of any work that you submit. Your use of AI tools or content must be acknowledged or cited. If you do not acknowledge or cite your use of an AI tool, what you submit will be considered a form of cheating or plagiarism. Please use the following guidelines for acknowledging/citing generative AI in your assignments:

• MLA
• APA

Absence and Class Participation Policy

Importance of attendance and class participation

Participating in course and attending lectures and other course events are vital to the learning process. As such, attendance is required at all lectures and discussion section meetings. Students who miss class due to illness or emergency are required to bring documentation from their healthcare provider or other relevant, professional third parties. Failure to submit third-party documentation will result in unexcused absences.

Missed Exams

Students are expected to be present for all exams. If a verifiable emergency arises which prevents you from taking an in-class exam at the regularly scheduled time, the instructor must be notified as soon as possible, and in any case, prior to the next regularly scheduled class. Make-up exams and quizzes will be administered only at the discretion of the instructor and only under extreme circumstances. If a student is allowed to make up a missed exam, (s)he must take it at a mutually arranged time. No further opportunities will be extended. Failure to contact your instructor as stated above or inability to produce sufficient evidence of a real emergency will result in a grade of zero on the exam. Other remedies, such as adjusting credit for other exams, may be considered.

COVID-19 related policies

As we enter the semester, the health and wellbeing of everyone in this class is the highest priority. Accordingly, we are all required to follow the university guidelines on COVID-19 mitigation. Please visit http://www.covid19.arizona.edu for the latest guidance.

UA policies

The UA's policy concerning Class Attendance, Participation, and Administrative Drops is available at: http://catalog.arizona.edu/2015-16/policies/classatten.htm The UA policy regarding absences for any sincerely held religious belief, observance or practice will be accommodated where reasonable, http://policy.arizona.edu/human-resources/religious-accommodation-policy . Absences pre-approved by the UA Dean of Students (or Dean Designee) will be honored. See: http://uhap.web.arizona.edu/policy/appointed-personnel/7.04.02

Required Texts or Readings

Required Textbook

Mathematical Statistics with Applications. 7th Edition. Dennis Wackerly, William Mendenhall, Richard L. Scheaffer

Optional Reference Textbook

Introduction to probability theory. Hoel, Paul G., Stone, Charles J., Port, Sidney C.

Assignments and Examinations

Notes on exam administration

All examinations are planned to be administered during the class time, either in person or on Zoom.

If, due to unforseen circumstances, they cannot be held in person, they are held on Zoom using the "gallery view" mode.The exam papers for not in-person tests will be distributed on-line by D2L and collected electronically using D2L "dropbox" feature.

Exam/assignment listing with date and grade contribution

Exam or Assignment	Date	Grade contribution
Midterm 1	February 15 (Thursday), 12:30pm - 1:45pm	20%
Midterm 2	April 11 (Thursday), 12:30pm - 1:45pm	20%
Final Examination	May 8 (Wednesday), 1:00pm - 3:00pm	30%
Homework	See D2L	30%

Homework Assignments

Written homework consists of approximately twelve assignments equally contributing to the grade, each worth 30/12 = 2.5% of the grade. The assignments are posted on line at this link: Homework. The assignment papers are collected via Gradescope, which is cloud-based software for semi-automatic grading. Things to keep in mind:

• The student will submit all graded work through Gradescope. Gradescope assignments, by their nature, are answer-oriented. The opportunity for partial credit is somewhat limited.
• Homework may not be submitted through e-mail, no exceptions.
• Due dates will be adjusted in Gradescope onlyas needed and the student is expected to check regularly for those adjustments. Due dates in D2L will not be updated.

Written homework is assigned regularly throughout the semester, for a total of approximately 80 problems. Two types of homework will be assigned:

1. Homework which consists of selected exercises in the required textbook.
2. Some custom homework will be composed by the instructor. Some of the custom problems will require programming.

Homework submission requirements

Using Gradescope for grading differs from other grading systems. Mainly, it uses AI to allow the instructor to accurately grade a larger number of problems than it would be possible otherwise. Some grading is completely automated (e.g., solutions to problems with a numerical answer). More comples answers may be grouped automatically by using Machine Learning, OCR and image analysis. However, it is possible to completely confuse the system by improperly structuring the submitted document. Therefore, please read the instructions below carefully and re-visit them as needed. Note that Gradescope supports automatic regrade requests which you can use if all fails.

The solutions must be structured in such a way that Gradescope can read them and that its 'AI' can interpret them. Your homework must be submitted as a PDF document, even if you use scanner or phone to capture images. Two typical workflows will be as follows:

1. Download the blank assignment (also called a 'template') from Gradescope.
2. Read and understand exactly what answers you need to provide. The space to enter the answer is a blue box, and marked with a label such as 'Q1.1' ("Question 1, part 1").
3. Work out the problem on "paper" (real or virtual), to obtain the answers. They must fit in the designated boxes in the 'template'. The size of the box is a hint from the instructor about the size of the answer (typically a number or a math formula) when entered by hand, using regular character size.
4. The recommended way to fill out the 'template' is paperless, by using suitable software and hardware (digital pen or tablet). I use a free program Xournal for this and it works great. You need to use it in combination with a digital pen or a tablet. It can produce a PDF easily, ready for submission to Gradescope.
5. You can also print the assignment on (real) paper, fill out the answers and scan the marked up document back to PDF format. However, the position of the boxes must be exactly (to a fraction of an inch) as in the original. Also, you may encounter a variety of "quality control" issues, especially if you are using a digital camera to scan the paper solution. All issues can be solved by a mix of the right hardware and software, but may not be the best time investment. The least troublesome way to scan is to use a real, flatbed scanner, e.g. in the library.
6. Upload the resulting document (a PDF of the 'template' marked up with your answers) to Gradescope. Your PDF must contain your name and student id in designated spaces. The Gradescope 'AI' will look for your name and student id, to properly associate it with your account.
7. After grading, the grade will be transmitted to D2L (Brightspace) and will be added to your 'Final Calculated Grade' automatically.
8. Do not reduce handwriting size! Reduce the size of your answer using
   • closed form expressions;
   • appropriate math functions, e.g., absolute value, min and max.
9. Under no circumstances write outside the provided space (boxes). Gradescope, and the grader only considers the content of the designated boxes.
10. IMPORTANT! Do not insert pages in the solution template. This will confuse Gradescope, and will result in reduced score and/or will require re-submission. However, you are encouraged to submit scratchwork. You should create pages at the end of the document. Similarly, if you run out of space in the template for your solution, you can continue the solution on a newly created page at the end of the document, adding a note in the template: "Solution continued on page 13" where page 13 will contain the continuation.

Programming and Software

The class will have small programming assignments. It is expected that you will be using software to gain insights into the assigned problems and subject matter. The programming assignments must be submitted in formats supported by Gradescope and the instructor. The number of programming languages will be limited two two or three. R will be supported and it is encouraged that you use it as it is most compatible with the course content.

For illustrating some aspects of the course, I will be using these programs (easy to download and free to use):

• A computer algebra system (CAS) called Maxima.
• The maxima GUI wxMaxima is a separate program and this is the preferred way to use Maxima if you are a beginner. During the previous semester students successfully applied Maxima to doing their homework, and I will provide examples of Maxima calculations in class.
• The R programming language and environment, with a rich set of tools to do simulations in probability and stochastic processes (and analyze data).
• The GUI (and more for R): RStudio. Makes use of R somewhat more pleasant, especially for a new user.

Final Examination

The final examination is scheduled for: May 8 (Wednesday), 1:00pm - 3:00pm.

The time, data and general exam rules are set by the University and can be found at these links:

• Final Exam Regulations
• Final Exam Schedule

Grading Scale and Policies

The student in the class normally receives a letter grade A, B, C, D or E.

The cut-offs for the grades are:

Grade	% Range
A	90%+
B	80-90%
C	70-80%
D	60-70%
E	0-60%

Normally, individual tests and assignments will not be "curved". However, grade cut-offs may be lowered at the end of the semester (but not raised!) to reflect the difficulty of the assignments and other factors that may cause abnormal grade distribution.

The grade will be computed by D2L and the partial grade will be updated automatically by the system as soon as the individual grades are recorded.

General UA policy regarding grades and grading systems is available at https://catalog.arizona.edu/policy-type/grade-policies

Safety on Campus and in the Classroom

For a list of emergency procedures for all types of incidents, please visit the website of the Critical Incident Response Team (CIRT):

https://cirt.arizona.edu/case-emergency/overview Also watch the video available at https://arizona.sabacloud.com/Saba/Web_spf/NA7P1PRD161/common/learningeventdetail/crtfy000000000003560

Classroom Behavior Policy

To foster a positive learning environment, students and instructors have a shared responsibility. We want a safe, welcoming and inclusive environment where all of us feel comfortable with each other and where we can challenge ourselves to succeed. To that end, our focus is on the tasks at hand and not on extraneous activities (i.e. texting, chatting, reading a newspaper, making phone calls, web surfing, etc).

Threatening Behavior Policy

The UA Threatening Behavior by Students Policy prohibits threats of physical harm to any member of the University community, including to one's self. See: http://policy.arizona.edu/education-and-student-affairs/threatening-behavior-students .

Accessibility and Accommodations

Our goal in this classroom is that learning experiences be as accessible as possible. If you anticipate or experience physical or academic barriers based on disability, please let me know immediately so that we can discuss options. You are also welcome to contact Disability Resources (520-621-3268) to establish reasonable accommodations. For additional information on Disability Resources and reasonable accommodations, please visit http://drc.arizona.edu/ .

If you have reasonable accommodations, please plan to meet with me by appointment or during office hours to discuss accommodations and how my course requirements and activities may impact your ability to fully participate. Please be aware that the accessible table and chairs in this room should remain available for students who find that standard classroom seating is not usable. Code of Academic Integrity Required language: Students are encouraged to share intellectual views and discuss freely the principles and applications of course materials. However, graded work/exercises must be the product of independent effort unless otherwise instructed. Students are expected to adhere to the UA Code of Academic Integrity as described in the UA General Catalog. See: http://deanofstudents.arizona.edu/academic-integrity/students/academic-integrity http://deanofstudents.arizona.edu/codeofacademicintegrity .

UA Nondiscrimination and Anti-harassment Policy

The University is committed to creating and maintaining an environment free of discrimination, http://policy.arizona.edu/human-resources/nondiscrimination-and-anti-harassment-policy . Our classroom is a place where everyone is encouraged to express well-formed opinions and their reasons for those opinions. We also want to create a tolerant and open environment where such opinions can be expressed without resorting to bullying or discrimination of others.

Additional Resources for Students

UA Academic policies and procedures are available at: http://catalog.arizona.edu/2015-16/policies/aaindex.html Student Assistance and Advocacy information is available at: http://deanofstudents.arizona.edu/student-assistance/students/student-assistance

Confidentiality of Student Records

http://www.registrar.arizona.edu/ferpa/default.htm .

Subject to Change Statement

Information contained in the course syllabus, other than the grade and absence policy, may be subject to change with advance notice, as deemed appropriate by the instructor.

Significant Dates (from the Registrar's Website)

Undergraduate

      


  

  

    
Date
    
Event
  
  
  

  

    
January 10, 2024
    
Classes Begin
  
  

    
January 15, 2024
    
Martin Luther King Jr. Holiday - No Classes
  
  

    
March 2 - 10, 2024
    
Spring Recess - No Classes
  
  

    
May 1, 2024
    
Last Day of Classes and Laboratory Sessions
  
  

    
May 2, 2024
    
Reading Day - No Classes or Finals
  
  

    
May 3 - 9, 2024
    
Final Examinations
  
  

    
May 10, 2024
    
Commencement
  
  

    
May 10, 2024
    
Degree Award Date for Students Completing by Close of Spring Semester
  






  
Spring 2024 - Undergraduate Regular Academic Session
  

    

      

        
Date
        
Event
      
    
    

      

        
10/1/2023
        
Shopping Cart available
      
      

        
1/9/2024
        
Last day to file Undergraduate Leave of Absence
      
      

        
1/9/2024
        
Last day for students to add to or drop from a waitlist
      
      

        
1/10/2024
        
FIRST DAY OF SPRING CLASSES

            UAccess still available for registration

            First day to file for the Grade Replacement Opportunity (GRO)

            First day to add classes for audit and instructor signature is required
      
      

        
1/15/2024
        
Martin Luther King Day, no classes
      
      

        
1/17/2024
        
Last day to use UAccess for adding classes, changing classes, or changing sections
      
      

        
1/18/2024
        
Instructor approval required on a Change of Schedule form to ADD or CHANGE classes
      
      

        
1/23/2024
        
Last day to drop without a grade of W (withdraw)

            Classes dropped on or before this date will remain on your UAccess academic record with a status of dropped, but will not appear on your transcript

            Last day to change from credit to audit, or vice versa, with only an instructor's signature
      
      

        
1/23/2024
        
Last day for a refund
      
      

        
1/24/2024
        
Beginning today, students may completely withdraw from all classes in the term
      
      

        
1/24/2024
        
W period begins a penalty grade of W will be awarded for each withdrawal and the class(es) will appear on your transcript

            Beginning today, a change from credit to audit requires instructor approval on a Change of Schedule form
      
      

        
2/1/2024
        
Last day to apply for Spring degree candidacy without a late fee After this date a $50 00 Late Candidacy Application fee will be assessed
      
      

        
2/6/2024
        
Last day for department staff to add or drop in UAccess
      
      

        
2/6/2024
        
Last day to change from pass/fail to regular grading or vice versa with only instructor approval on a Change of Schedule form
      
      

        
2/7/2024
        
Instructor's and dean's signatures are required on a Change of Schedule form to change from pass/fail to regular grades or vice versa
      
      

        
3/3/2024
        
Last day to make registration changes without the dean's signature
      
      

        
3/4/2024
        
Instructor's and dean's signatures are required on all Change of Schedule forms to ADD or CHANGE classes
      
      

        
3/4/2024
        
Spring recess begins
      
      

        
3/10/2024
        
Spring recess ends
      
      

        
3/26/2024
        
Last day to file for Grade Replacement Opportunity (GRO)
      
      

        
3/26/2024
        
Last day for students to withdraw from a class online through UAccess

            Last day for students to change to/from audit with only instructor approval

            Last day for instructors to administratively drop students
      
      

        
3/27/2024
        
Instructor and dean's signatures required on a Late Change Petition in order to withdraw from class and students must have an extraordinary reason for approval

            Beginning today, a change from credit to audit will be permitted only if the student is doing passing work on the course. Instructor and dean's permission required on a Change of Schedule form
      
      

        
4/9/2024
        
Last day for students to submit a Late Change Petition to their college
      
      

        
5/1/2024
        
Last day to request a complete withdraw from all classes in the term
      
      

        
5/1/2024
        
Last day of class--no registration changes can be made after the last day of class
      
      

        
5/2/2024
        
Reading day, no classes
      
      

        
5/3/2024
        
Final exams begin
      
      

        
5/9/2024
        
Final exams end

            Final grades are available in UAccess as soon as the instructor posts them

            Per Faculty Senate Policy, grades should be submitted within two business days after the final exam
      
      

        
5/10/2024
        
Degree award date
      
    
  



    

Material Covered

We will cover Chapters 7-10 of the book. Here is the approximate schedule with approximate dates when the particular sections shall be covered.

Chapter.Section	Title	Page	Covered	Date
	What Is Statistics?
1.1	Introduction	1
1.2	Characterizing a Set of Measurements: Graphical Methods	3
1.3	Characterizing a Set of Measurements: Numerical Methods	8
1.4	How Inferences Are Made	13
1.5	Theory and Reality	14
1.6	Summary	15
	Probability
2.1	Introduction	20
2.2	Probability and Inference	21
2.3	A Review of Set Notation	23
2.4	A Probabilistic Model for an Experiment: The Discrete Case	26
2.5	Calculating the Probability of an Event: The Sample-Point Method	35
2.6	Tools for Counting Sample Points	40
2.7	Conditional Probability and the Independence of Events	51
2.8	Two Laws of Probability	57
2.9	Calculating the Probability of an Event: The Event-Composition Method	62
2.10	The Law of Total Probability and Bayes’ Rule	70
2.11	Numerical Events and Random Variables	75
2.12	Random Sampling	77
2.13	Summary	79
	Discrete Random Variables and Their Probability Distributions
3.1	Basic Definition	86
3.2	The Probability Distribution for a Discrete Random Variable	87
3.3	The Expected Value of a Random Variable or a Function of a Random Variable	91
3.4	The Binomial Probability Distribution	100
3.5	The Geometric Probability Distribution	114
3.6	The Negative Binomial Probability Distribution (Optional)	121
3.7	The Hypergeometric Probability Distribution	125
3.8	The Poisson Probability Distribution	131
3.9	Moments and Moment-Generating Functions	138
3.10	Probability-Generating Functions (Optional)	143
3.11	Tchebysheff’s Theorem	146
3.12	Summary	149
	Continuous Variables and Their Probability Distributions
4.1	Introduction	157
4.2	The Probability Distribution for a Continuous Random Variable	158
4.3	Expected Values for Continuous Random Variables	170
4.4	The Uniform Probability Distribution	174
4.5	The Normal Probability Distribution	178
4.6	The Gamma Probability Distribution	185
4.7	The Beta Probability Distribution	194
4.8	Some General Comments	201
4.9	Other Expected Values	202
4.10	Tchebysheff’s Theorem	207
4.11	Expectations of Discontinuous Functions and Mixed Probability Distributions (Optional)	210
4.12	Summary	214
	Multivariate Probability Distributions
5.1	Introduction	223
5.2	Bivariate and Multivariate Probability Distributions	224
5.3	Marginal and Conditional Probability Distributions	235
5.4	Independent Random Variables	247
5.5	The Expected Value of a Function of Random Variables	255
5.6	Special Theorems	258
5.7	The Covariance of Two Random Variables	264
5.8	The Expected Value and Variance of Linear Functions of Random Variables	270
5.9	The Multinomial Probability Distribution	279
5.10	The Bivariate Normal Distribution (Optional)	283
5.11	Conditional Expectations	285
5.12	Summary	290
	Functions of Random Variables
6.1	Introduction	296
6.2	Finding the Probability Distribution of a Function of Random Variables	297
6.3	The Method of Distribution Functions	298
6.4	The Method of Transformations	310
6.5	The Method of Moment-Generating Functions	318
6.6	Multivariable Transformations Using Jacobians (Optional)	325
6.7	Order Statistics	333
6.8	Summary	341
	Sampling Distributions and the Central Limit Theorem
7.1	Introduction	346
7.2	Sampling Distributions Related to the Normal Distribution	353
7.3	The Central Limit Theorem	370
7.4	A Proof of the Central Limit Theorem (Optional)	377
7.5	The Normal Approximation to the Binomial Distribution	378
7.6	Summary	385
	Estimation
8.1	Introduction	390
8.2	The Bias and Mean Square Error of Point Estimators	392
8.3	Some Common Unbiased Point Estimators	396
8.4	Evaluating the Goodness of a Point Estimator	399
8.5	Confidence Intervals	406
8.6	Large-Sample Confidence Intervals	411
8.7	Selecting the Sample Size	421
8.8	Small-Sample Confidence Intervals for μ and μ1 − μ2	425
8.9	Confidence Intervals for σ^2	434
8.10	Summary	437
	Properties of Point Estimators and Methods of Estimation
9.1	Introduction	444
9.2	Relative Efficiency	445
9.3	Consistency	448
9.4	Sufficiency	459
9.5	The Rao–Blackwell Theorem and Minimum-Variance Unbiased Estimation	464
9.6	The Method of Moments	472
9.7	The Method of Maximum Likelihood	476
9.8	Some Large-Sample Properties of Maximum-Likelihood Estimators (Optional)	483
9.9	Summary	485
	Hypothesis Testing
10.1	Introduction	488
10.2	Elements of a Statistical Test	489
10.3	Common Large-Sample Tests	496
10.4	Calculating Type II Error Probabilities and Finding the Sample Size for Z Tests	507
10.5	Relationships Between Hypothesis-Testing Procedures and Confidence Intervals	511
10.6	Another Way to Report the Results of a Statistical Test: Attained Significance Levels, or p-Values	513
10.7	Some Comments on the Theory of Hypothesis Testing	518
10.8	Small-Sample Hypothesis Testing for μ and μ1 − μ2	520
10.9	Testing Hypotheses Concerning Variances	530
10.10	Power of Tests and the Neyman–Pearson Lemma	540
10.11	Likelihood Ratio Tests	549
10.12	Summary	556
	Linear Models and Estimation by Least Squares
11.1	Introduction	564
11.2	Linear Statistical Models	566
11.3	The Method of Least Squares	569
11.4	Properties of the Least-Squares Estimators: Simple Linear Regression	577
11.5	Inferences Concerning the Parameters βi	584
11.6	Inferences Concerning Linear Functions of the Model Parameters: Simple Linear Regression	589
11.7	Predicting a Particular Value of Y by Using Simple Linear Regression	593
11.8	Correlation	598
11.9	Some Practical Examples	604
11.10	Fitting the Linear Model by Using Matrices	609
11.11	Linear Functions of the Model Parameters: Multiple Linear Regression	615
11.12	Inferences Concerning Linear Functions of the Model Parameters: Multiple Linear Regression	616
11.13	Predicting a Particular Value of Y by Using Multiple Regression	622
11.14	A Test for H0 : βg+1 = βg+2 = · · · = βk = 0	624
11.15	Summary and Concluding Remarks	633
	Considerations in Designing Experiments
12.1	The Elements Affecting the Information in a Sample	640
12.2	Designing Experiments to Increase Accuracy	641
12.3	The Matched-Pairs Experiment	644
12.4	Some Elementary Experimental Designs	651
12.5	Summary	657
	The Analysis of Variance
13.1	Introduction	661
13.2	The Analysis of Variance Procedure	662
13.3	Comparison of More Than Two Means: Analysis of Variance for a One-Way Layout	667
13.4	An Analysis of Variance Table for a One-Way Layout	671
13.5	A Statistical Model for the One-Way Layout	677
13.6	Proof of Additivity of the Sums of Squares and E(MST) for a One-Way Layout (Optional)	679
13.7	Estimation in the One-Way Layout	681
13.8	A Statistical Model for the Randomized Block Design	686
13.9	The Analysis of Variance for a Randomized Block Design	688
13.10	Estimation in the Randomized Block Design	695
13.11	Selecting the Sample Size	696
13.12	Simultaneous Confidence Intervals for More Than One Parameter	698
13.13	Analysis of Variance Using Linear Models	701
13.14	Summary	705
	Analysis of Categorical Data
14.1	A Description of the Experiment	713
14.2	The Chi-Square Test	714
14.3	A Test of a Hypothesis Concerning Specified Cell Probabilities: A Goodness-of-Fit Test	716
14.4	Contingency Tables	721
14.5	r × c Tables with Fixed Row or Column Totals	729
14.6	Other Applications	734
14.7	Summary and Concluding Remarks	736
	Nonparametric Statistics
15.1	Introduction	741
15.2	A General Two-Sample Shift Model	742
15.3	The Sign Test for a Matched-Pairs Experiment	744
15.4	The Wilcoxon Signed-Rank Test for a Matched-Pairs Experiment	750
15.5	Using Ranks for Comparing Two Population Distributions: Independent Random Samples	755
15.6	The Mann–Whitney U Test: Independent Random Samples	758
15.7	The Kruskal–Wallis Test for the One-Way Layout	765
15.8	The Friedman Test for Randomized Block Designs	771
15.9	The Runs Test: A Test for Randomness	777
15.10	Rank Correlation Coefficient	783
15.11	Some General Comments on Nonparametric Statistical Tests	789
	Introduction to Bayesian Methods for Inference
16.1	Introduction	796
16.2	Bayesian Priors, Posteriors, and Estimators	797
16.3	Bayesian Credible Intervals	808
16.4	Bayesian Tests of Hypotheses	813
16.5	Summary and Additional Comments	816
Appendix 1.	Matrices and Other Useful Mathematical Results	821
A1.1	Matrices and Matrix Algebra	821
A1.2	Addition of Matrices	822
A1.3	Multiplication of a Matrix by a Real Number	823
A1.4	Matrix Multiplication	823
A1.5	Identity Elements	825
A1.6	The Inverse of a Matrix	827
A1.7	The Transpose of a Matrix	828
A1.8	A Matrix Expression for a System of Simultaneous Linear Equations	828
A1.9	Inverting a Matrix	830
A1.10	Solving a System of Simultaneous Linear Equations	834
A1.11	Other Useful Mathematical Results	835
Appendix 2.	Common Probability Distributions, Means, Variances, and Moment-Generating Functions	837
Appendix 2.Table 1	Discrete Distributions	837
Appendix 2.Table 2	Continuous Distributions	838
Appendix 3.	Tables	839
Appendix 3.Table 1	Binomial Probabilities	839
Appendix 3.Table 2	Table of e−x	842
Appendix 3.Table 3	Poisson Probabilities	843
Appendix 3.Table 4	Normal Curve Areas	848
Appendix 3.Table 5	Percentage Points of the t Distributions	849
Appendix 3.Table 6	Percentage Points of the χ 2 Distributions	850
Appendix 3.Table 7	Percentage Points of the F Distributions	852
Appendix 3.Table 8	Distribution Function of U	862
Appendix 3.Table 9	Critical Values of T in the Wilcoxon Matched-Pairs, Signed-Ranks Test; n = 5(1)50	868
Appendix 3.Table 10	Distribution of the Total Number of Runs R in Samples of Size (n 1 , n 2 ); P(R ≤ a)	870
Appendix 3.Table 11	Critical Values of Spearman’s Rank Correlation Coefficient	872
Appendix 3.Table 12	Random Numbers	873
	Index	896