Math 466 - Theory of Statistics

Syllabus for Math 466, Section 002, Fall 2023

Location and Times

Math 466, Section 002, meets in PSYCH 307, MWF 1-1:50pm.

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 Fall 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	September 27 (Wednesday), 1:00pm - 1:50pm	20%
Midterm 2	November 8 (Wednesday), 1:00pm - 1:50pm	20%
Final Examination	December 11 (Monday), 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: December 11 (Monday), 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

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

      Dates: 08/21/2023 - 12/06/2023
Date	Standard Class Dates: Fall 2023 - Undergraduate - Regular Academic Session
3/1/2023	Shopping Cart available
8/20/2023	Last day to file Undergraduate Leave of Absence
8/20/2023	Last day for students to add to or drop from a waitlist
8/21/2023	FIRST DAY OF FALL 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 

8/28/2023	Last day to use UAccess for adding classes, changing classes, or changing sections
8/29/2023	Instructor approval required on a Change of Schedule form to ADD or CHANGE classes
9/1/2023	Last day to apply for Fall degree candidacy without a late fee After this date a $50 00 Late Candidacy Application fee will be assessed
9/3/2023	

    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

9/3/2023	Last day for a refund
9/4/2023	Beginning today, students may completely withdraw from all classes in the term
9/4/2023	Labor Day, no classes
9/4/2023	

    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

9/15/2023	Last day to change from pass/fail to regular grading or vice versa with only instructor approval on a Change of Schedule form
9/16/2023	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
9/17/2023	Last day for department staff to add or drop in UAccess
9/29/2023	Honors Convocation - no classes from 3:00 PM to 5:00 PM (Family Weekend)
10/15/2023	Last day to make registration changes without the dean's signature
10/16/2023	Instructor's and dean's signatures are required on all Change of Schedule forms to ADD or CHANGE classes
10/29/2023	Last day to file for Grade Replacement Opportunity (GRO)
10/29/2023	

    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 

10/30/2023	

    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

11/10/2023	No classes in observance of Veteran's Day
11/19/2023	Last day for students to submit a Late Change Petition to their college
11/23/2023	Thanksgiving recess begins today with no classes until Monday
12/6/2023	Last day to request a complete withdraw from all classes in the term
12/6/2023	Last day of class--no registration changes can be made after the last day of class
12/7/2023	Reading day, no classes
12/8/2023	Final exams begin
12/14/2023	

    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 

12/15/2023	Degree award date

    

Graduate

      Date	Standard Class Dates: Spring 2022 - Graduate - Regular Academic Session
10/1/2021	Shopping Cart available
1/11/2022	Last day for students to add to or drop from a waitlist
1/12/2022	FIRST DAY OF CLASS

    UAccess still available for registration
    First day to add classes for audit; instructor signature is required 

1/17/2022	Martin Luther King Day, no classes
1/19/2022	Last day to use UAccess for adding classes, changing classes, or changing sections
1/20/2022	Instructor approval required on a Change of Schedule form to ADD or CHANGE classes
1/25/2022	Last day for a refund
1/26/2022	Beginning today, students may completely withdraw from all classes in the term
2/8/2022	

    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

2/9/2022	Last day for department staff to add or drop in UAccess
2/9/2022	Last day to change from pass/fail to regular grading or vice versa with only instructor approval on a Change of Schedule form
2/9/2022	

    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

2/10/2022	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
3/7/2022	Spring recess begins
3/9/2022	Last day to make registration changes without the dean's signature
3/10/2022	Instructor's and dean's signatures are required on all Change of Schedule forms to ADD or CHANGE classes
3/13/2022	Spring recess ends
3/29/2022	

    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 

3/30/2022	

    Instructor's and Graduate College dean's permission required on a Change of Schedule form to withdraw from a class--penalty grade of W will be awarded and the class will appear on your transcript
    Instructor's and dean's permission required on a Change of Schedule form to change to/from audit

5/4/2022	Last day to request a complete withdraw from all classes in the term
5/4/2022	Last day of class--no registration changes can be made after the last day of class and last day to file a Complete Withdraw
5/5/2022	Reading day, no classes
5/6/2022	Final exams begin
5/12/2022	

    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 

5/13/2022	Degree award date

    

Material Covered

We will cover Chapters 1-4 of the book ("Stochastic Processes up to Margingales"). Here is the approximate schedule with approximate dates when the particular sections shall be covered.

Id	Chapter.Section	Title	Page	Covered	Date
1		What Is Statistics?
2	1.1	Introduction	1
3	1.2	Characterizing a Set of Measurements: Graphical Methods	3
4	1.3	Characterizing a Set of Measurements: Numerical Methods	8
5	1.4	How Inferences Are Made	13
6	1.5	Theory and Reality	14
7	1.6	Summary	15
8		Probability
9	2.1	Introduction	20
10	2.2	Probability and Inference	21
11	2.3	A Review of Set Notation	23
12	2.4	A Probabilistic Model for an Experiment: The Discrete Case	26
13	2.5	Calculating the Probability of an Event: The Sample-Point Method	35
14	2.6	Tools for Counting Sample Points	40
15	2.7	Conditional Probability and the Independence of Events	51
16	2.8	Two Laws of Probability	57
17	2.9	Calculating the Probability of an Event: The Event-Composition Method	62
18	2.10	The Law of Total Probability and Bayes’ Rule	70
19	2.11	Numerical Events and Random Variables	75
20	2.12	Random Sampling	77
21	2.13	Summary	79
22		Discrete Random Variables and Their Probability Distributions
23	3.1	Basic Definition	86
24	3.2	The Probability Distribution for a Discrete Random Variable	87
25	3.3	The Expected Value of a Random Variable or a Function of a Random Variable	91
26	3.4	The Binomial Probability Distribution	100
27	3.5	The Geometric Probability Distribution	114
28	3.6	The Negative Binomial Probability Distribution (Optional)	121
29	3.7	The Hypergeometric Probability Distribution	125
30	3.8	The Poisson Probability Distribution	131
31	3.9	Moments and Moment-Generating Functions	138
32	3.10	Probability-Generating Functions (Optional)	143
33	3.11	Tchebysheff’s Theorem	146
34	3.12	Summary	149
35		Continuous Variables and Their Probability Distributions
36	4.1	Introduction	157
37	4.2	The Probability Distribution for a Continuous Random Variable	158
38	4.3	Expected Values for Continuous Random Variables	170
39	4.4	The Uniform Probability Distribution	174
40	4.5	The Normal Probability Distribution	178
41	4.6	The Gamma Probability Distribution	185
42	4.7	The Beta Probability Distribution	194
43	4.8	Some General Comments	201
44	4.9	Other Expected Values	202
45	4.10	Tchebysheff’s Theorem	207
46	4.11	Expectations of Discontinuous Functions and Mixed Probability Distributions (Optional)	210
47	4.12	Summary	214
48		Multivariate Probability Distributions
49	5.1	Introduction	223
50	5.2	Bivariate and Multivariate Probability Distributions	224
51	5.3	Marginal and Conditional Probability Distributions	235
52	5.4	Independent Random Variables	247
53	5.5	The Expected Value of a Function of Random Variables	255
54	5.6	Special Theorems	258
55	5.7	The Covariance of Two Random Variables	264
56	5.8	The Expected Value and Variance of Linear Functions of Random Variables	270
57	5.9	The Multinomial Probability Distribution	279
58	5.10	The Bivariate Normal Distribution (Optional)	283
59	5.11	Conditional Expectations	285
60	5.12	Summary	290
61		Functions of Random Variables
62	6.1	Introduction	296
63	6.2	Finding the Probability Distribution of a Function of Random Variables	297
64	6.3	The Method of Distribution Functions	298
65	6.4	The Method of Transformations	310
66	6.5	The Method of Moment-Generating Functions	318
67	6.6	Multivariable Transformations Using Jacobians (Optional)	325
68	6.7	Order Statistics	333
69	6.8	Summary	341
70		Sampling Distributions and the Central Limit Theorem
71	7.1	Introduction	346
72	7.2	Sampling Distributions Related to the Normal Distribution	353
73	7.3	The Central Limit Theorem	370
74	7.4	A Proof of the Central Limit Theorem (Optional)	377
75	7.5	The Normal Approximation to the Binomial Distribution	378
76	7.6	Summary	385
77		Estimation
78	8.1	Introduction	390
79	8.2	The Bias and Mean Square Error of Point Estimators	392
80	8.3	Some Common Unbiased Point Estimators	396
81	8.4	Evaluating the Goodness of a Point Estimator	399
82	8.5	Confidence Intervals	406
83	8.6	Large-Sample Confidence Intervals	411
84	8.7	Selecting the Sample Size	421
85	8.8	Small-Sample Confidence Intervals for μ and μ1 − μ2	425
86	8.9	Confidence Intervals for σ^2	434
87	8.10	Summary	437
88		Properties of Point Estimators and Methods of Estimation
89	9.1	Introduction	444
90	9.2	Relative Efficiency	445
91	9.3	Consistency	448
92	9.4	Sufficiency	459
93	9.5	The Rao–Blackwell Theorem and Minimum-Variance Unbiased Estimation	464
94	9.6	The Method of Moments	472
95	9.7	The Method of Maximum Likelihood	476
96	9.8	Some Large-Sample Properties of Maximum-Likelihood Estimators (Optional)	483
97	9.9	Summary	485
98		Hypothesis Testing
99	10.1	Introduction	488
100	10.2	Elements of a Statistical Test	489
101	10.3	Common Large-Sample Tests	496
102	10.4	Calculating Type II Error Probabilities and Finding the Sample Size for Z Tests	507
103	10.5	Relationships Between Hypothesis-Testing Procedures and Confidence Intervals	511
104	10.6	Another Way to Report the Results of a Statistical Test: Attained Significance Levels, or p-Values	513
105	10.7	Some Comments on the Theory of Hypothesis Testing	518
106	10.8	Small-Sample Hypothesis Testing for μ and μ1 − μ2	520
107	10.9	Testing Hypotheses Concerning Variances	530
108	10.10	Power of Tests and the Neyman–Pearson Lemma	540
109	10.11	Likelihood Ratio Tests	549
110	10.12	Summary	556
111		Linear Models and Estimation by Least Squares
112	11.1	Introduction	564
113	11.2	Linear Statistical Models	566
114	11.3	The Method of Least Squares	569
115	11.4	Properties of the Least-Squares Estimators: Simple Linear Regression	577
116	11.5	Inferences Concerning the Parameters βi	584
117	11.6	Inferences Concerning Linear Functions of the Model Parameters: Simple Linear Regression	589
118	11.7	Predicting a Particular Value of Y by Using Simple Linear Regression	593
119	11.8	Correlation	598
120	11.9	Some Practical Examples	604
121	11.10	Fitting the Linear Model by Using Matrices	609
122	11.11	Linear Functions of the Model Parameters: Multiple Linear Regression	615
123	11.12	Inferences Concerning Linear Functions of the Model Parameters: Multiple Linear Regression	616
124	11.13	Predicting a Particular Value of Y by Using Multiple Regression	622
125	11.14	A Test for H0 : βg+1 = βg+2 = · · · = βk = 0	624
126	11.15	Summary and Concluding Remarks	633
127		Considerations in Designing Experiments
128	12.1	The Elements Affecting the Information in a Sample	640
129	12.2	Designing Experiments to Increase Accuracy	641
130	12.3	The Matched-Pairs Experiment	644
131	12.4	Some Elementary Experimental Designs	651
132	12.5	Summary	657
133		The Analysis of Variance
134	13.1	Introduction	661
135	13.2	The Analysis of Variance Procedure	662
136	13.3	Comparison of More Than Two Means: Analysis of Variance for a One-Way Layout	667
137	13.4	An Analysis of Variance Table for a One-Way Layout	671
138	13.5	A Statistical Model for the One-Way Layout	677
139	13.6	Proof of Additivity of the Sums of Squares and E(MST) for a One-Way Layout (Optional)	679
140	13.7	Estimation in the One-Way Layout	681
141	13.8	A Statistical Model for the Randomized Block Design	686
142	13.9	The Analysis of Variance for a Randomized Block Design	688
143	13.10	Estimation in the Randomized Block Design	695
144	13.11	Selecting the Sample Size	696
145	13.12	Simultaneous Confidence Intervals for More Than One Parameter	698
146	13.13	Analysis of Variance Using Linear Models	701
147	13.14	Summary	705
148		Analysis of Categorical Data
149	14.1	A Description of the Experiment	713
150	14.2	The Chi-Square Test	714
151	14.3	A Test of a Hypothesis Concerning Specified Cell Probabilities: A Goodness-of-Fit Test	716
152	14.4	Contingency Tables	721
153	14.5	r × c Tables with Fixed Row or Column Totals	729
154	14.6	Other Applications	734
155	14.7	Summary and Concluding Remarks	736
156		Nonparametric Statistics
157	15.1	Introduction	741
158	15.2	A General Two-Sample Shift Model	742
159	15.3	The Sign Test for a Matched-Pairs Experiment	744
160	15.4	The Wilcoxon Signed-Rank Test for a Matched-Pairs Experiment	750
161	15.5	Using Ranks for Comparing Two Population Distributions: Independent Random Samples	755
162	15.6	The Mann–Whitney U Test: Independent Random Samples	758
163	15.7	The Kruskal–Wallis Test for the One-Way Layout	765
164	15.8	The Friedman Test for Randomized Block Designs	771
165	15.9	The Runs Test: A Test for Randomness	777
166	15.10	Rank Correlation Coefficient	783
167	15.11	Some General Comments on Nonparametric Statistical Tests	789
168		Introduction to Bayesian Methods for Inference
169	16.1	Introduction	796
170	16.2	Bayesian Priors, Posteriors, and Estimators	797
171	16.3	Bayesian Credible Intervals	808
172	16.4	Bayesian Tests of Hypotheses	813
173	16.5	Summary and Additional Comments	816
174	Appendix 1.	Matrices and Other Useful Mathematical Results	821
175	A1.1	Matrices and Matrix Algebra	821
176	A1.2	Addition of Matrices	822
177	A1.3	Multiplication of a Matrix by a Real Number	823
178	A1.4	Matrix Multiplication	823
179	A1.5	Identity Elements	825
180	A1.6	The Inverse of a Matrix	827
181	A1.7	The Transpose of a Matrix	828
182	A1.8	A Matrix Expression for a System of Simultaneous Linear Equations	828
183	A1.9	Inverting a Matrix	830
184	A1.10	Solving a System of Simultaneous Linear Equations	834
185	A1.11	Other Useful Mathematical Results	835
186	Appendix 2.	Common Probability Distributions, Means, Variances, and Moment-Generating Functions	837
187	Appendix 2.Table 1	Discrete Distributions	837
188	Appendix 2.Table 2	Continuous Distributions	838
189	Appendix 3.	Tables	839
190	Appendix 3.Table 1	Binomial Probabilities	839
191	Appendix 3.Table 2	Table of e−x	842
192	Appendix 3.Table 3	Poisson Probabilities	843
193	Appendix 3.Table 4	Normal Curve Areas	848
194	Appendix 3.Table 5	Percentage Points of the t Distributions	849
195	Appendix 3.Table 6	Percentage Points of the χ 2 Distributions	850
196	Appendix 3.Table 7	Percentage Points of the F Distributions	852
197	Appendix 3.Table 8	Distribution Function of U	862
198	Appendix 3.Table 9	Critical Values of T in the Wilcoxon Matched-Pairs, Signed-Ranks Test; n = 5(1)50	868
199	Appendix 3.Table 10	Distribution of the Total Number of Runs R in Samples of Size (n 1 , n 2 ); P(R ≤ a)	870
200	Appendix 3.Table 11	Critical Values of Spearman’s Rank Correlation Coefficient	872
201	Appendix 3.Table 12	Random Numbers	873
202		Index	896