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

According to the catalogue:
Sampling theory. Point estimation. Limiting distributions. Testing Hypotheses. Confidence intervals. Large sample methods.

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

Office Hours

Semester: Fall, 2023
Personnel Day of the Week Hour Room Comment
Marek Rychlik Wednesday 11:00am-12:00pm Math 466 Zoom Link Regular office hours (Zoom)
Marek Rychlik Wednesday 5:00pm-6:00pm Upper Division Tutoring via Teams (Zoom) Upper Division Tutoring
Marek Rychlik Friday 3:00pm-4:00pm Math 466 Zoom Link Regular office hours (Zoom)

Office hours by appointment are welcome. Please contact me by e-mail first, so that I can activate a Zoom link for the meeting.

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:

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:

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:

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):

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:

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
A90%+
B80-90%
C70-80%
D60-70%
E0-60%

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