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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

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
Instructor Homepage/Web Server
Course Homepage
Course Homepage (Mirror)

Office Hours

Semester: Spring, 2024
Personnel Day of the Week Hour Room Comment
Marek Rychlik Monday 4:00pm-5:00pm Upper Division Tutoring via Teams (Zoom) Upper Division Tutoring
Marek Rychlik Friday 3:00pm-4:00pm Math 466 Zoom Link Regular office hours (Zoom, Math 466)
Marek Rychlik Friday 4:00pm-5:00pm Math 589 Zoom Link Regular office hours (Zoom, Math 589)

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 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 for the latest guidance.

UA policies

The UA's policy concerning Class Attendance, Participation, and Administrative Drops is available at: The UA policy regarding absences for any sincerely held religious belief, observance or practice will be accommodated where reasonable, . Absences pre-approved by the UA Dean of Students (or Dean Designee) will be honored. See:

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:

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: 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:

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

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

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):
Also watch the video available at

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: .

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 .

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: .

UA Nondiscrimination and Anti-harassment Policy

The University is committed to creating and maintaining an environment free of discrimination, . 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: Student Assistance and Advocacy information is available at:

Confidentiality of Student Records .

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)


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
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
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
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