BENEDICTINE UNIVERSITY



BENEDICTINE UNIVERSITY

Course Outline MGT 251 STATISTICS II Spring, 2015

Text: Modern Business Statistics with Microsoft Office Excel, 5th edition, Anderson, Sweeney & Williams, South-Western/Cengage, 2015.

ISBN: 978-1-285-43330-1 (hard cover)

Other: Aplia interactive learning/assignment system.

TI-83 or TI-84 calculator.

Course Prerequisites: MATH 105 (Finite Math I) or MATH 110 (College Algebra)

Instructor: Jeffrey M. Madura, 160 Scholl Hall Office Hours: Announced in class or see web page at

B.A., University of Notre Dame ben.edu/faculty/jmadura/home.htm

M.B.A., Northwestern University e-mail: jmadura@ben.edu

C.P.A., State of Illinois

Course Description: This is a course in introductory statistics. The orientation is toward applications and problem-solving, not mathematical theory. The instructor intends that students gain an appreciation for the usefulness of statistical methods in analyzing data commonly encountered in business and the social and natural sciences. The course is a framework within which students may learn the subject matter. This framework consists of a program of study, opportunity for questions/discussion, explanation, and evaluation (quizzes). The major topics are:

Inferences About Population Variances

Tests of Goodness of Fit and Independence

Experimental Design and Analysis of Variance

Simple Linear Regression

Multiple Regression

Nonparametric Methods

The course addresses the following College of Business Program Objectives:

Students in this program will receive a thorough grounding in Mathematics and Statistics.

Your student evaluation of this course will be completed online using the IDEA system. This course emphasizes the following IDEA objectives:

Learning fundamental principles, generalizations, or theories.

Learning to apply course material to improve thinking, problem-solving, and decision-making.

Developing specific skills, competencies and points of view needed by professionals in the fields most closely

related to this course.

Quizzes and Grades: The course is divided into five three-week parts, with a quiz at the end

of each part. Dates are subject to change.

Quiz 1 Feb. 5

Quiz 2 Feb. 26 Quizzes will constitute 2/3 of your grade.

Quiz 3 Mar. 26 The other 1/3 will be your score on assignments,

Quiz 4 Apr. 16 Class participation may also be a factor.

Quiz 5 Finals Week

Grade requirements: A--90%, B--80%, C--60%, D--50%.

There may also be other assignments requiring analysis of data using Excel, and there may be a term project, with weight equal to one quiz.

There may also be other assignments requiring the use of Excel. It is the responsibility of any student who is unsure of the grading scale, course requirements, or anything else in this course outline to ask the instructor for clarification.

Homework Assignments: There will be about 10 Aplia homework assignments. Due dates are listed in the Aplia system.

The assignments will constitute 1/3 of the course grade. To accommodate the occasional instance when you cannot meet an Aplia deadline, the lowest assignment will be dropped. Assignments will be handled by Aplia. You will be required to access the Aplia website, which means you need to register for an account at:

Please register within 24 hours of the first class meeting.

Please note: The computer is absolutely unforgiving about accepting late assignments. Time is kept at Aplia, and not by the computer you are working on. You may appeal grading decisions made by the computer, if you can demonstrate that an error has been made.

Faculty have observed that the worst thing students do, in any course, is not think about course material every day. They sometimes let weeks go by and then try to learn all the material in one or two days. This usually does not work. The weekly assignments will require keeping up-to-date.

Calculators: Calculators will be required for the computational portion of each quiz. Bring your calculator to every class and verify each computation performed. The TI-83 is the standard for this course.

Recommended Exercises: Students should work as many as possible of the even-numbered exercises in the text. Proficiency gained from practice on these will help when similar problems appear on quizzes. Answers to even-numbered exercises are at the back of the text.

Attendance: Attendance will be taken occasionally and randomly. Frequent absences will be noticed, and they will have an adverse impact on quiz performance and your final grade. Two or more absences on days when quizzes are handed back will lower your grade by one letter grade.

Missed Quizzes: Make-up quizzes will be given only if a quiz was missed for a good and documented reason. If a

make-up is given. The quiz score will be reduced 20% in an effort to maintain some degree of fairness

to those who took the quiz at the proper time.

Use of Class Time: Come to class prepared to discuss the material assigned, and to contribute to the solution of the assigned problems.

Special Needs: If you have a documented learning, psychological, or physical disability, you may be eligible for reasonable academic accommodations or services.  To request accommodations or services, please contact Tina in the Student Success Center, 012 Krasa Student Center, 630-829-6512.  All students are expected to fulfill essential course requirements.  The University will not waive any essential skill or requirement of a course or degree program.

Academic Honesty Policy: The search for truth and the dissemination of knowledge are the central mission of a university. Benedictine University pursues these missions in an environment guided by our Roman Catholic tradition and our Benedictine heritage. Integrity and honesty are therefore expected of all members of the community, including students, faculty members, administration, and staff. Actions such as cheating, plagiarism, collusion, fabrication, forgery, falsification, destructions, multiple submission, solicitation, and misrepresentation, are violations of these expectations and constitute unacceptable behavior in the University community. The penalties for such actions can range from a private verbal warning, all the way to expulsion from the University. The University’s Academic Honesty Policy is available at .

In this course, academic honesty is expected of all class participants, myself included. If your name is on the work submitted, it is expected that you alone did the work. For example, in terms of quizzes, this means that copying from another paper, unauthorized collaboration of any sort, or the use of “cribs” of any kind is a breach of academic honesty. The penalties for a breach of academic honesty in this course are (1) a zero for the assignment or quiz for the first offense, and (2) an “F” for the course for a subsequent offense by the same person(s).

Electronic Devices Policy: One aspect of being a member of a community of scholars is to show respect for others by the way you behave. Do your part to create or maintain an environment that is conducive to learning. Turn off your cell phone or set it to mute/silence before you enter class. If you use your cell phone or any other electronic device in any manner during a quiz, you will receive a zero for that test or quiz. Using the TI-83/84 calculator is permitted.

Feel free to see me if there is anything else of concern to you. Your comments about this course or any course are always welcome and appreciated. The student is responsible for the information in the syllabus and should ask for clarification for anything in the syllabus about which they are unsure.

COURSE PHILOSOPHY -- STATISTICS

In an article in the Chronicle of Higher Education, Sharon Rubin, assistant dean at the University of Maryland, states that all course syllabi, in addition to providing the basic information on texts, topics, schedule, etc., should answer certain questions. The instructor of this course would like to share these questions with you, and provide some answers.

You are what you know. You are what you can do.

"What value can you add to our organization?"

1. WHY SHOULD A STUDENT WANT TO TAKE THIS COURSE?

As a decision-maker, you must learn how to analyze and interpret quantitative information. Such skills will improve your ability to adopt the questioning attitude and independence of thought that are essential to leadership and success in any field. You may also have the opportunity to introduce statistical data analyses in areas where they are not currently in use, thus improving the quality of your organization's decisions.

2. WHAT IS THE RELEVANCE OF THIS COURSE TO THE DISCIPLINE?

Statistics courses are part of the curriculum in many of BU's programs. But since this course is part of a program leading to a degree in business, let us interpret the word "discipline" in this question to mean "management." This can refer to marketing management, financial management, human resource management, etc., even the management of your personal affairs. To MANAGE something requires the ability to exert some CONTROL over it, and the ability to exert control requires identification of DEPENDENCIES. In order to manage sales performance, for example, you must find things upon which sales depends (e.g. advertising budget; product price; number, training, and compensation of salespersons; interest rates; and competitive factors), and learn something about the nature of the dependencies. Statistics is the major tool for identifying dependencies.

Another example of the importance of identifying dependencies: a new disease appears. Researchers immediately try to find things that enhance the occurrence rate or the severity of the illness (positive dependencies), and things that reduce them (negative dependencies). Only after such things are found can there be any hope of controlling the disease. Again, statistical analysis plays a major role.

Or, the objective may simply be to know more about how the world works. So-called "pure research" has no immediate application, but seeks to find relationships among things, thereby securing knowledge that may become useful in the future.

CAREFUL STATISTICAL ANALYSIS OF DATA OFTEN RESULTS IN THE IDENTIFICATION OF DEPENDENCIES, and this is the reason why statistics is an important tool in virtually all disciplines.

3. HOW DOES THIS COURSE FIT INTO THE "GENERAL EDUCATION" PROGRAM?

Statistics is a major way in which human beings learn about the world, and how to control it. To be familiar with a tool as fundamental and important as this is a responsibility of every educated person.

Statistics can be viewed as applied quantitative logic, usually seeking to make inferences about unknown parameters on the basis of observations and measurements of samples drawn from a target population.

The study of statistics can promote clear and careful thinking, enhance problem-solving skills, and strengthen one's ability to avoid premature conclusions. These are traits of the educated person, and are the mental qualities essential for "knowledge workers" in modern society.

4. WHAT ARE THE OBJECTIVES OF THE COURSE?

The most important objective is the development of your ability to learn this kind of material on your own, and to continue learning more about the subject after the course is over. Continuous and independent learning is an important activity of every successful person. In connection with the objective of independent learning, the instructor will expect students to study and learn certain topics in the course without formal discussion of them in class. Questions on these topics, of course, are always welcomed and encouraged.

With respect to specific objectives, they are: that students learn the terminology, theory, principles, and computational procedures related to basic descriptive and inferential statistics; and the careful cultivation of the logical processes involved in statistical inference. This will enable students to understand statistics and communicate statistical ideas using generally-accepted terminology.

Another important objective is that students become aware of the limitations of various statistical procedures. This is particularly important since most students in this course will be consumers rather than providers of statistical information and conclusions. Estimates and forecasts, for example, are generally regarded with too much faith, and relied upon to a degree not warranted in light of their inherent limitations.

5. WHAT MUST STUDENTS DO TO SUCCEED IN THIS COURSE?

Your activities in this course should include: reading and studying the relevant sections of the text; attending class and taking notes; rewriting, reviewing, and studying your notes; working the recommended exercises in the text; practicing and experimenting with various spreadsheet files supplied by the instructor; asking and answering questions in class; spending time just thinking about the procedures and their underlying logic; forming a study group with other students to review notes on terminology and concepts, and to practice problem-solving skills; and taking the quizzes.

These activities should help you to further develop your abilities to read, listen, record, and organize important information; and to communicate, analyze, compute, and learn independently the subject matter of statistics.

In order to do well, students must recognize a basic difference between courses like statistics and courses like history, philosophy, management or organizational strategy. In the latter type, the emphasis is often on general ideas in broad contexts, with grades based on essay exams and term papers in which students have considerable latitude to choose what they are going to discuss. The cogent expression and defense of well-reasoned opinion are highly valued. Students with good verbal, logical and writing skills often excel in this type of course. Statistics, on the other hand, is a skills course, requiring precise knowledge of concepts, terminology, and computational procedures. Verbal skills are still important, but now quantitative logic and computational competence are also critical. Grades are based on knowledge of terminology and concepts, and even more on the ability to get the right answers to problems.

Regarding study strategy, it is extremely important for most students to read about statistics, to think about statistics and to do a few problems every day. The most common error is to neglect the material until shortly before a quiz. But for most students, many of the concepts in statistics are new and strange, and there will be many places where they are stopped cold: "What?" "I just don't get this!" Then there is no time left to cultivate the understanding of new concepts and to refine the computational procedures. Anyone can learn statistics, but most cannot do it overnight.

As with most courses, this course is organized with the most fundamental material coming first. In learning a new language, or how to play a musical instrument, or any new set of skills, mastery of the basics is essential to success later on. The subject matter of statistics is not like history, where, if you did not study 14th century France, it probably did not affect your learning about 17th century England. In statistics, failure to obtain a good understanding of earlier material will have a serious adverse effect on your ability to make sense out of what comes later. It is therefore essential to build a solid foundation of fundamental knowledge early in the course in order to support the more elaborate logical and computational structures involved later.

6. WHAT ARE THE PREREQUISITES FOR THE COURSE?

The primary prerequisite is a logical mind. This course is computational, but it is not a "math" course. Mathematical theorems are not derived or proven; the need to solve equations is very rare. The emphasis is on concrete applications rather than abstract theory. Some students with good math backgrounds have done poorly, while others with little or no math experience have done very well.

The best MBA stats student I ever had was a philosophy major who did not have single math course at the college level. When asked about this, the he replied: "My philosophy major gave me excellent training in logic, and that's really what this course requires."

7. OF WHAT IMPORTANCE IS CLASS PARTICIPATION?

In this course, class participation means frequently asking relevant questions and supplying answers (right or wrong) to the instructor's and colleagues' questions as problems and examples are worked out and discussed. These behaviors are evidence of active involvement with the material and will result in better learning and an automatic positive effect on your grade. In grade border-line cases, a history of active participation will enable the instructor to award the higher grade to the deserving student.

8. WILL STUDENTS BE GIVEN ALTERNATIVE WAYS TO ACHIEVE SUCCESS, BASED ON DIFFERENT LEARNING STYLES?

Different learning styles do exist. Some prefer a deductive method (deriving specific knowledge from general principles), while others tend to prefer an inductive method (deriving the generalities from examples). The inductive learners may need to work a number of problems before seeing the patterns that are present. The deductive learners may never need to work a problem--they will know instinctively what to do. Some will not like the book, and will learn primarily from the class presentations and discussions, while others will learn mostly from the book and will find class time to be of lesser importance.

But the intended outcomes are the same for all--those in number 4 above.

9. WHAT IS THE PURPOSE OF THE ASSIGNMENTS?

Problems from the text may be suggested, for the purpose of providing practice in analyzing what must be done, and in performing the required computations. Even though computer software is available to perform calculations, students can gain insight into the logical structure of a sequence of computational steps if they go through them several times by hand (i.e. using simple calculators).

Computer assignments using instructor-supplied spreadsheet files will require students to become more familiar with spreadsheet software that they probably are or will be using in connection with their work. More importantly, the spreadsheets allow students to experiment with data in order to investigate the quantitative relationships involved. Such experimentation would be too tedious and time-consuming for manual or even calculator computation.

10. WHAT WILL THE TESTS TEST? -- MEMORY? UNDERSTANDING? ABILITY TO SYNTHESIZE? TO PRESENT EVIDENCE LOGICALLY? TO APPLY KNOWLEDGE IN A NEW CONTEXT?

The tests will test your ability to recognize and use statistical terminology correctly, and they will test your understanding of the logic and principles underlying various statistical procedures. In addition, you will have to demonstrate your ability to solve problems similar to those discussed in class, sometimes using computer spreadsheet files.

There is a place for memorization in learning. It is not a substitute for comprehension, but it is better than getting something wrong on a quiz that you were expected to know. As with prayers among small children, memorization is often a first step, eventually followed by understanding. But if the memorization (of terminology, for example) is not done, it is less likely that the comprehension will ever occur.

11. WHY HAS THIS PARTICULAR TEXT BEEN CHOSEN?

Our text is one of the most widely adopted introductory statistics books. It has gone through several editions, and its popularity remains high. It is relatively easy to read, and its exercise material is excellent.

12. WHAT IS THE RELATIONSHIP BETWEEN KNOWLEDGE LEVEL AND GRADES?

Consider this hypothetical but realistic situation.

Percentage Grade

Knowledge Course A Course B

100% 100% 100%

90% 90% 81%

80% 80% 64%

70% 70% 49%

60% 60% 36%

50% 50% 25%

40% 40% 16%

30% 30% 9%

20% 20% 4%

10% 10% 1%

Course A might be like philosophy, history, or management, where the grade is more-or-less proportional to knowledge level. Course B might be like statistics or other skills courses, where small deficiencies in knowledge can have disastrous effects on results. Overstudying is the best strategy for coping with this, with the dual payoffs of higher grades and, more importantly, greater knowledge.

| |QUIZ |HW | | | | | | | | | | | | | |0.667 |0.333 | |A |B |C |D |F | | | | | | | |HW | | | | | | | | | | | | | |  |10 |20 |30 |40 |50 |60 |70 |80 |90 |95 |100 |  | |QUIZ |100 | 70.0 | 73.4 | 76.7 | 80.0 | 83.4 | 86.7 | 90.0 | 93.3 | 96.7 | 98.3 | 100 |100 | | |95 | 66.7 | 70.0 | 73.4 | 76.7 | 80.0 | 83.3 | 86.7 | 90.0 | 93.3 | 95.0 | 96.7 |95 | | |90 | 63.4 | 66.7 | 70.0 | 73.4 | 76.7 | 80.0 | 83.3 | 86.7 | 90.0 | 91.7 | 93.3 |90 | | |85 | 60.0 | 63.4 | 66.7 | 70.0 | 73.3 | 76.7 | 80.0 | 83.3 | 86.7 | 88.3 | 90.0 |85 | | |80 | 56.7 | 60.0 | 63.4 | 66.7 | 70.0 | 73.3 | 76.7 | 80.0 | 83.3 | 85.0 | 86.7 |80 | | |75 | 53.4 | 56.7 | 60.0 | 63.3 | 66.7 | 70.0 | 73.3 | 76.7 | 80.0 | 81.7 | 83.3 |75 | | |70 | 50.0 | 53.4 | 56.7 | 60.0 | 63.3 | 66.7 | 70.0 | 73.3 | 76.7 | 78.3 | 80.0 |70 | | |65 | 46.7 | 50.0 | 53.3 | 56.7 | 60.0 | 63.3 | 66.7 | 70.0 | 73.3 | 75.0 | 76.7 |65 | | |60 | 43.4 | 46.7 | 50.0 | 53.3 | 56.7 | 60.0 | 63.3 | 66.7 | 70.0 | 71.7 | 73.3 |60 | | |55 | 40.0 | 43.3 | 46.7 | 50.0 | 53.3 | 56.7 | 60.0 | 63.3 | 66.7 | 68.3 | 70.0 |55 | | |50 | 36.7 | 40.0 | 43.3 | 46.7 | 50.0 | 53.3 | 56.7 | 60.0 | 63.3 | 65.0 | 66.7 |50 | | |45 | 33.3 | 36.7 | 40.0 | 43.3 | 46.7 | 50.0 | 53.3 | 56.7 | 60.0 | 61.7 | 63.3 |45 | | |40 | 30.0 | 33.3 | 36.7 | 40.0 | 43.3 | 46.7 | 50.0 | 53.3 | 56.7 | 58.3 | 60.0 |40 | | |30 | 23.3 | 26.7 | 30.0 | 33.3 | 36.7 | 40.0 | 43.3 | 46.7 | 50.0 | 51.6 | 53.3 |30 | | |20 | 16.7 | 20.0 | 23.3 | 26.7 | 30.0 | 33.3 | 36.7 | 40.0 | 43.3 | 45.0 | 46.6 |20 | |

I can use Excel to

perform basic computations

prepare tables

create charts and graphs

conduct common statistical procedures

create dashboards

I can use Word to

create various kinds of documents

I can

compute

means

medians

variances

standard deviations

confidence intervals for means and proportions

use the

binomial distribution to answer probability questions

normal distribution to answer probability questions

chi-square distribution to answer probability questions

F distribution to answer probability questions

conduct

hypothesis tests on

the means of one group or two

the proportions of one group or two

hypothetical vs. observed distributions

variances of one group or two

group means using ANOVA

regression analysis to examine correlation and make forecasts

I can

perform financial analysis

compute the NPV of various investment opportunities

decide between using debt or equity to raise new funds

determine the optimum mix of debt and equity financing

compute cost-of-capital

decide whether to make or buy components for our products

determine how much direct labor, direct materials, and overhead is going into our products

create cash budgets

conduct cost-volume-profit analyses

prepare a master budget

prepare prformance reports using standard costs and variances

employ the scientific method to study problems that may come up

PART TWO -- Essentials--Analysis of Enumerative Data

Enumerate: to count, usually after classification has been performed

Enumerative data: data obtained by classifying and counting occurrences

Multinomial experiment--like the binomial experiment, except each trial has more than two outcomes

n identical trials; k possible outcomes on each trial

Independence--the outcome of one trial does not affect the outcome of any other trial

Constant probabilities for each outcome from trial to trial

p1, p2, p3, . . ., pk are the probabilities of the various outcomes

Cell counts (number of times each outcome occurs) are the variables to be

analyzed

Chi-square (χ2) distribution: continuous, positively skewed

One-dimensional chi-square test--“goodness of fit” tests

Ho: that a population conforms to some expected distribution.

A cell consists of an expectation (E) and an observation (O).

Expected values (E) are derived from Ho.

The number of cells is denoted by k.

Calculated chi-square (test statistic, χ2c) for a cell is the squared deviation

(E-O)2 divided by E. The χ2c is the total of all the cells.

Degrees of freedom (df): the number of cells minus one (k-1)

(d.f. = k – 3 when the normal distribution is used.)

(d.f. = k – 2 when the Poisson distribution is used.)

Ho is rejected if χ2c ( χ2t , also if p ≤ α.

If Ho is rejected, additional information should be reported as to the nature of the deviation from the expected distribution.

Often used to test for normal distributions.

For the sample size to be sufficient, the expected number (e) in each cell

should equal or exceed 5.

Two-dimensional chi-square test

H0: in the population the row variable and column variable are independent.

Ha: in the population the row variable and column variable are dependent.

Contingency (dependency) table contains a matrix of cells

A cell consists of an expectation (E) and an observation (O).

Expected values (E) are derived from H0 using the multiplication rule for

intersections of independent events: P(A ∩ B) = P(A) * P(B).

Calculated chi-square for a cell is (E-O)2 / E (same as above).

Ho is rejected if χ2c ( χ2t , also if p ≤ α.

Degrees of freedom: number of rows minus one, times number of columns

minus one; (r-1)(c-1) where r and c are the numbers of rows and columns

If H0 is rejected, additional information should be reported as to the nature of the dependencies.

For the sample size to be sufficient, the expected number (e) in each cell

should equal or exceed 5.

Terminology--explain each of the following:

enumerative data, multinomial experiment, binomial experiment, identical trials, independence, one-dimensional or one-way chi-square test, “goodness-of-fit” test, two-dimensional or two-way chi-square test, dependency, contingency table, multiplication rule for intersections of independent events.

Skills and Procedures

• given appropriate data, conduct a one-way chi-square test and interpret the results

• given appropriate data, conduct a two-way chi-square test and interpret the results

Concepts

• describe what is meant by “goodness-of-fit”

• explain how expected values are determined in a one-way chi-square test

• explain how the concept of “deviation” applies in chi-square test computations

• explain how expected values are determined in a two-way chi-square test

• describe the application of the “multiplication rule for independent events” in two-way chi-square analysis

If the H0 is rejected:

One-Way: “The differences between the observations and the expectations are statistically significant at the ___α___ level. The population probably does not conform to the expected distribution.” (You should say more about the nature of the differences between the observations and the expectations.)

Two-Way: “There is statistically significant dependence between ______ and ______ at the ___α___ level.” (Give more information about the dependencies.)

If the H0 is not rejected:

One-Way: “The differences between the observations and the expectations are not statistically significant at the ___α___ level. The population could conform to the expected distribution.”

Two-Way: “The dependence between ______ and ______ is not statistically significant at the ___α___ level.”

PART THREE -- Essentials--Analysis of Variance (ANOVA)

Purpose: To test for differences between/among two or more population means.

H0: μ1 = μ2 = μ3 . . .; Population means are all equal.

Ha: not μ1 = μ2 = μ3 . . .; Population means are not all equal;

Note that Ha is not "all the population means are different."

Rejection of Ho means that there is a statistically significant difference between at

least two of the sample means.

Interval estimation of population means and differences between population means is

also possible.

Sums of squared deviations

TSS--total sum of squared deviations

SST--sum of squared deviations for treatments (between-group variation)

SSE--sum of squared deviations for error (within-group variation)

TSS = SST + SSE

Means of squared deviations--recall that a variance is a mean of squared deviations.

MST--mean of squared deviations for treatments (between-group variance)

MSE--mean of squared deviations for error (within-group variance)

Signal-to-noise analogy

Signal: between-group variance, MST

Noise: within-group variance, MSE

The more false Ho is (the larger the differences between/among population means),

the larger MST will be relative to MSE.

ANOVA table--standardized way of presenting computations and results

Calculated F ( test statistic, Fc ) is MST / MSE

Total degrees of freedom: the number of observations minus one

Degrees of freedom for treatments: number of treatments minus one

Degrees of freedom for error: the number of observations minus the number

of treatments

When there are only two groups and a t-test could be used, the Fc will be equal to the

square of the tc.

Reject Ho if Fc ( Ft and if p ( α.

Four assumptions (same as t-tests of chapter 9)

Samples

Random

Independent

Populations

Normally distributed

Equal variances

Moderate departures from the assumptions will not seriously affect validity (robust)

One-way ANOVA--completely randomized design

Two-way ANOVA--randomized block design

TSS = SST + SSB + SSE (B = "blocks")

Two calculated F's: treatments FT = MST / MSE and blocks FB = MSB / MSE

Total degrees of freedom: the number of observations minus one

Degrees of freedom for treatments: the number of treatments minus one

Degrees of freedom for blocks: the number of blocks minus one

Degrees of freedom for error: the number of observations minus the number of

treatments, minus the number of blocks, plus one

Estimation in One-Way ANOVA

tt in the following equations is based on the number of degrees of freedom for error.

[pic]

1

Single population mean

where

[pic]

2

Difference between two population means:

[pic]

3

where

[pic]

4

Estimation in two-way ANOVA (randomized block design)

Two-way ANOVA estimation -- valid only for differences between population means.

Confidence intervals cannot be obtained for individual treatment means.

tt in the following equations is based on the number of degrees of freedom for error,

Difference between two population means:

[pic]

5

where

[pic]

6

Three-way analysis of variance

"Latin square" design

Terminology--explain each of the following:

TSS--total sum of squared deviations, SST--sum of squared deviations for treatments (between-group variation), SSE--sum of squared deviations for error (within-group variation), variance, MST--mean of squared deviations for treatments (between-group variance), MSE--mean of squared deviations for error (within-group variance), signal-to-noise ratio, ANOVA table, calculated F (MST / MSE), degrees of freedom (treatments, blocks, error), four assumptions (same as t-tests of chapter 9), robust test--moderate departures from the assumptions will not seriously affect validity, completely randomized design, randomized block design, "Latin square" design

Skills and Procedures

• given appropriate data, conduct a one-way ANOVA and interpret the results; include all possible 95% confidence intervals

• given appropriate data, conduct a two-way ANOVA and interpret the results; include all possible 95% confidence intervals

Concepts

• explain why, when ANOVA deals with tests on means, it is called “analysis of variance”

• explain the “signal-to-noise ratio” concept in the context of ANOVA

• describe the shortcoming that ANOVA shares with small-sample t-tests

• show where the variances are found in the ANOVA table

If the H0 is rejected:

“The difference between at least two of the sample means of the __________ is statistically significant at the α level. The population means are probably not all equal.”

If the H0 is not rejected:

“The differences among the sample means of the __________ are not statistically significant at the α level. All the population means could be equal.”

PART FOUR -- Essentials--Linear Regression and Correlation

Major purpose in business: forecasting

In order for forecasting to be possible, the future must, in some way, be like the past.

Forecasting methods seek to identify relationships from the past, and use them to

predict the future (assuming that the identified relationship will persist).

Finding relationships is a way of identifying dependencies.

Dependent variable--one to be predicted

Independent variable--one used to make the prediction

Types of regression

Based on the number of independent variables

Simple regression--one predictor or independent variable (x)

E.g. y = a + bx

Multiple regression--two or more predictor or independent variables (x1, x2, . . . ,xn)

E.g. y = a + bx1 +cx2 +dx3 +ex4

Based on the type of regression line

Linear: y = a + bx a = y-intercept; b = slope

or y = mx + b: b = y-intercept; m = slope

or y = β0 + β1 x: β0 = y-intercept; β1 = slope

Slope is the coefficient (multiplier) of x, no matter what symbol is used or where

it appears in the equation.

Slope is the change in y for a one-unit change in x.

Usually regarded as the single most important result in regression, because it

describes the nature of the relationship between y and x.

In multiple regression, each independent variable has its own slope and its own measure of correlation.

Intercept is the other value, also known as the "constant".

Intercept is the value of y when x = 0.

Non-linear (curved): exponential e.g. y = abx or y = 35(1.06)x

logarithmic e.g. y = a log x or 3.2 log x

power e.g. y = axb or 60(x)5

trigonometric e.g. y = a sin x or 3.7 sin x

etc.

Over a restricted range (relevant range) a curve can be approximated with a straight line

Based on the nature of the suspected relationship between y and x

Causal regression: x may be an actual cause of y, or x may be related to something

else that is a cause of y

Time series regression--popular in business and economics

Time is the independent (x) variable, used to substitute for the actual causes of y.

In time series, it is often better to use less historical data rather than more.

The future is likely to be more like the recent past than the more distant past.

With less data x is closer to x-bar (see below).

Correlation--the degree of "relatedness" between dependent and independent variables

Types of correlation

positive: dependent variable increases as the independent variable increases

negative: dependent variable decreases as the independent variable increases

none: no apparent relationship between dependent variable and independent variable

Measures of correlation

Coefficient of non-determination, k2--always positive--range, 0 to 1

If there is perfect correlation, k2 is equal to zero.

If there is no correlation, k2 is equal to one.

Coefficient of determination, r2, equal to 1 - k2--always positive--range, 0 to 1

If there is perfect correlation, r2 is equal to one.

If there is no correlation, r2 is equal to zero.

Correlation coefficient, r, the square root of r2--positive or negative, depending on

the type of correlation--range -1 to +1

Note: ρ (rho) and ρ2 are the population parameters corresponding to r and r2

Correlation and causation

The presence of correlation does not, in itself, prove that x causes y.

Three things necessary to prove causation

Statistically significant correlation between the effect, y, and the alleged cause, x.

Alleged cause, x, must be present before or at the same time as the effect, y.

Explanation must be found as to how x causes y.

Prediction errors--five standard errors (sampling standard deviations)

Standard error of the slope, σb

Measure of uncertainty regarding the slope of the regression line

Used to find confidence interval for the slope: β = b ± ttσb

Note: β is the population slope, estimated by b.

Standard error of the intercept, σa

Measure of uncertainty regarding the intercept of the regression line

Used to find confidence interval for the intercept: α = a ± ttσa

Note: α is the population intercept, estimated by a.

Standard error of estimate, σd and standard error of prediction, σpred

Measures of uncertainty regarding predictions

Used in finding confidence interval for predictions: y = y' ± ttσpred

Predictions have the least uncertainty when the value of x is near x-bar.

Standard error of the correlation coeffiecient, σr

Measure of uncertainty regarding the correlation coefficient

Types of variation in regression

Initial or original variation

Sum of the squared deviations between the data y-values and the mean of the

y-values -- Σ(y-ybar)2

Residual variation

Sum of the squared deviations between the data y-values and the predicted

y-values -- Σ(y-y')2

Removed or explained variation

Initial variation minus residual variation

k2 is the ratio of residual variation to original variation, Σ(y-y')2 / Σ(y-ybar)2.

r2 is the ratio of removed variation to original variation.

Hypothesis testing in regression

Ho: No correlation (relationship) between y and x.

ρ = 0 or ρ2 = 0 or β = 0

Ha: Correlation between y and x (two-sided)

Positive correlation between y and x (one-sided)

Negative correlation between y and x (one-sided)

Reject Ho if tc ( tt (when n is small) or if zc ( zt (when n is large).

When n is small, df = (n-2)

Reject Ho if p ( α (hypothesis-test α, not intercept α)

If Ho is not rejected, there is no statistically significant correlation between x and y.

The regression equation should not be used--just use y-bar to predict y, or don't

make a prediction at all.

Exponential regression (not in the textbook)

Linear vs. exponential growth

Simple interest--example of linear growth

Interest is paid only on the initial deposit

E.g. $1,000 deposited today at 5% is worth $1,000 + $50(x) after x years.

$1,000 is the intercept (value of y today, when x = 0).

$50 is the slope (change in y each year (5% of $1,000)).

The slope, $50, is constant.

Compound interest--example of exponential growth

Interest paid not only on the initial deposit, but also on previously-earned interest.

E.g. $1,000 deposited today at 5% is worth $1,000 (1.05)x after x years

$1,000 is the intercept (value of y today, when x = 0)

1.05 is the growth factor (b), which is equal to 1 + the growth rate (r)

b = 1+ r and r = b - 1

In the above example r = 0.05 (5%) and b = 1.05

The slope is not constant, but increases as x increases.

Exponential equation: y'exp = a (b)x

a = y-intercept; b = compound growth factor

Growth rate r = b - 1, and compound growth factor b = 1+ r

"b" values compared:

Linear: y = a + b(x)

b < 0 negative correlation

b = 0 no correlation (y = intercept a, regardless of value of x)

b > 0 positive correlation

Exponential: y = a (b)x

b < 1 negative correlation

b = 1 no correlation (y = intercept a, regardless of value of x)

b > 1 positive correlation

Exponential regression computations

Procedure is based on the fact that if y is an exponential function of x, then ln y

(or log y) is a linear function of x

That is, if y = a(b)x, then ln y = a' + b'(x) or log y = a'' + b''(x).

(The three "a" and "b" values in the above equations are different.)

Procedure

Transform the y-values into the lns (or logs) of the y-values.

Math review

The logarithm of a number is the power to which a base number must

be raised in order to give the original number

Natural logarithms use the number e (2.718281828...) as the base.

ln 25 is 3.218876 because e3.218876 is 25

ln 100 is 4.605170 because e4.605170 is 100

Common logarithms use the number 10 as the base

log 25 is 1.397940 because 101.397940 is 25

log 100 is 2 because 102 is 100

Perform linear regression analysis on the lns (or logs) of the y-values.

Result is a linear equation for predicting the ln (or log) of y

ln y' = a'+b'x or log y' = a''+b''x

Determine a and b values in y' = a(b)x

a is the inverse ln of a' (or the inverse log of a'')

b is the inverse ln of b' (or the inverse log of b'')

Inverse ln of z = ez (or Inverse log of z = 10z)

Confidence intervals in exponential forecasting

Intervals are first computed for ln (or log) of y', then are converted to LCL and UCL

values using inverse lns (or logs)

Two-point regression--linear and exponential--quick forecasts (see examples at end of outline)

Linear

Slope (b) is the difference between y-values divided by the difference between

x-values.

Let y-axis be located at the first x-value (let first x-value correspond to zero

on the x-axis).

Intercept (a) is then the first y-value.

Equation y' = a + bx can then be written and used to make forecasts

Exponential

Growth factor (b) is the ratio of the y-values raised to the 1/n power, where n is the difference between x-values.

Let y-axis be located at the first x-value (let first x-value correspond to zero

on the x-axis).

Intercept (a) is then the first y-value.

Equation y' = abx can then be written and used to make forecasts

Confidence intervals cannot be computed for two-point forecasts.

Multiple Regression

More than one independent variable

Linear form: y' = a + bx1 + cx2 + dx3 + . . . (a coefficient for each variable)

Partial correlation coefficients and partial coefficients of determination

r1, r2, r3, . . . and r12, r22, r32, . . .

Terminology--explain each of the following:

forecasting (basic concept), dependent variable, independent variable, simple regression, multiple regression, linear regression, intercept, slope, non-linear regression, exponential regression, causal regression, time-series regression, correlation, positive correlation, negative correlation, k2, coefficient of non-determination, r2, coefficient of determination, r, correlation coefficient, causation, standard error of the slope, standard error of the intercept, standard error of estimate, standard error of prediction, standard error of the correlation coefficient, initial or original variation, residual variation, removed or explained variation, null hypothesis in regression, alternate hypotheses in regression, simple interest, compound interest, compound growth factor, growth rate, transformation, logarithm, natural logarithm, common logarithm, inverse logarithm, two-point regression, multiple regression, partial correlation, cross-products, degrees of freedom, table-t, calculated-t, signal-to-noise ratio

Skills and Procedures

• perform linear regression using the TI-83 and the spreadsheet , including predictions, error factors, hypothesis tests, and evaluation of the degree of correlation

• perform exponential regression using the TI-83 and the spreadsheet , including predictions, error factors, hypothesis tests and evaluation of the degree of correlation

• interpret, in nonmathematical terms, the intercept and slope in linear regression

• interpret, in nonmathematical terms, the intercept and growth factor in exponential regression

• interpret the coefficients of nondetermination and determination in linear and exponential regression

Concepts

• describe “intercept” as nonmathematically as possible

• describe “slope” as nonmathematically as possible

• describe “compound growth factor” as nonmathematically as possible

• explain the difference between simple regression and multiple regression

• explain the significance of the “sum of the squared deviations between the data points and their mean”

• explain the significance of the “sum of the squared deviations between the data points and the regression line”

• describe the relationship between the “coefficient of nondetermination” and the two items immediately above

• describe the relationship between the “coefficient of nondetermination” and the “coefficient of determination”

• identify the difference between linear growth and exponential growth in terms of what is constant in each case

• explain why the demonstrated correlation between smoking and lung cancer does not prove that smoking causes lung cancer

• describe the relationship among the three types of variation: “original,” “residual,” and “explained” (or “removed”)

• explain the relationship between regression hypothesis-test results and the ability (advisability) to make predictions

• in exponential growth, describe the relationship between the compound growth factor and the growth rate

• describe how a regression line, straight or exponential, may be fitted between two data points

If the Ho is rejected:

“The correlation between _____ and _____ is statistically significant at the _α_ level.”

If the Ho is not rejected:

“The correlation between _____ and _____ is not statistically significant at the _α_ level.”

Two-point regression examples: A city’s population was 234,000 in 1995, and 683,000 in 2005.

What are the growth rates and forecasts for 2010?

Linear: The b-value is (683,000 - 234,000) / 10 = 44,900 people per year.

Equation is y’ = 234,000 + 44,900(x)

Forecast for 2010 is y’ = 234,000 + 44,900(15) = 907,500.

Exponential: The b-value is (683,000 / 234,000) ^ (1/10) = 1.113065 or 11.31% annual growth.

Equation is y’ = 234,000 * 1.113065 ^ x

Forecast for 2010 is y’ = 234,000 * 1.113065 ^ 15 = 1,166,872.

PART FIVE -- Essentials--Nonparametric Statistics

Parameter: population characteristic

Nonparametric test: does not require any particular population characteristics

Advantage: no required population characteristics

Disadvantage: not as powerful as parametric tests (t-test, ANOVA)

Power: ability of a test to detect when Ho is false, and give the correct

conclusion (rejection of Ho)

Power of any test can be increased by increasing the sample size.

If two different tests are applied to the same data, the more powerful test will

produce a lower p-value.

Sign Test--for differences between population means, paired-difference design

Based on the binomial distribution

Ho: the population means are equal.

Ha: the population means are not equal (2-sided). 1-sided tests are also possible.

Wilcoxon Signed-Rank Test--test for differences between population means,

paired-difference design

Ho: the population means are equal.

Ha: the population means are not equal (2-sided).

1-sided tests are also possible.

Procedure--see “notes”

Mann-Whitney "U" Test--test for differences between population means, unpaired

design

Ho: the population means are equal.

Ha: the population means are not equal (2-sided).

1-sided tests are also possible.

Procedure--see ”notes”

Runs Test (not in textbook)--test for independence in a series of binomial events

Procedure--see ”notes”

Terminology--explain each of the following:

Sign Test--application, nonparametric, parametric, sign test, binomial distribution, paired-difference design, power, ranking, tied observations, tied rankings, Wilcoxon Signed-Rank Test--application, Mann-Whitney “U” Test--application, Runs Test--application, run, positive dependence, negative dependence, independence.

Skills and Procedures

• given appropriate data, conduct a Sign Test and interpret the results

• given appropriate data, conduct a Wilcoxon Signed-Rank Test and interpret the results

• given appropriate data, conduct a Mann-Whitney “U” Test and interpret the results

• given appropriate data, conduct a Runs Test and interpret the results

Concepts

• describe the advantage of nonparametric tests

• describe the disadvantage of nonparametric tests

• explain how the disadvantage of nonparametric tests may be overcome

• explain the theory of the sign test

• explain the concept of “power” and tell why nonparametric tests are generally less powerful than their parametric equivalents

• describe what is meant by “randomness” in a series of binomial events

• describe what is meant by “positive dependence” in a series of binomial events

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