# BENEDICTINE UNIVERSITY

BENEDICTINE UNIVERSITY

Course Outline MGT 150 STATISTICS I Fall, 2011

Text: Elementary Statistics, 11th edition, Mario F. Triola, Pearson/Addison-Wesley, 2010.

ISBN: 978-0-321-50024-3

TI-83 or TI-84 calculator also required.

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

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 evaluative activities (quizzes). The major topics are:

Descriptive Statistics

Probability and Probability Distributions

Inferential Statistics: Estimation and Hypothesis Testing

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: There are five quizzes, worth 100 points each.

Quiz 1 Chapter 1, 2, 3 Sep. 19, 20 Grades are determined primarily by your

Quiz 2 Chapter 4 Oct. 12, 13 percentage out of about 600 points.

Quiz 3 Chapter 5 Oct. 31, Nov. 1 Class participation may also be a factor.

Quiz 4 Chapter 6 Nov. 21, 22

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

Quiz 5 Chapter 7-9 * Final Exam week *certain sections

There will also be at least five 20-point assignments requiring the analysis of data using Excel spreadsheet files prepared by the instructor. 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.

Use of the Text: Some may find the text difficult to read, so the instructor will provide, in class, all the information

needed to solve the relevant problems. The text may thus be used primarily for its descriptions, discussions,

and illustrations. The formulas used in class may differ from those in the text, due to differences in notation,

and other changes made in the interest of consistency, simplification, and understandability. Where class

usage differs from the text, class usage will control.

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 odd-numbered exercises in the Basic Skills set at the end of each section of the text. Proficiency gained from practice on these will help when similar problems appear on quizzes. Answers to odd-numbered exercises are at the back of the text.

Assignments: Assignments must be turned in during class on the day they are due. Assignments turned in after this time but before the assignment is handed back may receive one-half credit. Assignments turned in after the hand-back can no longer be accepted for credit.

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 Sonderby 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 and students are expected to read it.

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. One way of showing respect for others in the educational community is to do your part to create or maintain an environment that is conducive to learning. That being said, allowing your cell phone to ring in class is completely inappropriate because it distracts your classmates and thus degrades their overall classroom experience. For the sake of your classmates, you are expected to 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.

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

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

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 extremely 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.

It may be significant that the highest grade in a recent MBA class in statistics was earned by a philosophy major who did not have single math course at the college level. When asked about this, the student 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 are 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?

The Triola text is one of the most widely adopted introductory statistics books. It has gone through ten 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 reward 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.

|PART ONE Essentials| -- Introduction and Fundamental Concepts

Two Types of Statistics: Descriptive and Inferential

Descriptive Statistics--purpose: to communicate characteristics of a set of data

Characteristics: Mean, median, mode, variance, standard deviation, skewness, etc.

Charts, graphs

Inferential Statistics--purpose: to make statements about population parameters based on sample statistics

Population--group of interest being studied; often too large to sample every member

Sample--subset of the population; must be representative of the population

Random sampling is a popular way of obtaining a representative sample.

Parameter--a characteristic of a population, usually unknown, often can be estimated

Population mean, population variance, population proportion, etc.

Statistic--a characteristic of a sample

Sample mean, sample variance, sample proportion, etc.

Two ways of conducting inferential statistics

Estimation

Point estimate--single number estimate of a population parameter, no recognition of uncertainty, such as

"40" to estimate the average age of the voting population

Interval estimation--point estimate with an error factor, as in "40 ± 5"

The error factor provides formal and quantitative recognition of uncertainty.

Confidence level (confidence coefficient)--the probability that the parameter being estimated actually

is in the stated range

Hypothesis testing

Null hypothesis--an idea about an unknown population parameter, such as: "In the population,

the correlation between smoking and lung cancer is zero."

Alternate hypothesis--the opposite idea about the unknown population parameter, such

as: "In the population, the correlation between smoking and lung cancer is not zero."

Data are gathered to see which hypothesis is supported. The result is either rejection

or non-rejection (acceptance) of the null hypothesis.

Four types of data

Nominal

Names, labels, categories (e.g. cat, dog, bird, rabbit, ferret, gerbil)

Ordinal

Suggests order, but computations on the data are impossible or meaningless (e.g. pets can be listed in order of popularity--1-cat, 2-dog, 3-bird, etc.--but the difference between cat and dog is not related to the difference between dog and bird)

Interval

Differences are meaningful, but not ratios; there is no natural zero point (e.g. clock time--the difference between noon and 1 p.m. is the same amount of time as the difference between 1 p.m. and 2 p.m. But 2 p.m. is not twice as late as 1 p.m. unless you define the starting point of time as noon, thereby creating a ratio scale)

Ratio

Differences and ratios are both meaningful; there is a natural zero point (e.g. length--8 feet is twice as long as 4 feet, and 0 feet actually does mean no length at all)

Two types of statistical studies

Observational study (naturalistic observation)

Researcher cannot control the variables under study; they must be taken as they are found (e.g. most research in astronomy).

Experiment

Researcher can manipulate the variables under study (e.g. drug dosage).

|Essentials| -- Characteristics of Data

Central tendency--attempt to find a "representative" or "typical" value

Mean--the sum of the data items divided by the number of items, or Σx / n

More sensitive to outliers than the median

Outlier--data item far from the typical data item

Median--the middle item when the items are ordered high-to-low or low-to-high

Equal to the 50th percentile

Less sensitive to outliers than the mean

Mode--most-frequently-occurring item in a data set

Dispersion (variation or variability)--the opposite of consistency

Variance--the Mean of the Squared Deviations (MSD), or Σ(x-xbar)2/n

Deviation--difference between a data item and the mean

The sum of the deviations in any data set is always equal to zero.

Standard Deviation--square root of the variance

Coefficient of Variation—measures relative dispersion—CV = ssd / x-bar

Range--difference between the highest and lowest value in a data set

Skewness--the opposite of symmetry

Positive skewness--mean exceeds median, high outliers

Negative skewness--mean less than median, low outliers

Symmetry--mean, median, mode, and midrange about the same

Kurtosis--degree of relative concentration or peakedness

Leptokurtic--distribution strongly peaked

Mesokurtic--distribution moderately peaked

Platykurtic--distribution weakly peaked

|Essentials| -- Symbols & "Formula Sheet No. 1"

Descriptive statistics

Sample Mean--"xbar" (x with a bar above it)

Sample Variance--"svar" (the same as MSD for the sample)

Also, the "mean of the squares less the square of the mean"

Sample Standard Deviation--"ssd"--square root of svar

Population parameters (usually unknown, but can be estimated)

Population Mean--"μ" (mu)

Population Variance--"σ2" (sigma squared) (MSD for the population)

Population Standard Deviation--"σ" (sigma)--square root of σ2

Inferential statistics--estimates of population parameters based on sample statistics

Estimated Population Mean--"μ^" (mu hat)

The sample mean is an unbiased estimator of the population mean.

Unbiased estimator--just as likely to be greater than as less than the parameter being estimated

If every possible sample of size n is selected from a population, exactly as many sample

means will be above as will be below the population mean.

Estimated Population Variance--"σ^2" (sigma hat squared)

The sample variance is a biased estimator of the population variance.

Biased estimator--not just as likely to be greater than as less than the parameter

being estimated

If every possible sample of size n is selected from a population, more of the sample

variances will be below than will be above the population variance.

The reason for this bias is the probable absence of outliers in the sample.

The variance is greatly affected by outliers.

The smaller a sample is, the less likely it is to contain outliers.

Note how the correction factor's [ n / (n-1) ] impact increases as the sample size decreases.

This quantity is also widely referred to as "s2" and is widely referred to as the "sample variance."

In this context "sample variance" does not mean variance of the sample; it is, rather, a shortening

of the cumbersome phrase "estimate of population variance computed from a sample."

Estimated Population Standard Deviation--"σ^" (sigma hat)--square root of σ^2

The bias considerations that apply to the estimated population variance also apply to the estimated population standard deviation.

This quantity is also widely referred to as "s", and is widely referred to as the "sample standard deviation."

In this context "sample standard deviation" does not mean standard deviation of the sample;

it is, rather, a shortening of the cumbersome phrase "estimate of population standard

deviation computed from a sample."

Calculator note--some calculators, notably TI's, compute two standard deviations

The smaller of the two is the one we call "ssd"

TI calculator manuals call this the "population standard deviation."

This refers to the special case in which the entire population is included in the sample; then the sample standard deviation (ssd) and the population standard deviation are one and the same. (This statement also applies to means and variances.) There is no need for inferential statistics in such cases.

The larger of the two is the one we call σ^ (sigma-hat) (estimated population standard deviation).

TI calculator manuals call this the "sample standard deviation."

This refers to the more common case in which "sample standard deviation" really means estimated

population standard deviation, computed from a sample.

|Essentials| -- Significance of the Standard Deviation

Normal distribution (empirical rule)--empirical: derived from experience

Two major characteristics: symmetry and center concentration

Two parameters: mean and standard deviation

"Parameter," in this context, means a defining characteristic of a distribution.

Mean and median are identical (due to symmetry) and are at the high point.

Standard deviation--distance from mean to inflection point

Inflection point--the point where the second derivative of the normal curve equation is equal to zero,

or, the point where the curvature changes from "right" to "left" (or vice-versa), as when

you momentarily travel straight on an s-curve on the highway

z-value--distance from mean, measured in standard deviations

Areas under the normal curve can be computed using integral calculus.

Total area under the curve is taken to be 1.000

Tables enable easy determination of these areas.

68-1/4%, 95-1/2%, and 99-3/4% of the area under a normal curve lie within

one, two, and three standard deviations from the mean, respectively

Many natural and economic phenomena are normally distributed.

Tchebyshev's Theorem (or Chebysheff P. F., 1821-1894)

What if a distribution is not normal? Can any statements be made as to what percentage of the area lies

within various distances (z-values) of the mean?

Tchebysheff proved that certain minimum percentages of the area must lie within various

distances (z-values) of the mean.

The minimum percentage for a given z-value, stated as a fraction, is [ (z2-1) / z2 ]

Tchebysheff's Theorem is valid for all distributions.

|Essentials| -- Other measures of relative standing

Percentiles--A percentile is the percentage of a data set that is below a specified value.

Percentile values divide a data set into 100 parts, each with the same number of items.

The median is the 50th percentile value.

Z-values can be converted into percentiles and vice-versa.

A z-value of +1.00, for example, corresponds to the 84.13 percentile.

The 95th percentile, for example, corresponds to a z-value of +1.645.

A z-value of 0.00 is the 50th percentile, the median.

Deciles

Decile values divide a data set into 10 parts, each with the same number of items (10%).

The median is the 5th decile value.

The 9th decile value, for example, separates the upper 10% of the data set from the

lower 90%. (Some would call this the 1st decile value.)

Quartiles

Quartile values divide a data set into 4 parts, each with the same number of items (25%).

The median is the 2nd quartile value.

The 3rd quartile value (Q3), for example, separates the upper 25% of the data set from the lower 75%.

Q3 is the median of the upper half; Q1 (lower quartile) is the median of the lower half

Other possibilities: quintiles (5 parts), stanines (9 parts)

Some ambiguity in usage exists, especially regarding quartiles--For example, the phrase "first quartile" could mean one of two things: (1) It could refer to the value that separates the lower 25% of the data set from the upper 75%, or (2) It could refer to the members, as a group, of the lower 25% of the data.

Example (1): "The first quartile score on this test was 60."

Example (2): "Your score was 55, putting you in the first quartile."

Also the phrase "first quartile" is used by some to mean the 25th percentile value, and by others to mean the 75th percentile value. To avoid this ambiguity, the phrases "lower quartile," "middle quartile," and "upper quartile" may be used.

Terminology Review

Statistics, population, sample, parameter, statistic, qualitative data, quantitative data, discrete data, continuous data, nominal measurements, ordinal measurements, interval measurements, ratio measurements, observational study (naturalistic observation), experiment, precision, accuracy, sampling, random sampling, stratified sampling, systematic sampling, cluster sampling, convenience sampling, representativeness, inferential statistics, descriptive statistics, estimation, point estimation, interval estimation, hypothesis testing, dependency, central tendency, dispersion, skewness, frequency table, mutually exclusive, collectively exhaustive, relative frequencies, cumulative frequency, histogram, Pareto chart, bell-shaped distribution, uniform distribution, skewed distribution, pie chart, pictogram, mean, median, mode, bimodal, midrange, reliability, symmetry, skewness, positive skewness, negative skewness, range, MSD, variance, deviation, standard deviation, z-value, Chebyshev's theorem, empirical rule, normal distribution, quartiles, quintiles, deciles, percentiles, interquartile range, stem-and-leaf plot, boxplot, biased, unbiased.

Skills and Concepts

Given a data set, compute or identify the

Sample mean, median, mode, variance, standard deviation, and range

Estimated population mean, variance, and standard deviation

Kind of skewness, if any, present in the data set

z-value of any data item

Upper, middle, and lower quartiles

Percentile of any data item

Percentile of any integer z-value from -3 to +3

Identify circumstances under which the median is a more suitable measure of central tendency than the mean

Explain when the normal distribution (empirical rule) may be used

Explain when Chebyshev's Theorem may be used; when it should be used

Give an example (create a data set) in which the mode fails as a measure of central tendency

Give an example (create a data set) in which the mean fails as a measure of central tendency

Explain why the sum of the deviations fails as a measure of dispersion, and describe how this failure is overcome

Distinguish between unbiased and biased estimators of population parameters

Describe how percentile scores are determined on standardized tests like the SAT or the ACT

Explain why the variance and standard deviation of a sample are likely to be lower than the variance and standard deviation of the population from which the sample was taken

Identify when the sample mean, variance, and standard deviation are identical to the population mean, variance, and standard deviation

|PART TWO Essentials| -- Basic Probability Concepts

Probability--the likelihood of an event

Probability is expressed as a decimal or fraction between zero and one, inclusive.

An event that is certain has a probability of 1.

An event that is impossible has a probability of 0.

If the probability of rain today (R) is 30%, it can be written P(R) = 0.3.

Objective probabilities--calculated from data according to generally-accepted methods

Relative frequency method--example: In a class of 25 college students there are 14 seniors. If a student is selected at random from the class, the probability of selecting a senior is 14/25 or 0.56. Relative to the number in the class, 25, the number of seniors (frequency), 14, is 56% or 0.56.

Subjective probabilities--arrived at through judgment, experience, estimation, educated guessing, intuition, etc.

There may be as many different answers as there are people making the estimate. (With objective

probability, all should get the same answer.)

|Essentials| -- Boolean operations--Boolean algebra--(George Boole, 1815-1864)

Used to express various logical relationships; taught as "symbolic logic" in college philosophy and mathematics departments; important in computer design

Complementation--translated by the word "not"--symbol: A¯or A-bar

Complementary events are commonly known as "opposites."

Examples: Heads/Tails on a coin-flip; Rain/No Rain on a particular day; On Time/Late for work

Complementary events have two properties

Mutually exclusive--they cannot occur together, each excludes the other

Collectively exhaustive--there are no other outcomes, the two events are a complete or exhaustive list of the possibilities

Partition--a set of more than two events that are mutually exclusive and collectively exhaustive

Examples: A, B, C, D, F, W, I--grades received at the end of a course; Freshman, Sophomore, Junior, Senior--traditional college student categories

The sum of the probabilities of complementary events, or of the probabilities of all the events in a partition is 1.

Intersection--translated by the words "and," "with," or "but"--symbol: ( or, for convenience, n

A day that is cool (C) and rainy (R) can be designated (CnR).

If there is a 25% chance that today will be cool (C) and rainy (R), it can be written P(CnR) = 0.25.

Intersections are often expressed without using the word "and."

Examples: "Today might be cool with rain." or "It may be a cool, rainy day."

Two formulas for intersections:

For any two events A and B: P(AnB) = P(A|B)*P(B) ("|" is defined below.)

For independent events A and B: P(AnB) = P(A)*P(B) (This will appear later as a test for independence).

This formula may be extended to any number of independent events

P(AnBnCn . . . X) = P(A)*P(B)*P(C)* . . . P(X), where X is any letter

The intersection operation has the commutative property

P(AnB) = P(BnA)

"Commutative" is related to the word "commute" which means "to switch."

The events can be switched without changing anything.

In our familiar algebra, addition and multiplication are commutative, but

subtraction and division are not.

Intersections are also called "joint probabilities" (Joint in the sense of "together.")

Union--translated by the word "or"--symbol: ( or, for convenience, u

A day that is cool (C) or rainy (R) can be designated (CuR).

If there is a 25% chance that today will be cool (C) or rainy (R), it can be written

P(CuR) = 0.25.

Unions always use the word "or."

Addition rule to compute unions: P(AuB) = P(A) + P(B) - P(AnB)

The deduction of P(AnB) eliminates the double counting that occurs when P(A) is added to P(B).

The union operation is commutative: P(AuB) = P(BuA)

Condition--translated by the word "given"--symbol: |

A day that is cool (C) given that it is rainy (R) can be designated (C|R). The event R is called the condition.

If there is a 25% chance that today will be cool (C) given that it is rainy (R), it can be written P(C|R) = 0.25.

Conditions are often expressed without using the word "given."

Examples: "The probability that it will be cool when it is rainy is 0.25." [P(C|R) = 0.25.]

"The probability that it will be cool if it is rainy is 0.25." [P(C|R) = 0.25.]

"25% of the rainy days are cool." [P(C|R) = 0.25.]

All three of the above statements are the same, but this next one is different:

"25% of the cool days are rainy." This one is P(R|C) = 0.25.

The condition operation is not commutative: P(A|B) ≠ P(B|A)

For example, it is easy to see that P(rain|clouds) is not the same as P(clouds|rain).

Conditional probability formula: P(A|B) = P(AnB) / P(B)

|Essentials| -- Occurrence Tables and Probability Tables

Occurrence table--table that shows the number of items in each category and in the intersections of categories

Can be used to help compute probabilities of single events, intersections, unions, and conditional probabilities

Probability table--created by dividing every entry in an occurrence table by the total number of occurrences.

Probability tables contain marginal probabilities and joint probabilities.

Marginal probabilities--probabilities of single events, found in the right and bottom margins of the table

Joint probabilities--probabilities of intersections, found in the interior part of the table where the rows and columns intersect

Unions and conditional probabilities are not found directly in a probability table, but they can be computed easily from values in the table.

Two conditional probabilities are complementary if they have the same condition and the events before the "bar" (|) are complementary. For example, if warm (W) is the opposite of cool, then (W|R) is the complement of (C|R), and P(W|R) + P(C|R) = 1.

In a 2 x 2 probability table, there are eight conditional probabilities, forming four pairs of complementary conditional probabilities.

It is also possible for a set of conditional probabilities to constitute a partition (if they all have the same condition, and the events before the "bar" are a partition).

|Essentials| -- Testing for Dependence/Independence

Statistical dependence

Events are statistically dependent if the occurrence of one event affects the probability of the other event.

Identifying dependencies is one of the most important tasks of statistical analysis.

Tests for independence/dependence

Conditional probability test--posterior/prior test

Prior and posterior are, literally, the Latin words for "before" and "after."

A prior probability is one that is computed or estimated before additional information is obtained.

A posterior probability is one that is computed or estimated after additional information is obtained.

Prior probabilities are probabilities of single events, such as P(A).

Posterior probabilities are conditional probabilities, such as P(A|B).

Independence exists between any two events A and B if P(A|B) = P(A)

If P(A|B) = P(A), the occurrence of B has no effect on P(A)

If P(A|B) ≠ P(A), the occurrence of B does have an effect on P(A)

Positive dependence if P(A|B) > P(A) -- posterior greater than prior

Negative dependence if P(A|B) < P(A) -- posterior less than prior

Multiplicative test--joint/marginal test

Independence exists between any two events A and B if P(AnB) = P(A)*P(B)

Positive dependence if P(AnB) > P(A)*P(B) -- intersection greater than product

Negative dependence if P(AnB) < P(A)*P(B) -- intersection less than product

|Essentials| -- Bayesian Inference

Thomas Bayes (1702-1761)

Bayes developed a technique to compute a conditional probability, given the reverse conditional probability

Computations are simplified, and complex formulas can often be avoided, if a probability table is used.

Basic computation is: P(A|B) = P(AnB) / P(B), an intersection probability divided by single-event probability. That is, a joint probability divided by a marginal probability.

Bayesian analysis is very important because most of the probabilities upon which we base decisions are conditional probabilities.

|Essentials| -- Other Probability Topics:

Matching-birthday problem

Example of a "sequential" intersection probability computation, where each probability is revised slightly and complementary thinking is used

Complementary thinking--strategy of computing the complement (because it is easier) of what is

really sought, then subtracting from 1

Redundancy

Strategy of using back-ups to increase the probability of success

Usually employs complementary thinking and the extended multiplicative rule for independent events to compute the probability of failure. P(Success) is then equal to 1 - P(Failure).

|Essentials| -- Permutations and Combinations

Permutation--a set of items in which the order is important

Without replacement--duplicate items are not permitted

With replacement--duplicate items are permitted

Combination--a set of items in which the order is not important

Without replacement--duplicate items are not permitted

With replacement--duplicate items are permitted

In the formulas, "n" designates the number of items available, from which "r" is the number that will be chosen.

(Can r ever exceed n?)

To apply the correct formula when confronting a problem, two decisions must be made:

Is order important or not?

Are duplicates permitted or not?

Permutations, both with and without replacement, can be computed by using the "sequential" method instead of

the formula. This provides way of verifying the formula result.

Lotteries

Usually combination ("Lotto") or permutation ("Pick 3 or 4") problems

Lotto games are usually without replacement--duplicate numbers are not possible

Pick 3 or 4 games are usually with replacement--duplicate numbers are possible

Poker hands

Can be computed using combinations and the relative frequency method

Can also be computed sequentially

Terminology--Probability--explain each of the following:

probability, experiment, event, simple event, compound event, sample space, relative frequency method, classical approach, law of large numbers, random sample, impossible event probability, certain event probability, complement, partition, subjective probability, occurrence table, probability table, addition rule for unions, mutually exclusive, collectively exhaustive, redundancy, multiplicative rule for intersections, tree diagram, statistical independence/dependence, conditional probability, Bayes' theorem, acceptance sampling, simulation, factorial, permutations, combinations, with replacement, without replacement, risk assessment, Boolean algebra, complementation, intersection, union, condition, marginal probabilities, joint probabilities, prior probabilities, posterior probabilities, two tests for independence, triad, complementary thinking, commutative.

Skills and Procedures--given appropriate data,

prepare an occurrence table

prepare a probability table

compute the following 20 probabilities

4 marginal probabilities (single simple events)

4 joint probabilities (intersections)

4 unions

8 conditional probabilities--identify the 4 pairs of conditional complementary events

identify triads (one unconditional and two conditional probabilities in each triad)

conduct the conditional (prior/posterior) probability test for

independence / dependence

conduct the multiplication (multiplicative) (joint/marginal) test for

independence / dependence

identify positive / negative dependency

identify Bayesian questions

use the extended multiplicative rule to compute probabilities

use complementary thinking to compute probabilities

compute the probability of "success" when redundancy is used

Concepts--

give an example of two or more events that are not mutually exclusive

give an example of two or more events that are not collectively exhaustive

give an example of a partition--a set of three or more events that are mutually exclusive and collectively exhaustive

express the following in symbolic form using F for females and V for voters in a retirement community

60% of the residents are females

30% of the residents are female voters

50% of the females are voters

75% of the voters are female

70% of the residents are female or voters

30% of the residents are male non-voters

25% of the voters are male

40% of the residents are male

identify which two of the items above are a pair of complementary probabilities

identify which two of the items above are a pair of complementary conditional probabilities

from the items above, comment on the dependency relationship between F and V

if there are 100 residents, determine how many female voters there would be if gender and voting were independent

explain why joint probabilities are called "intersections"?

identify which two of our familiar arithmetic operations and which two Boolean operations are commutative

tell what Thomas Bayes is known for (not English muffins)

|PART THREE Essentials| -- Permutations and Combinations

Permutation--a set of items in which the order is important

Without replacement--duplicate items are not permitted

With replacement--duplicate items are permitted

Permutations, both with and without replacement, can be computed by using the "sequential" method instead of the formula. This provides way of verifying the formula result.

Combination--a set of items in which the order is not important

Without replacement--duplicate items are not permitted

With replacement--duplicate items are permitted

In the formulas, "n" designates the number of items available, from which "r" is the number that will be chosen.

(Can r ever exceed n?)

To apply the correct formula when confronting a problem, two decisions must be made:

Is order important (permutations) or not (combinations)?

Are duplicates permitted (with replacement) or not (without replacement)?

Lotteries

Usually combination ("Lotto") or permutation ("Pick 3” or “Pick 4") cases

Lotto games are usually without replacement--duplicate numbers are not possible

Pick 3 or 4 games are usually with replacement--duplicate numbers are possible

Poker hands

Can be computed using combinations and the relative frequency method

Can also be computed sequentially

|Essentials| -- Mathematical Expectation

Discrete variable--one that can assume only certain values (often the whole numbers)--there is only a finite

countable number of values between any two specified values

Examples: the number of people in a room, your score on a quiz in this course, shoe sizes (certain fractions permitted), hat sizes (certain fractions permitted)

Continuous variable--one that can take on any value--there is an infinite number of values between any two specified values

Examples: your weight (can be any value, and changes as you breathe), the length of an object, the

amount of time that passes between two events, the amount of water in a container (but if you look

at the water closely enough, you find that it is made up of very tiny chunks--molecules--so this last

example is really discrete at the submicroscopic level, but in ordinary everyday terms we would call it

continuous)

Mean (expected value) of a discrete probability distribution

Probability distribution--a set of outcomes and their likelihoods

Probability-weighted average of the outcomes

Each outcome is multiplied by its probability, and these are added.

The result is not an estimate. It is the actual population value, because the probability distribution specifies

an entire population of outcomes. ("μ" may be used, without the estimation caret above it.)

The mean need not be a possible outcome, and for this reason the term "expected value" is misleading.

Variance of a discrete probability distribution

Probability-weighted average of the squared deviations (similar to MSD)

Each squared deviation is multiplied by its probability, and these are added.

The result is not an estimate. It is the actual population value, because the probability distribution specifies

an entire population of outcomes. ("σ2" may be used, without the estimation caret above it.)

Standard deviation of a discrete probability distribution--the square root of the variance

("σ" may be used, without the estimation caret ^ above it.)

|Essentials| -- The Binomial Distribution

Binomial experiment requirements

Two possible outcomes on each trial

The two outcomes are (often inappropriately) referred to as "success" and "failure."

n identical trials

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

Constant p and q from trial to trial

p is the probability of the "success" event

q is the probability of the "failure" event; (q = 1-p )

"x" is the number of "successes" out of the n trials.

Symmetry is present when p = q

When p < .5, the distribution is positively skewed (high outliers).

When p > .5, the distribution is negatively skewed (low outliers).

Binomial formula--for noncumulative probabilities

Cumulative binomial probabilities--computed by adding the noncumulative probabilities

Binomial probability tables--may show cumulative or noncumulative probabilities

If cumulative, compute noncumulative probabilities by subtraction

Parameters of the binomial distribution--n and p

Terminology--explain each of the following:

PERMUTATIONS AND COMBINATIONS: permutations, permutations with replacement, sequential method, combinations, combinations with replacement. MATHEMATICAL EXPECTATION: random variable, discrete variable, continuous variable, probability distribution, probability histogram, mean of a probability distribution, variance and standard deviation of a probability distribution, probability-weighted average of outcomes (mean), probability-weighted average of squared deviations (variance). BINOMIAL DISTRIBUTION: binomial experiment, requirements for a binomial experiment, independent trials, binomial probabilities, cumulative binomial probabilities, binomial distribution symmetry conditions, binomial distribution skewness conditions, binomial distribution parameters, mean and variance of a binomial distribution

Skills and Procedures--given appropriate data,

PERMUTATIONS AND COMBINATIONS:

decide when order is and is not important

decide when selection is done with replacement and without replacement

compute permutations with and without replacement using the permutation formula

compute combinations with and without replacement using the combination formula

use the sequential method to compute permutations with and without replacement

solve various applications problems involving permutations and combinations

MATHEMATICAL EXPECTATION:

compute the mean, variance, and standard deviation of a discrete random variable

solve various applications problems involving discrete probability distributions

BINOMIAL DISTRIBUTION:

compute binomial probabilities and verify results with table in textbook

compute cumulative binomial probabilities

compute binomial probabilities with p = q and verify symmetry

solve various application problems using the binomial distribution

Concepts

PERMUTATIONS AND COMBINATIONS:

give an example of a set of items that is a permutation

give an example of a set of items that is a combination

tell if, in combinations/permutations, "r" can ever exceed "n"

MATHEMATICAL EXPECTATION

give an example (other than water) of something that looks continuous at a distance, but, when you get up close, turns out to be discrete

explain why "expected value" may be a misleading name for the mean of a probability distribution

describe how to compute a weighted average

BINOMIAL DISTRIBUTION:

explain why rolling a die is or is not a binomial experiment

explain why drawing red/black cards from a deck of 52 without replacement is or is not a binomial experiment

explain why drawing red/black cards from a deck of 52 with replacement is or is not a binomial experiment

|PART FOUR Essentials| -- The Normal Distribution -- The red section may be omitted

Normal distribution characteristics--center concentration and symmetry

Parameters of the normal distribution--μ (mu), mean; and σ (sigma), standard deviation

Z-value formula (four arrangements--for z, x, μ, and σ)

Normal distribution problems have three variables given, and the fourth must be

computed and interpreted

Z-values determine areas (probabilities) and areas (probabilities) determine z-values--the normal

table converts from one to the other

Normal distribution probability tables--our text table presents one-sided central areas

Two uses of the normal distribution

Normally-distributed phenomena

To approximate the binomial distribution--this application is far less important now that computers

and even small calculators can generate binomial probabilities

Binomial parameters (n and p) can be converted to normal parameters μ and σ

μ = np; σ2 = (npq); σ = ((npq)

Continuity correction--needed when a continuous distribution (e.g. normal) is used to

Approximate a discrete (e.g. binomial) distribution

Three causes for the normal-binomial discrepancy

Normal is continuous; binomial is discrete

Normal is infinite; binomial is finite

Normal is always symmetrical; binomial is symmetrical only when p = q

Conditions for good normal-binomial approximation: large n; p near q

Poisson distribution a better approximator of the binomial when p is not near q

Poisson is discrete, like the binomial

Poisson is finite, like the binomial

Poisson is skewed, like the binomial when p and q are not equal

Sampling Distributions

Sampling distribution of the mean--the distribution of the means of many samples of the same size

Drawn from the same population

Central Limit Theorem--three statements about the sampling distribution of the means:

1. Sampling distribution of the means is normal in shape, regardless of the population distribution

shape when the sample size, n, is large. (When n is small, the population must be normal in

order for the sampling distribution of the mean to be normal.) ("Large" n is usually taken

to mean 30 or more.)

2. Sampling distribution of the means is centered at the true population mean.

3. Sampling distribution of the means has a standard deviation equal to σ / (n.

This quantity is called the sampling standard deviation or the standard error (of the mean).

(The full name is "standard deviation of the sampling distribution.”)

This quantity is represented by the symbol σxbar.

σxbar is less than σ because of the offsetting that occurs within the sample. The larger

the sample size n, the smaller the σxbar (standard error), because the larger the

n, the greater the amount of offsetting that can occur, and the sample means will

cluster more closely around the true population mean μ.

Sampling standard deviation (σxbar or standard error)--key value for inferential statistics

Two uses of the standard error

Computing the error factor in interval estimation

Computing the test statistic (zc or tc) in hypothesis testing

Terminology--explain each of the following:

normal distribution, normal distribution parameters, mean, standard deviation, standard normal distribution, z-value, reliability, validity, sampling distribution, central limit theorem (three parts), sampling standard deviation, standard error, offsetting, effect of the sample size on the sampling standard deviation (standard error).

Skills and Procedures--given appropriate data,

( determine a normal probability (area), given x, μ, and σ

( determine x, given μ, σ, and the normal probability (area)

( determine μ, given x, σ, and the normal probability (area)

( determine σ, given x, μ, and the normal probability (area)

( solve various applications problems involving the normal distribution

( compute the sampling standard deviation (standard error) from the population standard deviation and the sample size

( solve various applications problems involving the central limit theorem

Concepts--

( describe conditions under which the normal distribution is symmetric

( describe the kind of shift in the graph of a normal distribution caused by a change in the mean

( describe the kind of shift in the graph of a normal distribution caused by a change in the standard deviation

( explain why, as the sample size increases, the distribution of sample means clusters more and more closely around the population mean

|PART FIVE Essentials|--Interval Estimation--Large Samples

Four Types of Problems

Means--one-group; two-group

Columns one and two of the four-column formula sheet

Proportions--one-group; two-group

Columns three and four of the four-column formula sheet

Confidence level (confidence coefficient)--the probability that a confidence interval will actually contain the

population parameter being estimated (confidence interval is a range of values that is likely to contain

the population parameter being estimated).

90%, 95%, and 99% are the most popular, and correspond to z-values of 1.645, 1.960, and 2.576,

respectively.

Of these, 95% is the most popular, and is assumed unless information to the contrary is provided.

Error (uncertainty) factors express precision, as in 40 ± 3.

Upper confidence limit--the point estimate plus the error factor, 43 in this example

Lower confidence limit--the point estimate minus the error factor, 37 in this example

Error factor is the product of the relevant z-value and the standard error: zt * σxbar

Required sample sizes for desired precision may be computed

Increased precision means a lower error factor.

Precision can be increased by increasing the sample size, n.

Increasing n lowers the standard error, since the standard error = σ / (n.

Taken to the extreme, every member of the population may be sampled, in which case the error factor

becomes zero--no uncertainty at all--and the population parameter is determined exactly.

Economic considerations--the high cost of precision

The required increase in n is equal to the square of the desired increase in precision.

To double the precision--to cut the error factor in half--the sample size must be quadrupled.

Doubling the precision may thus quadruple the cost.

To triple the precision--to cut the error factor to 1/3 of its previous value, n must be multiplied by 9.

Hypothesis Testing--Large Samples

Four Types of Problems--Four-column formula sheet

Means--one-group; two-group

Proportions--one-group; two-group

Null (Ho) and alternate (Ha) hypotheses

Means, one-group

Ho: μ = some value

Ha: μ ≠ that same value (two-sided test)

μ > that same value (one-sided test, high end, right side)

μ < that same value (one-sided test, low end, left side)

Means, two-group

Ho: μ1 = μ2

Ha: μ1 ≠ μ2 (two-sided test)

μ1 > μ2 (one-sided test, high end, right side)

μ1 < μ2 (one-sided test, low end, left side)

Proportions, one-group

Ho: π = some value

Ha: π ≠ that same value (two-sided test)

π > that same value (one-sided test, high end, right side)

π < that same value (one-sided test, low end, left side)

Proportions, two-group

Ho: π1 = π2

Ha: π1 ≠ π2 (two-sided test)

π1 > π2 (one-sided test, high end, right side)

π1 < π2 (one-sided test, low end, left side)

Type I error

Erroneous rejection of a true Ho

Probability of a Type I error is symbolized by α.

Type II error

Erroneous acceptance of a false Ho

Probability of a Type II error is symbolized by β.

Selecting α--based on researcher’s attitude toward risk

α--the researcher's maximum tolerable risk of committing a type I error

0.10, 0.05, and 0.01 are the most commonly used.

Of these, 0.05 is the most common--known as "the normal scientific standard of proof."

Table-z (critical value); symbolized by zt; determined by the selected α value

α 2-sided z 1-sided z

0.10 1.645 1.282

0.05 1.960 1.645

0.01 2.576 2.326

Calculated-z (test statistic); symbolized by zc

Fraction--"signal-to-noise" ratio

Numerator ("signal")--strength of the evidence against Ho

Denominator ("noise")--uncertainty factor for the numerator

Rejection criteria

Two-sided test: |zc| >= |zt|; also p = |zt|, AND zc and zt have the same sign; also p ................

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