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BENEDICTINE UNIVERSITY
Course Outline
Text:
MGT 150
STATISTICS I
Fall, 2011
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
B.A., University of Notre Dame
M.B.A., Northwestern University
C.P.A., State of Illinois
Office Hours: Announced in class or see web page at
http://www.ben.edu/faculty/jmadura/home.htm
e-mail: [email protected]
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 http://www.ben.edu/AHP 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 wellreasoned 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.
Knowledge
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
Percentage Grade
Course A
Course B
100%
100%
90%
81%
80%
64%
70%
49%
60%
36%
50%
25%
40%
16%
30%
9%
20%
4%
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, zvalue, 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 <= α
One-sided test: |zc| >= |zt|, AND zc and zt have the same sign; also p <= α
Significance level (p-value) ("p" stands for probability)
Actual risk (probability) of a Type I error if Ho is rejected on the basis of the experimental evidence
Graphically, the area beyond the calculated z-value, zc.
Terminology--explain each of the following:
inferential statistics, sample mean, population mean, estimator, estimate, unbiased estimator, point
estimate, interval estimate, confidence interval, degree of confidence, confidence level, table-z, error
factor, required sample size, upper confidence limit, lower confidence limit, hypothesis test, null
hypothesis, alternate hypothesis, type I error, α, type II error, β, calculated-z (test statistic), critical region,
table-z (critical value of z), rejection of the null hypothesis, non-rejection of the null hypothesis, p-value,
hypothesis-test conclusion, independent samples, standard error of the difference, sample proportion,
population proportion, pooled proportion (two-group proportion cases)
Skills and Procedures
given appropriate data, conduct estimation and hypothesis testing on the population mean of one group,
involving these steps:
 make a point estimate of a population mean
 compute the sampling standard deviation (standard error) of the sample means
 compute and interpret the error factor for the interval estimate for the 90%, 95% and 99% confidence
levels
 determine the sample size needed to obtain a given desired error factor
 state the null and alternate hypotheses regarding the population mean
 determine the table-z (critical value of z) for alpha levels of 0.10, 0.05 and 0.01
 compute the calculated-z (test statistic)
 draw the appropriate hypothesis-test conclusion based on the given level of α, the table-z (critical
value) and the calculated-z (test statistic)
 interpret the conclusion
 determine and interpret the p-value
given appropriate data, conduct estimation and hypothesis testing on the population means of two groups,
involving these steps:
 make a point estimate of the difference between population means
 compute the sampling standard deviation (standard error) of the difference between sample means
 compute and interpret the error factor for the interval estimate for the 90%, 95% and 99% confidence
levels







determine the sample size needed to obtain a given desired error factor
state the null and alternate hypotheses regarding the difference between population means
determine the table-z (critical value of z) for alpha levels of 0.10, 0.05 and 0.01
compute the calculated-z (test statistic)
draw the appropriate hypothesis-test conclusion based on the given level of α, the table-z and the
calculated-z
interpret the conclusion
determine and interpret the p-value
given appropriate data, conduct estimation and hypothesis testing on the population proportion of one group,
involving these steps:
 make a point estimate of a population proportion
 compute the sampling standard deviation (standard error) of the sample proportions
 compute and interpret the error factor for the interval estimate for the 90%, 95% and 99% confidence
levels
 determine the sample size needed to obtain a given desired error factor
 state the null and alternate hypotheses regarding the population proportion
 determine the table-z (critical value of z) for alpha levels of 0.10, 0.05 and 0.01
 compute the calculated-z (test statistic)
 draw the appropriate hypothesis-test conclusion based on the given level of α, the table-z and the
calculated-z
 interpret the conclusion
 determine and interpret the p-value
given appropriate data, conduct estimation and hypothesis testing on the population proportions of two
groups, involving these steps:
 make a point estimate of the difference between population proportions
 compute the sampling standard deviation (standard error) of the difference between sample
proportions
 compute and interpret the error factor for the interval estimate for the 90%, 95% and 99% confidence
levels
 determine the sample size needed to obtain a given desired error factor
 state the null and alternate hypotheses regarding the difference between population proportions
 determine the table-z (critical value of z) for alpha levels of 0.10, 0.05 and 0.01
 compute the calculated-z (test statistic)
 draw the appropriate hypothesis-test conclusion based on the given level of α, the table-z and the
calculated-z
 interpret the conclusion
 determine and interpret the p-value
Concepts-





explain why a confidence interval becomes larger as the confidence level increases
explain why a confidence interval becomes smaller as the sample size increases
describe the nature of the trade-off between precision and cost
identify the type of error that is made if the null hypothesis is "the defendant is innocent," and an
innocent defendant is erroneously convicted
identify the type of error that is made if the null hypothesis is "the defendant is innocent," and a guilty
defendant is erroneously acquitted
explain why a researcher seeking to reject a null hypothesis may tend to prefer a one-sided alternative
hypothesis