Benedictine University
Informing today – Transforming tomorrow
SYLLABUS
Course:
Instructor:
MGT 150–Business Statistics I–Spring, 2015
Jeffrey M. Madura
B.A. University of Notre Dame, 1967
M.B.A. Northwestern University, 1971
C.P.A. State of Illinois, 1979
630-829-6467
jmadura@ben.edu
www.ben.edu/faculty/jmadura/home.htm
Phone:
Email:
Website:
Text:
Modern Business Statistics with Microsoft Office Excel, 5th edition, Anderson, Sweeney & Williams,
South-Western/Cengage, 2015.
ISBN: 978-1-285-43330-1 (hard cover)
Other Required: Aplia interactive learning/assignment system.
TI-83 or TI-84 calculator.
Course Objectives:
The course addresses the following formal College of Business Program Objectives:
Students in this program will receive a thorough grounding in: Mathematics and
Statistics.
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.
Course Description: (from the Catalog)
Basic course in statistical technique, includes measures of central tendency, variability, probability theory,
sampling, estimation and hypothesis testing. Prerequisite: MATH 105 or MATH 110. Three semester hours.
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:
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Data and Statistics
Descriptive Statistics: Tabular and Graphical Presentations
Descriptive Statistics: Numerical Measures
Introduction to Probability
Discrete Probability Distributions
Continuous Probability Distributions
Sampling and Sampling Distributions
Interval Estimation, Means and Proportions
Hypothesis Tests, Means and Proportions
Quizzes and Grades: The course is divided into five three-week parts, with a quiz at the end
of each part. Dates are subject to change.
Quiz 1 Feb. 5
Quiz 2 Feb. 26
Quizzes will constitute 2/3 of your grade.
Quiz 3 Mar. 26
The other 1/3 will be your score on assignments,
Quiz 4 Apr. 16
Class participation may also be a factor.
Quiz 5 Finals Week
Grade requirements: A--90%, B--80%, C--60%, D--50%.
There may also be other assignments requiring analysis of data using Excel, and there may be a term project,
with weight equal to one quiz.
It is the responsibility of any student who is unsure of the grading scale, course requirements,
or anything else in this course outline to ask the instructor for clarification.
Homework Assignments: There will be 10 Aplia homework assignments. Due dates are listed
in the Aplia system.
The assignments will constitute 1/3 of the course grade. To accommodate the occasional instance when you
cannot meet an Aplia deadline, the lowest assignment will be dropped. Assignments will be handled by Aplia.
You must access the Aplia website, which means you must register for an account at: http://www.aplia.com.
Please register within 24 hours of the first class meeting.
The computer is absolutely unforgiving about accepting late assignments. Time is kept at Aplia, and not by
the computer you are working on. You may appeal grading decisions made by the computer, if you can
demonstrate that an error has been made.
Faculty members have observed that the worst thing some students do in a course is not think about course
material every day. They sometimes let weeks go by and then try to learn all the material in one or two days.
This usually does not work. The weekly assignments will require keeping up-to-date.
Calculators: Calculators will be required for the computational portion of each quiz. Bring your calculator to
every class and verify each computation performed. The TI-83 is the standard for this course.
Recommended Exercises: Students should work as many as possible of the even-numbered exercises in the
text. Proficiency gained from practice on these will help when similar problems appear on quizzes.
Answers to even-numbered exercises are at the back of the text.
Assignments: Non-Aplia 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 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 Jennifer Rigor in the Student Success Center, 015 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: 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.
In this course, academic honesty is expected of all class participants. 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).
Exception: Activities in the course that are designated as "group work."
Electronic Devices: Bring your TI83/84 to every class. Turn off or mute your phone before class. Using your
laptop or tablet to follow class examples in Excel is encouraged, but only the TI calculator is permitted for
quizzes.
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.
Other Grading Policies:
Students on Academic Probation are not eligible for a grade of (I) incomplete.
Students who are not enrolled in class (either for credit or audit) cannot attend the
class and cannot receive credit for the course.
Students cannot submit additional work after grades have been submitted to alter
their grade (except in cases of temporary grades such as I, X, IP).
Make up exams or assignments must be completed within one week of the schedule due date. Failure to
attend class does not excuse the student from meeting deadlines for assigned work. Any student who is
unsure of the grading scale or course requirements is responsible for clarifying questions with the
instructor.
Essential Ideas, Terminology, Skills/Procedures, and Concepts for Each Part of the Course
Part I
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 they are 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).
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
Also called 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
Range--difference between the highest and lowest value in a data set
Coefficient of Variation—measures relative dispersion—CV = ssd / x-bar (or est.  / )
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
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--estimating 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, 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 the same.
(This 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.
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 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 or 100%
Tables enable easy determination of these areas.
about 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
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.
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.
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.
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
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, kurtosis, leptokurtic, mesokurtic, platykurtic, 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/Procedures--given appropriate data, 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
Concepts
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 II
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 results as there are people making the estimate.
(With objective probability, all should get the same answer.)
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 typing 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 . . . nZ) = P(A)*P(B)*P(C)* . . . P(Z)
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 (together) probabilities."
Union--translated by the word "or"--symbol:  or, for typing 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 the 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)
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).
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 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
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
a 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.
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 needed, 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).
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:
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, risk assessment, redundancy, Boolean algebra, complementation, intersection, union, condition,
marginal probabilities, joint probabilities, prior probabilities, posterior probabilities, two tests for independence,
triad, complementary thinking, commutative.
PERMUTATIONS AND COMBINATIONS:
permutations, permutations with replacement, sequential method, combinations, combinations with
replacement.
Skills/Procedures--given appropriate data, prepare an occurrence table
PROBABILITY
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
compute permutations and combinations with and without replacement
Concepts
PROBABILITY
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)
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"
Part III
Permutations and Combinations (outline, etc. Repeated from Part II)
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
PERMUTATIONS AND COMBINATIONS:
permutations, permutations with replacement, sequential method, combinations, combinations with
replacement.
Skills/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
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
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 pieces--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
Mean is the 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" can be misleading.
Variance of a discrete probability distribution
Variance is the probability-weighted average of the squared deviations
similar to MSD, except it's a weighted average
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.)
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
Binomial formula: P(x) = n!/(x!(n-x)! * p^x * q^(n-x)
Note that when x=n, the formula reduces to p^n, and when x=0, the formula reduces to q^n.
These are just applications of the multiplicative rule for independent events.
The Normal Distribution
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)
Terminology
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
NORMAL DISTRIBUTION
normal distribution, normal distribution parameters, mean, standard deviation, standard normal distribution, zvalue, reliability, validity
Skills/Procedures
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
NORMAL DISTRIBUTION -- 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
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
NORMAL DISTRIBUTION
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 IV
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 sample 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 of the mean(s).”)
This quantity is represented by the symbol σx bar.
σx bar is less than σ because of the offsetting that occurs within the sample. The larger
the sample size n, the smaller the σx bar (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 (σx bar 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
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/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 V
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 confidence levels, 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 another value is given.
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 * σx bar.
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
H0: 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
H0: 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
H0: 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
H0: 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 H0
Probability of a Type I error is symbolized by α.
Type II error
Erroneous acceptance of a false H0
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 H0
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 H0 is rejected on the basis of the experimental evidence
Graphically, the area beyond the calculated z-value, zc.
Treatment--in a column-2 test, the difference that the experimenter introduces between the two groups
Terminology
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), treatment
Skills/Procedures
given appropriate data, conduct estimation and hypothesis testing on the population mean of one group,
involving these ten 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 ten 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 ten 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