SHAW UNIVERSITY
MAT 313 Mathematical Probability & Statistics
COURSE OUTLINE – FALL 2005
Instructor: Simon Ugwuoke
Office Hrs: TBA
Office: Graphics Room 4
Phone: 546-8543
Email: sugwuoke@shawu.edu
Website: http://faculty.shawu.edu/sugwuoke
TEXT: Dennis D.Wakerly, William Mendenhall, &
Richard L. Scheaffer
Mathematical Statistics with Applications, Wadsworth
Publishing Company, California, Sixth Edition, 2002.
References: DeGroot, M.H., Probability and Statistics, 1986.
Blommers & Forsyth, Elementary Statistical Methods, 1977.
Hogg, R.V. & Tanis, E.A., Prob. and Statistical Inference, 1983.
Technology: Minitab, PcCalculator, and/or other statistical packages
will be used for analysis. Graphing Calculators will also be
used (TI-92 recommended).
General Description
This course is intended to provide a solid undergraduate foundation in
statistical and probability theory, and at the same time, to equip the student
in solving practical problems in the real world. It is a required course for
most majors in the mathematical sciences, including students majoring in
mathematics, mathematics education, computer science, physics, and chemistry.
Advisably, a student should have completed a first course in calculus prior to
taking this course.
Course Objectives
After completing this course successfully, the students should be
able to do the following:
Represent and characterize a set of data using:
1) distributions: frequency , relative frequency, and cumulative
relative frequency
2) the graphical methods: histograms, polygons, ogives, bar and pie
charts, pictograms, stem-and-leaf, box plots
3) numerical methods: measures of central tendencies (mean, mode, median),
measures of dispersion or variation (variance, standard deviation, and
range)
Use computer and/or graphing calculator to obtain graphs of data and analyze
data to obtain measures of central tendencies, and measures of variation.
Understand the meaning and use of the empirical rule for normal curves.
Explain the reason for studying probability theory.
Understand definitions and basic concepts of probability theory such as
sample space, sample point, an event, probability p(E) of an event E,
random samples/experiments, equally-likely events, mutually exclusive
events.
Clearly state the definition of probability distribution of a random
variable.
Calculate the probability (including conditional probability) of an event.
Clearly state the definitions of factorials, combinations, and permutations,
and apply the formulas in solving problems.
Explain the meanings of dependent & independent events and apply them in
solving probability problems.
Apply the multiplicative and additive laws of probability in solving
problems.
Explain the difference between a discrete and a continuous random variable.
Give at least four examples of discrete probability distributions using
formulas, graphs, or tables, and be able to represent at least one example
in these three different ways.
State the range of a discrete probability distribution of a random variable.
State the sum of the probabilities of individual elements in any discrete
probability distribution.
Find the probability distribution of a given random variable in a given
situation.
Find the probability of a given event in a given situation.
Find the expected value of a given event in a given situation.
Find the mean, variance, and standard deviation of a probability
distribution.
Using the appropriate formula, solve at least two problems involving
the following probability distributions:
1) Binomial probability distribution;
2) Geometric probability distribution;
3) Negative binomial probability distribution;
4) Hypergeometric probability distribution;
5) Poison probability distribution;
Find the distribution function of a random variable in a given domain.
Using the appropriate formula, solve at least one problem involving
uniform probability distribution
Find the mean, variance, and standard deviation of a normal probability
distribution
Given a score value in a normal distribution, find the corresponding value
of Z, and vice-versa, where Z is the standardized normal random variable.
Give some examples of the uses of Z-values in a normal distribution.
Solve various problems involving normal distribution,and, in particular,
apply these to educational measurement.
Give at least one example of a bivariate and multivariate probability
distributions, respectively.
Illustrate the meaning of marginal probability using the toss of two dice.
Solve problems similar to 5.13, 5.15, and 5.17.
Find and give the interpretation of the value of the covariance between two
random variables.
Define Central Limit Theorem, and state its importance in statistical
experiments.
Demonstrate an understanding of the characteristics and the use of normal
distributions.
Demonstrate an understanding and use of confidence intervals and point
estimators.
State the importance of large samples in interval estimates.
Demonstrate an understanding of the meaning of Type I and Type II error
statistics, and significance levels or p-values.
Test hypotheses concerning means of data using the Z-statistic and the
t-statistic.
Test hypotheses concerning variance of data using the chi-square statistic.
Compare variances of two normal distributions using the F-statistic.
Fit a straight line to a set of data by using the method of least squares.
Fit the linear model by using matrices.
Determine the correlation coefficient between two sets of data, and
interpret the results.
Topic Outline
1. What Is Statistics (NCDPI 4.1 - 4.4, 4.6)
1.2 Characterizing a set of Measurements: Graphical Methods
(NCDPI 4.1, 4.6)
1.3 Characterizing a set of Measurements: Numerical Methods
(NCDPI 4.1, 4.6)
2. Probability (NCDPI 4.3 - 4.5)
2.5 Calculating the probability of an event: The sample point method
2.6 Tools for use when counting sample points
2.7 Conditional probability and the independent events
2.8 Two laws of probability
2.9 Calculating the probability of an event:
The event-composition method
2.11 Numerical events and random variables; 2.12 Random sampling
3. Discrete Random Variables and
Their Probability Distributions (NCDPI 4.3 - 4.5)
3.1 Basic Definition
3.2 The probability distribution of a discrete random variable
3.3 The expected value of a random variable
3.4 The binomial probability distribution
3.5 The geometric probability distribution
3.6 The negative binomial probability distribution
3.7 The hypergeometric probability distribution
3.8 The Poisson probability distribution
4. Continuous Random Variables and their Probability Distributions
(NCDPI 4.3, 4.4)
4.2 The probability distribution a continuous random variable
4.3 The expected value for a continuous random variable
4.4 The uniform probability distribution
4.5 The normal probability distribution
5. Multivariate Probability Distribution (NCDPI 4.3, 4.4)
5.2 Bivariate and multivariate probability distributions
5.3 Marginal and conditional probability distributions
5.6 Special theorems involving expected values
5.7 The covariance of two random variables
5.9 Multinomial probability distribution
7. Sampling Distributions and the Central Limit Theorem (NCDPI 4.1 - 4.4)
7.2 Sampling distributions related to the normal distribution:
The t, F, and Chi-square distributions. (SDPI 10.4)
7.3 The central limit theorem
7.5 The normal approximation of the central limit theorem
8 Estimation (NCDPI 4.2)
8.2 Some properties of point estimators
8.3 Some common unbiased point estimators
8.4 Evaluating the goodness of a point estimator
8.5-8.9 Confidence intervals
10 Hypothesis Testing (NCDPI 4.1, 4.2, 4.3, 4.4, 4.6)
10.2 Elements of a statistical test
10.3 Common large-sample test
10.4 Type I and Type II error statistics
10.5 Another way to report the results of a statistical test:
Attained significance levels or p-values.
10.6 Some comments on the theory of hypothesis testing
10.7 Testing hypotheses concerning means - Two-tailed and one-tailed
tests: The Z-value and the students t-distribution
10.8 Testing hypotheses concerning variances: The chi-square
distribution
11 Linear Models and Estimation by Least Squares (NCDPI 4.2)
11.1 Introduction 11.2 Linear statistical models
11.3 The method of least squares
11.8 Correlation
11.10 Fitting the linear model by using matrices
11.12 Multiple linear regression
13* The analysis of variance procedure
14* Analysis of categorical data; The chi-square test
* If time permits.
Evaluation
Attendance and Class Participation 10% of final grade
Projects, Quizzes, Assignments 20% of final grade
Tests 40% of final grade
Final exam 30% of final grade
Attendance Policy
Regular class attendance and punctuality in this course are stressed.
More than two unexcused absences may lead to a grade of NC (an F grade).
Note
that this class meets only two times a week (75 min each).
Two absences are equivalent (class time-wise) to a whole week lost!!
Student Classroom Decorum Expectations
To enhance the learning atmosphere of the classroom, students are expected to
dress and behave in a fashion conducive to learning in the classroom. More
specifically, students will refrain from disruptive classroom behavior (i. e.,
talking to classmates, disrespectful responses to teacher instructions;
swearing; wearing clothes that impede academic learning such as but not limited
to, wearing body-revealing clothing and excessively baggy pants; hats/caps,
and/or headdress. Students will turn off telephones prior to entering the
classroom. Students who exhibit the behaviors described above, or similar
behaviors will be immediately dismissed from class at the third documented
offense. The student will be readmitted to class only following a decision by
the department chair. The student may appeal the decision of the department
chair to the Dean of the College offering the course, and, subsequently, to the
Office of the Vice President for Academic Affairs, and then to the President of
Shaw University. The decision of the President will be final. Failure to follow
the procedures herein outlined will result in termination of the appeal, and
revert to the decision of the department chair.
Each behavior construed by the teacher/professor as noncontributive to learning
will be recorded, properly documented, and appropriately reported to the student
and to the chair of the academic department offering the course. The report will
be in written form with a copy provided to both the student and the department
chair. The faculty member should retain a copy for his/her own records.
Additional student behavior codes may be found in Student Affairs.