Approved by University Studies Sub-Committee.  A2C2 action pending.

University Studies Course Approval Proposal

 

Flagged Courses Writing

 

 

The Department of Mathematics and Statistics proposes the following course for inclusion in University Studies, Flagged Courses, Mathematics and Statistics at Winona State University.  This was approved by the full department on Thursday, January 18, 2001.

 

 

Course: Introduction to Mathematical Statistics II (STAT 460), 3 S.H.

 

Catalog Description: A mathematical approach to probability and statistics.  Prerequisite:   STAT 450, MATH 260, and completion or concurrent enrollment in MATH 220.

 

Department Contact Person for this course:

      Christopher J. Malone

      Department of Mathematics and Statistics

      Email:  cmalone@winona.edu

 

Comment:  This is the second course of a two-semester sequence that is a requirement for a major in statistics.   A writing flag proposal for the first course, Introduction to Mathematical Statistics I (STAT 450), is being submitted separately.  

 

 

Writing Flag

 

The purpose of the Writing Flag requirement is to reinforce the outcomes specified for the basic skills area of writing. These courses are intended to provide contexts, opportunities, and feedback for students writing with discipline-specific texts, tools, and strategies. These courses should emphasize writing as essential to academic learning and intellectual development.

 

Courses can merit the Writing Flag by demonstrating that section enrollment will allow for clear guidance, criteria, and feedback for the writing assignments; that the course will require a significant amount of writing to be distributed throughout the semester; that writing will comprise a significant portion of the students’ final course grade; and that students will have opportunities to incorporate readers’ critiques of their writing.

 

 


The writing flag requirement in relation to STAT 460 

 

… to reinforce the outcome for the basic skills area of writing

 

An important writing skill for a student of statistics is that of summarizing the general strategy of a result or derivation in a non-rigorous fashion.  In particular, a student within the field of statistics must be able to communicate complex derivations to non-statisticians.  A student will receive several opportunities to engage in such writing throughout the semester.  In each case, the student will receive feedback on the appropriateness their writing style.

 

... to provide contexts, opportunities, and feedback for students writing with discipline-specific texts, tools, and strategies

 

In addition to summarizing a general strategy in a non-rigorous fashion, a student is frequently asked to engage in discipline-specific writing commonly found in mathematical statistics.  The discipline-specific writings often involve rigorous proofs of various theorems, lemmas, and corollaries.  Such writings require a student to go beyond the basic understanding of some previously discussed concept.   The student will receive feedback on the appropriateness of their writing style.  However, unlike the non-rigorous writings, these writings are more technical in nature and may be understandable only to a mathematical statistician.

 

 

The types of writing discussed above require the student to extend their basic understanding of a particular concept.  The student is asked repeatedly to convey complex concepts in a precisely written format. 

 

 


Details to how STAT 460 satisfies the writing flag requirements  

 

Writing Flag courses must include requirements and learning activities that promote students’ abilities to:

 

a.      practice the processes and procedures for creating and completing successful writing in their fields

 

This course is the second of a two-semester sequence in which a rigorous introduction to theoretical statistics is established.  To successfully complete this course, a student is required to demonstrate not only an understanding of the concepts involved in a derivation, but also an ability to convey those concepts in concisely written form.  The student is asked to communicate concepts in a non-rigorous writing style with the intent of being understood by the lay-person.  In addition to these non-rigorous writings, proof writing techniques are communicated and these writings are most likely accessible only to mathematical statisticians.    These writing styles require a complete understanding of the concepts involved, knowledge of good grammatical structure, and knowledge of good sentence structure.   The student will receive several opportunities to engage in these writing styles throughout the semester.  In each case, the student will receive significant feedback on the appropriateness of their writing style.

 

 

b.      understand the main features and uses of writing in their fields

 

A student of statistics must be able to communicate concepts to a non-practitioner of statistics, as well as articulate their understanding of a difficult concept to an individual with significant training in mathematical statistics.  The material covered in this course naturally allows the student to participate in commonly found writing styles found within the field of Statistics.   The instructor will spend a great deal of time discussing the technical correctness and appropriateness of the different writing styles.

 

 

c.      adapt their writing to the general expectations of readers in their fields

 

As previously mentioned, the student will receive ample opportunities to engage in various writing styles throughout this required course sequence.   Upon getting continual feedback form the instructor as the semester progresses, the student’s ability to write effectively should improve and eventually be well-aligned with common expectations within the field of Statistics.

 

 

 

 


d.      make use of technologies commonly used for research and writing in their fields

This course introduces students to theoretical statistics and so the use of software packages is somewhat limited.  However, several statistical concepts taught in this course can be elaborated upon through the use of statistical software packages.  These elaborations are left to the discretion of the instructor.  In addition to statistical software packages, students should be exposed to equation editors which are commonly found in word processing packages.

 

e.      learn the conventions of evidence, format, usage, and documentations in their fields

 

The usage of language and formatting issues is considerable different when writing to a non-statistician versus an individual with considerable training in mathematical statistics.  The instructor will address such issues and provide feedback to the students as needed.

 

 

 

 


Writing flag requirements in relation to the course outline

 

Writing Flag:

 

Writing flag courses must include requirements and learning activities that promote students' abilities to...

 

a.      practice the processes and procedures for creating and completing successful writing in their fields;

b.      understand the main features and uses of writing in their fields;

c.      adapt their writing to the general expectations of readers in their fields;

d.      make use of the technologies commonly used for research and writing in their fields; and

e.      learn the conventions of evidence, format, usage, and documentation in their fields.

 

Outline of Topics

 

A.      Sampling Distributions including Cauchy, Chi-Square, F, and t distributions (a),(b),(c),(d),(e)

B.     Law of Large Numbers and Central-limit Theorem (a),(b),(c),(d),(e)

C.     Order Statistics (a),(b),(c),(d),(e)

D.     Asymptotic Distributions (a),(b),(c),(d),(e)

E.     Least Squares Estimation (a),(b),(c),(d),(e)

F.      Method of Moments (a),(b),(c),(d),(e)

G.     Maximum Likelihood (a),(b),(c),(d),(e)

H.     Mean-squared Error, Bias, Consistency, Loss and Risk Functions (a),(b),(c),(d),(e)

I.          Sufficiency (a),(b),(c),(d),(e)

J.       Unbiased Estimation including BLUE s and UMVUEs (a),(b),(c),(d),(e)

K.     Bayes Estimation (a),(b),(c),(d),(e)

L.      Confidence Intervals (a),(b),(c),(d),(e)

M.     Statistical Theory behind Hypothesis Testing including Power (a),(b),(c),(d),(e)

N.     Composite Hypotheses (a),(b),(c),(d),(e)

O.     Tests on Means and Variances (a),(b),(c),(d),(e)

P.      Tests of Goodness of Fit (a),(b),(c),(d),(e)

Q.    Likelihood Ratio Estimation (a),(b),(c),(d),(e)

R.     Sequential Testing (a),(b),(c),(d),(e)

Additional topics as time permits

 

Basic instructional Plan: The basic methods of instruction will be lecture and discussion.

 

Course requirements and means of evaluation: Possible methods include homework problems (a),(b),(c),(d),(e), examinations (a),(b),(c),(e), quizzes, and/or a final examination (a),(b),(c),(d),(e)

 

 


Possible textbooks:

         Casella, George & Berger, Roger.  Statistical Inference.  Wadsworth/Brooks-Cole, 1990.

         Bain, Lee J. & Engelhardt, Max.  Introduction to Probability and Mathematical Statistics.  PWS-KENT, 1992.

         Hogg, Robert V. & Craig, Allen T.  Introduction to Mathematical Statistics (Fifth Edition).Prentice Hall, 1995

         Larsen, Richard J. & Marx, Morris L.  An Introduction to Mathematical Statistics and its Applications (Third Edition).    Prentice Hall, 2001.

 

 


An Example:

 

In accordance with criteria a., b., c., d., and e., this course requires the student to write using writing styles commonly found within the field of Statistics.  A student may be asked to discuss a general strategy or derivation to a non-statistician.  On the other hand, a student may be asked to provide a rigorous proof of a concept that may be accessible only to a mathematical statistician.

 

 

Example:  In class we derived a test for a single population mean along with the appropriate reference distribution for our test statistic.  We also discussed how to use this reference distribution to obtain a confidence interval for the true population mean.

 

 

Part I: Discuss the difference between a hypothesis test and a conference interval so that a non-statistician would understand.

 

The population mean is the average measurement for everyone in a particular population.  For example, if the population of interest is all Winona State students and their grade point average was the variable of interest, then the average GPA  of all Winona State students would be the population mean. 

 

A hypothesis test allows one to make a simple yes/no decision about a particular statement (or hypothesis) involving a population mean.  This procedure does not require collecting everyone’s GPA, but uses information from a subset of the population.   For example, suppose we wanted to know whether or not the average GPA of all WSU students was greater than 2.75.  After reviewing information from your subset of students, we would be able to say either: 1) yes, there is enough evidence to conclude the average GPA of WSU students is higher than 2.75, or 2) no, there is not enough evidence to say average GPA of WSU students is higher than 2.75.

 

On the other hand, a confidence interval does not require a pre-determined statement about the population mean.  A confidence interval gives a likely range of values where we would expect to find the population mean.  Again, this range of values is determined using a subset of WSU students.  For example, a confidence interval might indicate that average GPA of all students at WSU is between 2.8 and 3.0.

 

 

 


Part II: Derive the distribution of the test statistic for testing a single population mean.

 

 

Claim:  The reference distribution of the test statistic for testing a single population mean is a t-distribution with degrees of freedom equal to n-1.   That is, the quantity follows a t-distribution with (n-1) degrees of freedom.

 

 

Proof:

Recall these previously derived results: (i)  If  and , then , (ii), the distribution of   is , and (iii) the distribution of  is  .  We must show that the quantity  has a reference distribution given by . 

Now, realize that  can be written as

.

By letting and , we find that the test statistic

simplifies to which is known to follow a t-distribution with (n-1) degrees of freedom.   Hence, the reference distribution for the test statistic used in testing a single population mean follows a t-distribution with (n-1) degrees of freedom.