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

University Studies Course Approval
Proposal

**Flagged Courses Writing**

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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.

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__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**

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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.

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**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.

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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.

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__Writing
Flag:__

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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)**

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__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.

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__An
Example:__

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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.

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**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.

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

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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.

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

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**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.

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**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.