# MATH 470 Syllabus

### Probability and Statistics II

Revised: September 2014

### Course Description

Point and interval estimation, hypothesis testing, likelihood ratio and sequential testing, correlation and regression. Prerequisite: MATH 370. Three semester hours.

### Student Learning Objectives

By the end of the semester, students will be able to:

• Construct and interpret estimation procedures including confidence intervals;
• Explain phenomena using null and alternative hypotheses involving population parameters and test these hypotheses using routine methods for assessing statistical significance;
• Investigate relationships in univariate and multivariate data using linear models;
• Design experiments to extract optimal information from data;
• Analyze proportions and independence of categorical variables for statistical significance; and
• Connect hypothesis testing procedures with associated sampling distributions, where appropriate.

### Text

Wackerley, Dennis, Mendenhall, William and Richard Scheaffer. Introduction to Mathematical Statistics with Applications, Seventh Edition. Brooks/Cole Cengage, 2008.

Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy.

### Attendance Policy

Attendance policy is left to the discretion of individual instructors, subject to general university policy.

### Course Outline

• Since Chapters 1 through 7 are covered in MATH 370, they should be briefly reviewed as needed.

Chapter 1: Introduction

Chapter 2: Probability

Chapter 3: Discrete Random Variables and Their Probability Distributions

Chapter 4: Continuous Random Variables and Their Probability Distributions

Chapter 5: Multivariate Probability Distributions

Chapter 6: Functions of Random Variables

Chapter 7: Sampling Distributions and the Central Limit Theorem

• Chapter 8: Estimation (5 days)
Introduction,.The Bias and Mean Square of Point Estimators. Some Common Unbiased Point Estimators. Evaluating the Goodness of a Point Estimator. Confidence Intervals. Large-Sample Confidence Intervals. Selecting the Sample Size. Small-Sample Confidence Intervals for $\mu$ and $\mu_1 - \mu_2$. Confidence Intervals for $\sigma^2$. Summary.
• Chapter 9: Properties of Points Estimators and Methods of Estimation (4 days)
Introduction. Relative Efficiency. Consistency. Sufficiency. The Rao-Blackwell Theorem and Minimum-Variance Unbiased Estimation. The Method of Moment. The Method of Maximum Likelihood. Some Large-Sample Properties of Maximum-Likelihood Estimators (Optional). Summary.
• Chapter 10: Hypothesis Testing (8 days)
Introduction. Elements of a Statistical Test. Common Large-Sample Tests. Calculating Type II Error Probabilities and Find the Sample Size for Z Tests. Relationships Between Hypothesis-Testing Procedures and Confidence Intervals. Another Way to Report the Results of a Statistical Test: Attained Significance Levels, or p-values. Some Comments on the Theory of Hypothesis Testing. Small-Sample Hypothesis Testing for $\mu$ and $\mu_1 - \mu_2$. Testing Hypotheses Concerning Variance. Power of Tests and the Neyman-Pearson Lemma. Likelihood Ratio Tests. Summary.
• Chapter 11: Linear Models and Estimation by Least Squares (8 days)
Introduction. Linear Statistical Models. The Method of Least Squares. Properties of the Least-Squares Estimators: Simple Linear Regression. Inferences Concerning the Parameters $\beta_1$. Inferences Concerning Linear Functions of the Model Parameters: Simple Linear Regression. Predicting a Particular Value of $Y$ by Using Simple Linear Regression. Correlation. Some Practical Examples. Fitting the Linear Model by Using Matrices. Linear Functions of the Model Parameters: Multiple Linear Regression. Inferences Concerning Linear Functions of the Model Parameters: Multiple Linear Regression. Predicting a Particular Value of $Y$ by Using Multiple Regression. A Test for $H_0: \beta_{g+1} = \beta_{g+2} = \ldots = \beta+{k} = 0$. Summary and Concluding Remarks .
• Chapter 12: Considerations in Designing Experiments (2 days)
The Elements Affecting the Information in a Sample. Designing Experiments to Increase Accuracy. The Matched-Pairs Experiment. Some Elementary Experimental Designs. Summary.
• Chapter 13: The Analysis of Variance (5 days)
Introduction. The Analysis of Variance Procedure. Comparison of More Than Two Means: Analysis of Variance for a One-Way Layout. An Analysis of Variance Table for a One-Way Layout. A Statistical Model for the One-Way Layout. Proof of Additivity of the Sums of Squares and $E$(MST) for a One-Way Layout. Estimation in the One-Way Layout. A Statistical Model for the Randomized Block Design. The Analysis of Variance for a Randomized Block Design. Estimation in the Randomized Block Design. Selecting the Sample Size. Simultaneous Confidence Intervals for More Than One Parameter. Analysis of Variance Using Linear Models. Summary.
• Chapter 14: Analysis of Categorical Data (3 days)
A Description of the Experiment. The Chi-Square Test. A Test of a Hypothesis Concerning Specified Call Probabilities: A Goodness-of-Fit Test. Contingency Tables. $r \times c$ Tables with Fixed Row or Column Totals. Other Applications. Summary and Concluding Remarks.