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 Procedure

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.
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