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MATH 470 Syllabus

Probability and Statistics II

Revised: January 7, 2020

Course Description

Point and interval estimation, hypothesis testing, likelihood ratio and sequential testing, correlation and regression.

Prerequisite: MATH 370

Student Learning Objectives

The following are consistent with the learning objectives established by the Society of Actuaries for the VEE in Mathematical Statisics, effective July 2018.

By the end of the course, students should be able to

  1. Explain the concepts of random sampling, statistical inference and sampling distribution, and state and use basic sampling distributions.
  2. Describe the main methods of estimation and the main properties of estimators, and apply them. Methods include matching moments, percentile matching, and maximum likelihood, and properties include bias, variance, mean squared error, consistency, efficiency and UMVUE.
  3. Construct confidence intervals for unknown parameters, including the mean, difference of two means, variances, and proportions.
  4. Test hypotheses. Concepts to be covered include Neyman-Pearson lemma, significance and power, likelihood ratio test, and information criteria. Tests should include for mean, variance, contingency tables, and goodness-of-fit.

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

Previous chapters and material can be covered as needed.

  • Chapter 7: Sampling Distributions and the Central Limit Theorem (2 weeks)

Introduction. Sampling Distributions Related to the Normal Distribution. The Central Limit Theorem. A Proof of the Central Limit Theorem (Optional). The Normal Approximation to the Binomial Distribution. Summary.

  • Chapter 8: Estimation (3 weeks)

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 (3 weeks)

Introduction. Relative Efficiency. Consistency. Sufficiency. The Rao-Blackwell Theorem and Minimum-Variance Unbiased Estimation. The Method of Moments. The Method of Maximum Likelihood. Some Large-Sample Properties of Maximum-Likelihood Estimators. Summary.

  • Chapter 10: Hypothesis Testing (3 weeks)

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 14: The Analysis of Categorical Data (1 week)

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.