Comprehensive Examination

Parts –I-A and I-B

Part-IA

This is a three-hour closed-book exam.  It covers analytical and theoretical concepts of probability and statistical inference. This exam is given between 9:00 A.M.-12:00 noon. No tablets or cell phones allowed.

Part-IB

This is a three-hour open-book exam.  It covers topics of statistical inference described in items 6-11 of the following list of topics.  Notes, books, tables, and calculators may be used. No tablets or cell phones allowed. The exam is given between 1:00-4:00 P.M. the same day as part I-A.

List of Major Topics in Part 1 A  

The following  covers topics generally from the first ten Chapters of the Mendenhall text including the following:

· Elements of probability.  Review conditional  probabilities and independent event properties  and be able to sketch proof of Baye’s theorem and apply it to specific problems  Review probability problems assigned for homework.

· Probability distributions: their moments, moment-generating functions, functions of random variables, and transformation of variables.  Review m.g.f and be able to identify them for the different distributions along with their mean and variance.  Review summary in inside back cover of textbook.

· Discrete Probability models: Bernoulli, Binomial, Poisson, Geometric, negative binomial, hyper-geometric, and multinomial distributions.  Review development of means and variances for commonly used functions and review simpler exercises in this chapter.

· Continuous Probability Models: Uniform, Normal, the Gamma family of distributions [including the exponential & χ²];  the Beta-family, Student-t, and Fisher’s F-distributions.  Be familiar with simple proofs from these sections.

· De able to sampling distributions of means, variances, proportions.  Laws of large numbers, and know the Central Limit Theorem as applied to means and proportions.

· Estimation:  Properties of estimators, methods of estimation, point and interval estimation.  Be familiar with simple proofs from these  sections.  Review Type II error and power of a test and be able to find them for specific problems.  Be able to find M.L.E.’s for certain functions.

· Hypothesis testing: Neyman-Pearson lemma, most powerful tests, simple and composite hypothesis tests of means, variances and proportions.  Go over simple proofs and examples  from this section.   Be familiar with simple proofs from homework  problems assigned in these sections.

List of Major Topics in Part 1 B  

Part 1B of this exam is open book and will focus on select problems taken from Chapters 11through 15 in the Mendenhall text.   Be familiar with all of the following topics even though you may have reviewed them quickly in the course.   Calculators can be used to minimize calculations where needed.  Be sure to define variables clearly along with purpose of problem.  State the test you used and interpret findings relevant to the problem variables.  Br sure hypotheses of tests are clearly stated in terms of problem variables.

· Two sample testing  for differences in  means  (Independent and Dependent samples)

· Analysis of Variance (one-way ANOVA).

· χ² -tests of Goodness of Fit, tests of Independence and Homogeneity.

· Nonparametric methods of inference: One and two-sample Wilcoxon-Mann-Whitney and Sign-tests; Kruskall-Wallis K-sample test;  Runs test, and Spearman’s Correlation.