Math 445—Mathematical Statistics: Review Sheet for Final Exam

The final exam is Tuesday, March 16 at 8:30am. It will cover the textbook material in Chapters 10 (no Section 10.5), 11, and 12 (no Section 12.5), as well as any supplementary material/details I provided in lecture. Even though this test covers only the material in the last half of the class, it draws on some previous results (and you are expected to know the applicable previous results).

Be sure you understand all the homework problems and examples we did in class (if you don’t, please see me in office hours to talk about your questions—I’ll have an office hour Monday, 1:30 – 3). Furthermore, for the exam you must know the definitions (conceptual and formulaic), terminologies, and notations in your bones (that is, there won’t be time to leisurely recall definitions during the exam). You must actively practice for the exam. That is, do not simply read through your solutions to lecture examples and homework problems. Actually re-work through the problems. (You can do additional textbook problems for extra practice—some answers are included in the back of the book, and you can talk with me if you have questions.)

Specific Topics

• Definitions of and properties of  and (both are unbiased); When sampling from a  distribution  is , and  is Chi-squared with (n – 1) degrees of freedom—furthermore,  and  are independent

• Definition of a t random variable and using the definition to show that a random variable has a t distribution; Using Table A.5 to find areas and percentiles; Definition of an F random variable and using the definition to show that a random variable has a F distribution; Using Table A.9 to find areas and percentiles

• Two-sample t-test and confidence interval (no assumption of equal variances) for a difference in population means; two-sample z-test for a difference in population proportions (using the pooled estimate in the standard error); paired t-test; in all of these situations, 1) recognize situation, 2) define hypotheses, 3) check conditions, 4) determine test statistic, 5) determine p-value, 6) define the p-value in the context of the problem and provide a conclusion in the context of the problem, and 7) if there is statistically significance, investigate (via a confidence interval and whatever knowledge of the problem you might have) the practical significance.

• Relationship between confidence intervals and significance testing; interpretation of confidence

• Bootstrap idea and bootstrap (percentile) confidence interval for a difference in means or for a mean of paired data

• One-factor ANOVA model: Definition of the model (including conditions), definitions of the sums of squares (SSTr, SSE, SST), degrees of freedom for each sum-of-squares, definition of mean squares, definition of the null hypothesis and use of the F statistic to test that hypothesis, understanding of the issue of multiple comparisons (and adjusting the error rate)

• One-factor ANOVA data analysis: Check of conditions, interpretation of the p-value on the F statistic, ability to interpret Tukey’s confidence intervals (adjusted for multiple comparisons) to find where the specific significant differences are, discussion of possible practical significance

• Two-factor ANOVA model: Definition of the model (including conditions), definition of null hypotheses and use of different F statistics to test the hypotheses

• Two-factor ANOVA data analysis: Check conditions, interpretation of the interaction effect (if significant) via a plot and words, interpretation of main effects (if significant) via Tukey’s intervals (indicating where the significant differences are, whether the results are practically significant, and how these main-effects feed into the interaction effect—if there is an interaction effect)

• In both the one-factor and two-factor ANOVA situation, working with sums and expectations of sums (like in the homework and in the derivations we did in class)

• Simple linear regression model: Definition of the model (including conditions); idea of least-squares estimation; derivations of the mean, variance, and distribution of the slope estimator, and the use of these results to create and interpret a significance test (or confidence interval) for the population slope; confidence interval for a mean response and prediction interval for a new response (if I ask you to do derivations—not just give interpretations—then I’ll provide the standard error formulas)

• Simple linear regression analysis: Check conditions, check significance of the slope (possibly determine a confidence interval to assess the practical significance), interpret the value of the slope, interpret the value of , interpret a confidence interval for a mean or a prediction interval for a new value (depending on which is most applicable to the research question)