# Math 302

## Introduction to Probability and Statistics II

### Study guide for the final exam

For the final examination, you may use your textbook. This will primarily be so you can access tables and formulae. Do not memorize the formula for the information in the Rao-Cramer lower bound or for the various probability distributions, for example. However, you should know how to do most problems without needing to spend the time to look for similar examples in the textbook. Also, topics such as the bootstrap that were discussed in class but not in the textbook are fair game for the final examination.
1. Sampling:
• Be prepared to answer at most one question on sampling techniques.
• Review different types of bias associated with various sampling methods.
• Review the handout on sampling from the textbook "Statistics" by Freedman, Pisani, and Purves.
• Know the central limit theorem and characteristics of populations that affect the sample size necessary for the sampling distribution to become approximately normal.
2. Estimation:
• Know what the likelihood function is.
• Know how to find maximum likelihood estimates.
• Know how to define and evaluate properties of estimators, such as bias, standard error, and mean square error.
• Know the Rao-Cramer lower bound for unbiased estimators and the conditions under which it holds.
• Know how to calculate the information of a parameter.
• Know how to find a confidence interval for a population mean, and know when the t distribution and the normal distributions are appropriate.
• Review using the bootstrap (both nonparametric and parametric) to estimate standard errors or confidence intervals.
3. Hypothesis Testing:
• Know the basic terminology including the terms null hypothesis, alternative hypothesis, p-value, test statistic, type I and type II errors, power, simple and composite hypotheses, etcetera.
• Know how to do standard problems about a single population mean or comparisons between population means.
• Be able to sketch the power of various hypothesis tests.
• Know how to do a likelihood ratio test.
• Recall how to calculate the Wilcoxon test statistic and how to find a p-value in principle in a test for the median of a distribution.
• Recall how to calculate the Wilcoxon test statistic for comparisons between two populations and know in principle how to compute a p-value.
• Recall the sign test for a median.
• Recall how to use the permutation test in various situations.
4. Goodness-of-fit Tests:
• Know how to apply the chi-square goodness-of-fit tests to test the agreement with sample data with hypothesized probability distributions.
• Know in principle how to test for goodness-of-fit with a likelihood ratio test.
• Know how to calculate the Kolmogorov-Smirnov goodness-of-fit test statistic and how to compute a p-value in principle.
• Know how to do a chi-square test to test the independence of two discrete random variables.
5. Regression:
• Know how to interpret S-PLUS regression output.
• Know how to examine residual plots to look for deviation from regression assumption.
• Know the formula for the least squares line slope and intercept in terms of the correlation coefficient and the standard deviations.
6. Analysis of Variance:
• Know how to interpret S-PLUS ANOVA output.
• Know how to complete and ANOVA table from summary sample data.
• Know the assumptions for the F statistic to have an F distribution.
• Know how to find the various sums of squares and how to find their expectations.
Good studying! You can expect a mixture of numerical problems, derivations, and short answer questions.