INTUITIVE INTRODUCTION TO THE IMPORTANT IDEAS OF INFERENCE

Concepts of statistical inference, such as margin of error when estimating a parameter and p-value when testing a hypothesis, are notoriously difficult for students to grasp. In traditional approaches, these ideas typically come as the culmination of a long development of prerequisite material on sampling distributions, formulas for standard errors, standard reference distributions, central limit theorems, and formulas for standardizing values. Simulation methods, such as bootstrap intervals and randomization tests, require minimal background knowledge and highlight the underlying logic of statistical inference, giving students an intuitive appreciation for the key ideas early in a course. But are such methods accessible and understandable to beginning students? We argue that advances in technology make this approach both feasible and desirable. So how does one go about modifying a course to incorporate these ideas? That question is the main focus of this paper. BACKGROUND George Cobb in his address to the first United States Conference On Teaching Statistics (USCOTS 2005) and subsequent lead paper in the first edition of TISE (Cobb 2007) challenged statistics educators to move away from reliance on inferential methods based on standard normal and t-distributions and embrace simulation-based randomization methods as a more authentic way to introduce the core ideas of statistical inference. Also, these methods are now part of new Common Core Standards for Mathematics in the United States. In a frequently quoted conclusion to his paper, Cobb states “Before computers, there was no alternative. Now, there is no excuse. ” But many statistics teachers still had some “excuses”. Where could we find guidance on how to implement such an approach within an introductory curriculum? What about materials (textbooks, activities, sample assessments, etc.) to support these methods? Is there technology that is easily accessible and appropriate for use with beginning students? Should we use simulation methods exclusively or still include formula-based methods using traditional approximating distributions? How should simulation methods fit within a syllabus for an entire course? We address some of these questions below, addressing first What, then Why, then How. WHAT SIMULATION METHODS SHOULD WE TEACH? We broadly categorize inference questions in an introductory statistics course into two groups: estimation and testing. These two types of inference lead naturally to two general classes of simulation procedures: bootstrapping to estimate sampling errors and create confidence intervals, and randomization tests to measure strength of evidence and find p-values for significance tests. Bootstrap Distributions for Estimating Standard Error and Creating Confidence Intervals Suppose we have a sample statistic estimating a population parameter. We'd like to assess how accurate that statistic is likely to be for our sample size and population. Generating multiple samples from the population is generally not feasible, so instead we generate multiple bootstrap samples from the original sample. To generate one bootstrap sample we choose values with replacement from the original sample using the same sample size. We then compute a bootstrap statistic for that sample. Repeating this process for many (1000's) of bootstrap samples gives a bootstrap distribution, which we can use to measure the variability in the sample statistic. We show students two different ways to produce a confidence interval from a bootstrap distribution. The first uses the standard deviation of the bootstrap statistics to estimate the standard error (SE) of the statistic. Assuming that the bootstrap distribution is relatively symmetric and bellshaped, we create the interval using