Presentation on theme: "Panel at 2013 Joint Mathematics Meetings"— Presentation transcript:
1Panel at 2013 Joint Mathematics Meetings Randomization and Bootstrap Methods in the Introductory Statistics CourseKari Lock Morgan Robin LockDuke University St. Lawrence UniversityPanel at 2013 Joint Mathematics MeetingsSan Diego, CA
2Revised Curriculum Data production (samples/experiments) Descriptive Statistics – one and two samplesBootstrap confidence intervalsRandomization-based hypothesis testsNormal and t-based inferenceChi-square, ANOVA, Regression
3Why start with Bootstrap CI’s? Minimal prerequisites:Population parameter vs. sample statisticRandom samplingDotplot (or histogram)Standard deviation and/or percentilesNatural progression/questionSample estimate ==> How accurate is the estimate?Same method of randomization in most casesSample with replacement from original sampleIntervals are more useful?A good debate for another session…
4What new content is needed to teach bootstrapping?
5Bootstrapping Why does this work? Key ideas: Sample with replacement from the original sample using the same sample size.Compute the sample statistic.Collect lots of such bootstrap statistics.Use the distribution of bootstrap statistics to assess the sampling variability of the statistic.Why does this work?
6Sampling Distribution PopulationBUT, in practice we don’t see the “tree” or all of the “seeds” – we only have ONE seedRobin
7Bootstrap Distribution What can we do with just one seed?Bootstrap“Population”Estimate the distribution and variability (SE) of 𝑥 ’s from the bootstrapsGrow a NEW tree!Robin𝑥
8Golden Rule of Bootstraps The bootstrap statistics are to the original statisticasthe original statistic is to the population parameter.Robin
9How does teaching with randomization/bootstrap methods change technology needs?
10Desirable Technology Features Ability to simulate one to many samplesVisual display of resultsHelp students distinguish and keep straight the original data, a single simulated data set, and the distribution of simulated statisticsAllow students to interact with the bootstrap/randomization distributionConsistent interface for different parameters, tests, and intervals
12Example: Find a 95% confidence interval for the slope when using the size of bill to predict tip at a restaurant.Data: n=157 bills at First Crush Bistro (Potsdam, NY)𝑇𝑖𝑝 =− ∙𝐵𝑖𝑙𝑙r=0.915
13Example: Mustang Prices Data: Sample of 25 Mustangs listed on Autotrader.comFind a confidence interval for the slope of a regression line to predict prices of used Mustangs based on their mileage.
14How does the use of randomization/bootstrap methods for statistical inference change the assessments used?
15Assessment with Technology Given a question and corresponding data:Generate and interpret a CIGenerate a p-value and make a conclusionHow is this assessment different?Answers will vary slightlyTip: ask students to include a screenshot of their bootstrap/randomization distributionOR provide a distribution and ask students to label the x-axis according to their bootstrap/randomization distribution
16Assessment: ProjectsRelatively early in the course, students can do confidence intervals and hypotheses tests for many different parameters!Can find their own data and pick their own parameter of interest
17Assessment: Free Response Give students context and a picture of a bootstrap distribution and ask them to…Explain how to generate one of the dotsEstimate the sample statisticEstimate the standard errorUse these estimates to calculate a 95% CIOR have them estimate a 90% (or other) CIInterpret the CI in context
18Assessment: Free Response Give students context, the sample statistic, and a picture of a randomization distribution and ask them to…State the null and alternative hypothesesExplain how to generate one of the dotsEstimate the p-valueUse the p-value to evaluate strength of evidence against H0 / for HaUse the p-value to make a formal decisionMake a conclusion in context
19Assessment TipsShow bootstrap or randomization distributions as dotplots with a manageable number of dotsOR have students circle relevant part of distribution
21Assessment: Multiple Choice You have sample data on weight consisting of these data values: 121, 136, 160, 185, 203Is each of the following a valid bootstrap sample?121, 121, 160, 185, 203 (a) Yes (b) No121, 121, 136, 160, 185, 203 (a) Yes (b) No121, 160, 185, (a) Yes (b) No121, 142, 160, 185, 190 (a) Yes (b) No
22Assessment: Multiple Choice If would changing the following aspects of the study or analysis, change the confidence interval:Increase the sample size?Increase the number of bootstrap samples?Increase the confidence level?(a) It would get wider(b) It would get narrower(c) It would stay about the sameCan do similar questions for SE, p-value, etc.
23Assessment: Multiple Choice Randomizing in a randomized experiment breaks the link between(a) explanatory and response variables(b) explanatory and confounding variables(c) response and confounding variablesRe-randomizing (reallocating) in a randomization test breaks the link between
24Assessment: Connecting with Traditional Once students have learned formulas for standard errors….Give context, summary statistics, and an unlabeled bootstrap/randomization distribution, then ask students to label at least 3 points on the x-axis
25Conceptual Assessment Almost all conceptual assessment items used in the past regarding confidence intervals and hypothesis tests still work!The concepts we want our students to understand are the same!