Investigations for Introducing Mathematically Inclined Students to Statistics Allan Beth Chance

Student Audience Introductory statistics course for mathematically inclined students  mathematics and statistics majors  future secondary teachers  perhaps strong science, engineering, computer science majors

Goals for the Course? Brainstorm your goals for these students, particularly with attention to whether and how these goals differ from service courses (5 min) Reporter summarize top three goals

Summary of Goals

Efforts for Math Stat/Prob Sequence Supplement with data analysis component  Witmer’s Data Analysis: An Introduction Infuse data and applications  Rice’s Mathematical Statistics and Data Analysis Use lab activities  Nolan and Speed’s StatLabs  Baglivo’s Mathematica Laboratories for Mathematical Statistics

Our Project To develop and provide a: Data-Oriented, Active Learning, Post-Calculus Introduction to Statistical Concepts, Applications, Theory Supported by the NSF DUE/CCLI # ,

Guiding Principles Put students in the role of active investigator Motivate with real studies, genuine data Emphasize connections among study design, inference technique, scope of conclusions Use simulations frequently Use a variety of computational tools Investigate mathematical underpinnings Introduce probability “just in time” Experience entire statistical process over and over Provide a combination of immediate corrective formative and summative evaluation of key concepts

Chapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6 Data CollectionObservation vs. experiment, confounding, randomization Random sampling, bias, precision, nonsampling errors Paired dataIndependent random samples Bivariate Descriptive Statistics Conditional proportions, segmented bar graphs, odds ratio Quantitative summaries, transformations, z-scores, resistance Bar graphModels, Probability plots, trimmed mean Scatterplots, correlation, simple linear regression ProbabilityCounting, random variable, expected value empirical ruleBermoulli processes, rules for variances, expected value Normal, Central Limit Theorem Sampling/ Randomization Distribution Randomization distribution for Sampling distribution for X, Large sample sampling distributions for, Sampling distributions of, OR, Chi-square statistic, F statistic, regression coefficients ModelHypergeometricBinomialNormal, tNormal, t, log- normal Chi-square, F, t Statistical Inference p-value, significance, Fisher’s Exact Test p-value, significance, effect of variability Binomial tests and intervals, two-sided p- values, type I/II errors z-procedures for proportions t- procedures, robustness, bootstrapping Two-sample z- and t- procedures, bootstrap, CI for OR Chi-square for homogeneity, independence, ANOVA, regression Outline

Example Investigations Full versions available at Investigation 1: Sleep Deprivation and Visual Learning (randomization tests) Investigation 2: Sampling Words (random samples, variability) Investigation 3: Kissing the Right Way (confidence intervals) Investigation 4: Sleepless Drivers (CI for Odds Ratio)

Investigation 1: Sleep Deprivation Physiology Experiment  Stickgold, James, and Hobson (2000) studied the long-term effects of sleep deprivation on a visual discrimination task sleep condition n Mean StDev Median IQR deprived unrestricted (3 days later!)

Investigation 1: Sleep Deprivation How often would such an extreme experimental difference occur by chance, if there was no sleep deprivation effect? Set of 21 index cards with the improvement scores (positive and negative). Randomly assign 11 of the cards to the sleep deprived group. Calculate the difference in group means (deprived – unrestricted)

Investigation 1: Sleep Deprivation After this reminder of the randomization process, students then use a Minitab macroMinitab macro sample 21 c2 c3 unstack c1 c4 c5; subs c3. let c6(k1)=mean(c4)-mean(c5) let k1=k1+1

Investigation 1: Sleep Deprivation Students investigate this question through  Hands-on simulation (index cards)  Computer simulation (Minitab)  Exact distribution p-value= p-value .002

Investigation 1: Sleep Deprivation Experience the entire statistical process again  Develop deeper understanding of key ideas (randomization, significance, p-value)  Effect of variability Tools change, but reasoning remains same  Tools based on research study, question – not for their own sake Simulation as a problem solving tool  Empirical vs. exact p-values

Investigation 2: Sampling Words Four score and seven years ago, our fathers brought forth upon this continent a new nation: conceived in liberty, and dedicated to the proposition that all men are created equal. Now we are engaged in a great civil war, testing whether that nation, or any nation so conceived and so dedicated, can long endure. We are met on a great battlefield of that war. We have come to dedicate a portion of that field as a final resting place for those who here gave their lives that that nation might live. It is altogether fitting and proper that we should do this. But, in a larger sense, we cannot dedicate, we cannot consecrate, we cannot hallow this ground. The brave men, living and dead, who struggled here have consecrated it, far above our poor power to add or detract. The world will little note, nor long remember, what we say here, but it can never forget what they did here. It is for us the living, rather, to be dedicated here to the unfinished work which they who fought here have thus far so nobly advanced. It is rather for us to be here dedicated to the great task remaining before us, that from these honored dead we take increased devotion to that cause for which they gave the last full measure of devotion, that we here highly resolve that these dead shall not have died in vain, that this nation, under God, shall have a new birth of freedom, and that government of the people, by the people, for the people, shall not perish from the earth.

Investigation 2: Sampling Words Examine the average length of words in the sample

Example Class Results The population mean of all 268 words is letters

Investigation 2: Sampling Words Students use Minitab to select sample and compare results Example results

Investigation 2: Sampling Words Then turn to technology (applet)applet  What is the long-term behavior of this (random) sampling method? Unbiased method?  What happens if we change sample size? Population size?

Investigation 2: Sampling Words Using various forms of technology to support student conceptual learning  Tailored to the context  Dynamic, interactive, and visual  Easy to use Confront most common student misconceptions directly Distinguish randomization from random sampling

Investigation 3: Kissing the Right Way Biopsychology observational study  Güntürkün (2003) recorded the direction turned by kissing couples to see if there was also a right- sided dominance.

Investigation 3: Kissing the Right Way Is 1/2 a plausible value for  the probability a kissing couple turns right? Binomial Simulation applet  Introduce idea of two-sided p-value Is 2/3 a plausible value for  the probability a kissing couple turns right?  Discuss calculation of non-symmetric two-sided p- values

Investigation 3: Kissing the Right Way Have students explore and develop an “interval” of plausible values for 

Later Investigations Use another applet to explore the meaning of confidence level  Wald vs. adjusted Wald Wald vs. adjusted Wald  z vs. t  Robustness of t-intervals Robustness of t-intervals

Investigation 3: Kissing the Right Way Encourage students to make predictions and test their knowledge Use the technology to minimize computational burden so students focus on concepts Return to key ideas often, increasing the level of complexity each time Give them a taste for the modern flavor of statistical practice and methodology

Investigation 4: Sleepless Drivers Sociology case-control study  Connor et al (2002) investigated whether those in recent car accidents had been more sleep deprived than a control group of drivers No full night’s sleep in past week At least one full night’s sleep in past week Sample sizes “case” drivers (crash) “control” drivers (no crash)

Investigation 4: Sleepless Drivers Sample proportion that were in a car crash Sleep deprived:.581 Not sleep deprived:.466 Odds ratio: 1.59 How often would such an extreme observed odds ratio occur by chance, if there was no sleep deprivation effect?

Investigation 4: Sleepless Drivers Students investigate this question through  Computer simulation (Minitab) Empirical sampling distribution of odds-ratio Empirical p-value  Approximate mathematical model 1.59

Investigation 4: Sleepless Drivers SE(log-odds) = Confidence interval for population log odds:  sample log-odds + z* SE(log-odds)  Back-transformation 90% CI for odds ratio: 1.13 – 2.24

Investigation 4: Sleepless Drivers Students understand process through which they can investigate statistical ideas Students piece together powerful statistical tools learned throughout the course to derive new (to them) procedures  Concepts, applications, methods, theory

Expectations of Students (Midterm Qs) Issues in sampling, nonsampling errors Understand the implications of improper sampling Analyze data numerically and graphically, communicate their results Be able to explain how random variability affects the conclusions we should draw Verbalize student conclusions that follow based on study design – Causation? Generalizability? Explain the idea behind randomization/sampling distributions, think statistically  Increasing understanding of confidence, p-value

Discussion Are these worthy goals?  Recruiting students into statistics (2 nd course…)  Preparing future teachers Is such a course feasible?  Learning environment  Course structure  Integration of technology What are the essential components in students’ ability and understanding to assess?

For More Information Applets, data files, other resources: Faculty development workshop (July 18-22, 2005): Review copies of text:

Thank you Allan Rossman Beth Chance