investigations for introducing mathematically inclined students to statistics allan...
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Investigations for Introducing Mathematically Inclined Students to StatisticsAllan Rossman ([email protected])
Beth Chance ([email protected])
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
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 #9950476, 0321973
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 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6
Data Collection Observation vs. experiment, confounding, randomization
Random sampling, bias, precision, nonsampling errors
Paired data Independent random samples
Bivariate
Descriptive Statistics
Conditional proportions, segmented bar graphs, odds ratio
Quantitative summaries, transformations, z-scores, resistance
Bar graph Models, Probability plots, trimmed mean
Scatterplots, correlation, simple linear regression
Probability Counting, random variable, expected value
empirical rule Bermoulli processes, rules for variances, expected value
Normal, Central Limit Theorem
Sampling/ Randomization Distribution
Randomization distribution for
Randomization distribution for
Sampling distribution for X,
Large sample sampling distributions for
,
Sampling distributions of , OR,
Chi-square statistic, F statistic, regression coefficients
Model Hypergeometric Binomial Normal, t Normal, 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
21 ˆˆ pp 21 xx p̂
x p̂21 ˆˆ pp
21 xx
Outline
Example Investigations
Full versions available at www.rossmanchance.com/iscam/uscots/
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 11 3.90 12.17 4.50 20.7unrestricted 10 19.82 14.73 16.55 19.53
(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 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=.0072
15.92
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 Students use Minitab to select sample and
compare results
Example results
Investigation 2: Sampling Words Then turn to technology (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 z vs. t 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) 61 474 535
“control” drivers (no crash) 44 544 588
Investigation 4: Sleepless DriversSample proportion that were in a car crash
Sleep deprived: .581Not 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?
0%10%20%30%40%50%60%70%80%90%
100%
No full night’s sleep in pastweek
At least one full night’ssleep in past week
no crash
crash
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
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1111
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 (2nd 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:www.rossmanchance.com/iscam/
Faculty development workshop (July 18-22, 2005):www.rossmanchance.com/prep/workshop.html
Review copies of text:www.duxbury.com