CAUSE Webinar: Introducing Math Majors to Statistics

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CAUSE WebinarIntroducing Math Majors to
Statistics

• Allan Rossman and Beth Chance
• Cal Poly San Luis Obispo
• April 8, 2008

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Outline

• Goals
• Guiding principles
• Content of an example course
• Assessment
• Examples (four)

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Goals

• Redesign introductory statistics course for
  mathematically inclined students in order to
• Provide balanced introduction to the practice of
  statistics at appropriate mathematical level
• Better alternative than Stat 101 or Math Stat
  sequence for math majors first statistics course

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Guiding principles (Overview)

1. Put students in role of active investigator
2. Motivate with real studies, genuine data
3. Repeatedly experience entire statistical process
   from data collection to conclusion
4. Emphasize connections among study design,
   inference technique, scope of conclusions
5. Use variety of computational tools
6. Investigate mathematical underpinnings
7. Introduce probability just in time

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Principle 1 Active investigator

• Curricular materials consist of investigations
  that lead students to discover statistical
  concepts and methods
• Students learn through constructing own
  knowledge, developing own understanding
• Need direction, guidance to do that
• Students spend class time engaged with these
  materials, working collaboratively, with
  technology close at hand

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Principle 2 Real studies, genuine data

• Almost all investigations focus on a recent
  scientific study, existing data set, or student
  collected data
• Statistics as a science
• Frequent discussions of data collection issues
  and cautions
• Wide variety of contexts, research questions

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Real studies, genuine data

• Popcorn and lung cancer
• Historical smoking studies
• Night lights and myopia
• Effect of observer with vested interest
• Kissing the right way
• Do pets resemble their owners
• Who uses shared armrest
• Halloween treats
• Heart transplant mortality

• Lasting effects of sleep deprivation
• Sleep deprivation and car crashes
• Fan cost index
• Drive for show, putt for dough
• Spock legal trial
• Hiring discrimination
• Comparison shopping
• Computational linguistics

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Principle 3 Entire statistical process

• First two weeks
• Data collection
• Observation vs. experiment (Confounding, random
  assignment vs. random sampling, bias)
• Descriptive analysis
• Segmented bar graph
• Conditional proportions, relative risk, odds
  ratio
• Inference
• Simulating randomization test for p-value,
  significance
• Hypergeometric distribution, Fishers exact test
• Repeat, repeat, repeat,
• Random assignment ? dotplots/boxplots/means/media
  ns ? randomization test
• Sampling ? bar graph ? binomial ? normal
  approximation

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Principle 4 Emphasize connections

• Emphasize connections among study design,
  inference technique, scope of conclusions
• Appropriate inference technique determined by
  randomness in data collection process
• Simulation of randomization test (e.g.,
  hypergeometric)
• Repeated sampling from population (e.g.,
  binomial)
• Appropriate scope of conclusion also determined
  by randomness in data collection process
• Causation
• Generalizability

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Principle 5 Variety of computational tools

• For analyzing data, exploring statistical
  concepts
• Assume that students have frequent access to
  computing
• Not necessarily every class meeting in computer
  lab
• Choose right tool for task at hand
• Analyzing data statistics package (e.g.,
  Minitab)
• Exploring concepts Applets (interactivity,
  visualization)
• Immediate updating of calculations spreadsheet
  (Excel)

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Principle 6 Mathematical underpinnings

• Primary distinction from Stat 101 course
• Some use of calculus but not much
• Assume some mathematical sophistication
• E.g., function, summation, logarithm,
  optimization, proof
• Often occurs as follow-up homework exercises
• Examples
• Counting rules for probability
• Hypergeometric, binomial distributions
• Principle of least squares, derivatives to find
  minimum
• Univariate as well as bivariate setting
• Margin-of-error as function of sample size,
  population parameters, confidence level

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Principle 7 Probability just in time

• Whither probability?
• Not the primary goal
• Studied as needed to address statistical issues
• Often introduced through simulation
• Tactile and then computer-based
• Addressing how often would this happen by
  chance?
• Examples
• Hypergeometric distribution Fishers exact test
  for 22 table
• Binomial distribution Sampling from random
  process
• Continuous probability models as approximations

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Content of Example Course (ISCAM)
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, Fishers 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
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Assessments

• Investigations with summaries of conclusions
• Worked out examples
• Practice problems
• Quick practice, opportunity for immediate
  feedback, adjustment to class discussion
• Homework exercises
• Technology explorations (labs)
• e.g., comparison of sampling variability with
  stratified sampling vs. simple random sampling
• Student projects
• Student-generated research questions, data
  collection plans, implementation, data analyses,
  report

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Example 1 Friendly Observers

• Psychology experiment
• Butler and Baumeister (1998) studied the effect
  of observer with vested interest on skilled
  performance

A vested interest B no vested interest Total
Beat threshold 3 8 11
Do not beat threshold 9 4 13
Total 12 12 24
How often would such an extreme experimental
difference occur by chance, if there was no
vested interest effect?
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Example 1 Friendly Observers

• Students investigate this question through
• Hands-on simulation (playing cards)
• Computer simulation (Java applet)
• Mathematical model
• counting techniques

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Example 1 Friendly Observers

• Focus on statistical process
• Data collection, descriptive statistics,
  inferential analysis
• Arising from genuine research study
• Connection between the randomization in the
  design and the inference procedure used
• Scope of conclusions depends on study design
• Cause/effect inference is valid
• Use of simulation motivates the derivation of the
  mathematical probability model
• Investigate/answer real research questions in
  first two weeks

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Example 2 Sleep Deprivation

• Physiology Experiment
• Stickgold, James, and Hobson (2000) studied the
  long-term effects of sleep deprivation on a
  visual discrimination task

(3 days later!)
sleep condition n Mean StDev Median IQR
deprived 11 3.90 12.17 4.50
20.7 unrestricted 10 19.82 14.73 16.55
19.53
How often would such an extreme experimental
difference occur by chance, if there was no sleep
deprivation effect?
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Example 2 Sleep Deprivation

• Students investigate this question through
• Hands-on simulation (index cards)
• Computer simulation (Minitab)
• Mathematical model

p-value.0072
p-value? .002
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Example 2 Sleep Deprivation

• Experience the entire statistical process again
• Develop deeper understanding of key ideas
  (randomization, significance, p-value)
• 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

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Example 3 Infants Social Evaluation

• Sociology study
• Hamlin, Wynn, Bloom (2007) investigated whether
  infants would prefer a toy showing helpful
  behavior to a toy showing hindering behavior
• Infants were shown a video with these two kinds
  of toys, then asked to select one
• 14 of 16 10-month-olds selected helper
• Is this result surprising enough (under null
  model of no preference) to indicate a genuine
  preference for the helper toy?

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Example 3 Infants Social Evaluation

• Simulate with coin flipping
• Then simulate with applet

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Example 3 Infants Social Evaluation

• Then learn binomial distribution, calculate exact
  p-value

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Example 3 Infants Social Evaluation

• Learn probability distribution to answer
  inference question from research study
• Again the analysis is completed with
• Tactile simulation
• Technology simulation
• Mathematical model
• Modeling process of statistical investigation
• Examination of methodology, further questions in
  study
• Follow-ups
• Different number of successes
• Different sample size

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Example 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 nights sleep in past week At least one full nights sleep in past week Sample sizes
case drivers (crash)  61 510  571
control drivers (no crash) 44  544  588
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Example 4 Sleepless Drivers
Sample proportion that were in a car crash Sleep
deprived .581 Not sleep deprived .484
Odds ratio 1.48
How often would such an extreme observed odds
ratio occur by chance, if there was no sleep
deprivation effect?
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Example 4 Sleepless Drivers

• Students investigate this question through
• Computer simulation (Minitab)
• Empirical sampling distribution of odds-ratio
• Empirical p-value
• Approximate mathematical model

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Example 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 1.05 2.08

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Example 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

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For more information

• Investigating Statistical Concepts, Applications,
  and Methods (ISCAM), Cengage Learning,
  www.cengage.com
• Instructor resources www.rossmanchance.com/iscam/
  
• Solutions to investigations, practice problems,
  homework exercises
• Instructors guide
• Sample syllabi
• Sample exams