Teaching Statistical Concepts with Activities, Data, and Technology

1
Teaching Statistical Concepts with Activities,
Data, and Technology

• Beth L. Chance and Allan J. Rossman
• Dept of Statistics, Cal Poly San Luis Obispo

2
Goals

• Acquaint you with recent recommendations and
  ideas for teaching introductory statistics
• Including some very modern approaches
• On top of some issues we consider essential
• Provide specific examples and activities that you
  might plug into your courses
• Point you toward online and print resources that
  might be helpful

3
Schedule

• Introductions
• Opening Activity
• Activity Sessions
• Data Collection
• Data Analysis
• ltlt lunchgtgt
• Randomness
• Statistical Inference
• Resources and Assessment
• QA, Wrap-Up

4
Requests

• Participate in activities
• 23 of them!
• Well skip/highlight some
• Play role of student
• Good student, not disruptive student!
• Feel free to interject comments, questions

5
GAISE

• Emphasize statistical literacy and develop
  statistical thinking
• Use real data
• Stress conceptual understanding rather than mere
  knowledge of procedures
• Foster active learning in the classroom
• Use technology for developing conceptual
  understanding and analyzing data
• Use assessments to improve and evaluate student
  learning
• www.amstat.org/education/gaise

6
Opening Activity

• Naughty or nice? (Nature, 2007)
• Videos http//www.yale.edu/infantlab/socialevalua
  tion/Helper-Hinderer.html
• Flip 16 coins, one for each infant, to decide
  which toy you want to play with (headshelper)
• Coin Tossing Applet http//www.rossmanchance.com/
  applets

7
3S Strategy

• Statistic
• Simulate
• Could have been distribution of data for each
  repetition (under null model)
• What if distribution of statistics across
  repetitions (under null model)
• Strength of evidence
• Reject vs. plausible

8
Summary

• Use real data/scientific studies
• Emphasize the process of statistical
  investigation
• Stress conceptual understanding
• Idea of p-value on day 1/in one day!
• Foster active learning
• You are a dot on the board
• Use technology
• Could this have happened by chance alone?
• What if only 10 infants had picked the helper?

9
Data Collection Activities Activity 2 Sampling
Words

• Circle 10 representative words in the passage
• Record the number of letters in each word
• Calculate the mean number of letters in your
  sample
• Dotplot of results

10
Sampling Words

• The population mean of all 268 words is 4.295
  letters
• How many sample means were too high?
• Why do you think so many sample means are too
  high?

11
Sampling Words

• Tactile simulation
• Ask students to use computer or random number
  table to take simple random samples
• Determine the sample mean in each sample
• Compare the distributions

12
Sampling Words

• Java applet
• www.rossmanchance.com/applets/
• Select Sampling words applet
• Select individual sample of 5 words
• Repeat
• Select 98 more samples of size 5
• Explore the effect of sample size
• Explore the effect of population size

13
Morals Selecting a Sample

• Random Sampling eliminates human selection bias
  so the sample will be fair and unbiased/representa
  tive of the population.
• While increasing the sample size improves
  precision, this does not decrease bias.

14
Activity 3 Night Lights and Near-Sightedness

• Quinn, Shin, Maguire, and Stone (1999)
• 479 children
• Did your child use a night light (or room light
  or neither) before age 2?
• Eyesight Hyperopia (far-sighted), emmetropia
  (normal) or myopia (near-sighted)?

15
Night Lights and Near-Sightedness
Darkness Night light Room light
Near-sighted 18 78 41
Normal refraction 114 115 22
Far-sighted 40 39 12
16
Night Lights and Near-Sightedness
17
Morals Confounding

• Students can tell you that association is not the
  same as causation!
• Need practice clearly describing how confounding
  variable
• Is linked to both explanatory and response
  variables
• Provides an alternative explanation for observed
  association

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Activity 4 Have a Nice Trip

• Can instruction in a recovery strategy improve an
  older persons ability to recover from a loss of
  balance?
• 12 subjects have agreed to participate in the
  study
• Assign 6 people to use the lowering strategy and
  6 people to use the elevating strategy
• What does random assignment gain you?

19
Have a Nice Trip

• Randomizing subjects applet
• How do the two groups compare?

20
Morals

• Goal of random assignment is to be willing to
  consider the treatment groups equivalent prior to
  the imposition of the treatment(s).
• This allows us to eliminate all potential
  confounding variables as a plausible explanation
  for any significant differences in the response
  variable after the treatments are imposed.

21
Activity 5 Cursive Writing

• Does using cursive writing cause students to
  score better on the SAT essay?

22
Morals Scope of Conclusions
The Statistical Sleuth, Ramsey and Schafer
Allocation of units to groups Allocation of units to groups
By random assignment No random assignment
Selection of units Random sampling A random sample is selected from one population units are then randomly assigned to different treatment groups Random samples are selected from existing distinct populations Inferences to populations can be drawn
Selection of units Not random sampling A groups of study units is found units are then randomly assigned to treatment groups Collections of available units from distinct groups are examined
Cause and effect conclusions can be drawn
23
Activity 6 Memorizing Letters

• You will be asked to memorize as many letters as
  you can in 20 seconds, in order, from a sequence
  of 30 letters
• Variables?
• Type of study?
• Comparison?
• Random assignment?
• Blindness?
• Random sampling?
• More to come

24
Morals Data Collection

• Quick, simple experimental data collection
• Highlighting critical aspects of effective study
  design
• Can return to the data several times in the
  course

25
Data Analysis ActivitiesActivity 7 Matching
Variables to Graphs

• Which dotplot belongs to which variable?
• Justify your answer

26
Morals Graph-sense

• Learn to justify opinions
• Consistency, completeness
• Appreciate variability
• Be able to find and explain patterns in the data

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Activity 8 Rower Weights

• 2008 Mens Olympic Rowing Team

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Rower Weights
Mean Median Full Data Set 197.96 205.00 Wi
thout Coxswain 201.17 207.00 Without Coxswain
or 209.65 209.00 lightweight rowers With
heaviest at 249 210.65 209.00 With heaviest at
429 219.70 209.00 Resistance....
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Morals Rower Weights

• Think about the context
• Data are numbers with a context -Moore
• Know what your numerical summary is measuring
• Investigate causes for unusual observations
• Anticipate shape

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Activity 9 Cancer Pamphlets

• Researchers in Philadelphia investigated whether
  pamphlets containing information for cancer
  patients are written at a level that the cancer
  patients can comprehend

31
Cancer Pamphlets
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Morals Importance of Graphs

• Look at the data
• Think about the question
• Numerical summaries dont tell the whole story
• median isnt the message - Gould

33
Activity 10 Draft Lottery

• Draft numbers (1-366) were assigned to birthdates
  in the 1970 draft lottery
• Find your draft number
• Any 225s?

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Draft Lottery
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Draft Lottery

• month median
• January 211.0
• February 210.0
• March 256.0
• April 225.0
• May 226.0
• June 207.5

• month median
• July 188.0
• August 145.0
• September 168.0
• October 201.0
• November 131.5
• December 100.0

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Draft Lottery
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Morals Statistics matters!

• Summaries can illuminate
• Randomization can be difficult

38
Activity 11Televisions and Life Expectancy

• Is there an association between the two
  variables?
• So sending televisions to countries with lower
  life expectancies would cause their inhabitants
  to live longer?

r .743
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Morals Confounding

• Dont jump to conclusions from observational
  studies
• The association is real but consider carefully
  the interpretation of graph and wording of
  conclusions (and headlines)

40
Activity 6 Revisited (Memorizing Letters)

• Produce, interpret graphical displays to compare
  performance of two groups
• Does research hypothesis appear to be supported?
• Any unusual features in distributions?

41
Lunch!

• Questions?
• Write down and submit any questions you have thus
  far on the statistical or pedagogical content

42
Exploring RandomnessActivity 12 Random Babies

• Last Names First Names
• Jones Jerry
• Miller Marvin
• Smith Sam
• Williams Willy

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Random Babies

• Last Names First Names
• Jones Marvin
• Miller
• Smith
• Williams

44
Random Babies

• Last Names First Names
• Jones Marvin
• Miller Willy
• Smith
• Williams

45
Random Babies

• Last Names First Names
• Jones Marvin
• Miller Willy
• Smith Sam
• Williams

46
Random Babies

• Last Names First Names
• Jones Marvin
• Miller Willy
• Smith Sam
• Williams Jerry

47
Random Babies

• Last Names First Names
• Jones Marvin
• Miller Willy
• Smith Sam 1 match
• Williams Jerry

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Random Babies

• Long-run relative frequency
• Applet www.rossmanchance.com/applets/
• Random Babies

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Random Babies Mathematical Analysis

• 1234 1243 1324 1342 1423 1432
• 2134 2143 2314 2341 2413 2431
• 3124 3142 3214 3241 3412 3421
• 4123 4132 4213 4231 4312 4321

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Random Babies

• 1234 1243 1324 1342 1423 1432
• 4 2 2 1 1 2
• 2134 2143 2314 2341 2413 2431
• 2 0 1 0
  0 1
• 3124 3142 3214 3241 3412 3421
• 1 0 2 1 0
  0
• 4123 4132 4213 4231 4312 4321
• 0 1 1 2 0 0

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Random Babies

• 0 matches 9/243/8
• 1 match 8/241/3
• 2 matches 6/241/4
• 3 matches 0
• 4 matches 1/24

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Morals Treatment of Probability

• Goal Interpretation in terms of long-run
  relative frequency, average value
• 30 chance of rain
• First simulate, then do theoretical analysis
• Able to list sample space
• Short cuts when are actually equally likely
• Simple, fun applications of basic probability

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Activity 13 AIDS Testing

• ELISA test used to screen blood for the AIDS
  virus
• Sensitivity P(AIDS).977
• Specificity P(-no AIDS).926
• Base rate P(AIDS).005
• Find P(AIDS)
• Initial guess?
• Bayes theorem?
• Construct a two-way table for hypothetical
  population

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AIDS Testing
Positive Negative Total AIDS No
AIDS Total 1,000,000
55
AIDS Testing
Positive Negative Total AIDS
5,000 No AIDS 995,000 Total
1,000,000
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AIDS Testing
Positive Negative Total AIDS 4885 115
5,000 No AIDS 995,000 Total
1,000,000
57
AIDS Testing
Positive Negative Total AIDS 4885 115
5,000 No AIDS 73,630 921,370
995,000 Total 1,000,000
58
AIDS Testing
Positive Negative Total AIDS 4885 115
5,000 No AIDS 73,630 921,370
995,000 Total 78,515 921,485 1,000,000
59
AIDS Testing
Positive Negative Total AIDS 4885 115
5,000 No AIDS 73,630 921,370
995,000 Total 78,515 921,485 1,000,000 P(AIDS)
4885/78,515.062
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AIDS Testing
Positive Negative Total AIDS 4885 115
5,000 No AIDS 73,630 921,370
995,000 Total 78,515 921,485 1,000,000 P(AIDS)
4885/78,515.062 P(No AIDS-)
921,370/921,485 .999875
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Morals Surprise Students!

• Intuition about conditional probability can be
  very faulty
• Confront misconception head-on
• Conditional probability can be explored through
  two-way tables
• Treatment of formal probability can be minimized

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Activity 14 Reeses Pieces
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Reeses Pieces

• Take sample of 25 candies
• Sort by color
• Calculate the proportion of orange candies in
  your sample
• Construct a dotplot of the distribution of sample
  proportions

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Reeses Pieces

• Turn over to technology
• Reeses Pieces applet
• (www.rossmanchance.com/applets/)

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Morals Sampling Distributions

• Study randomness to develop intuition for
  statistical ideas
• Not probability for its own sake
• Always precede technology simulations with
  physical ones
• Apply more than derive formulas

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Activity 15 Which Tire?

• Left Front Right Front
• Left Rear Right Rear

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Which Tire?

• People tend to pick right front more than ¼ of
  the time
• Variable which tire pick
• Categorical (binary)
• How often would we get data like this by chance
  alone?
• Determine the probability of obtaining at least
  as many successes as we did if there were
  nothing special about this particular tire.

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Which Tire?

• Let p proportion of all who pick right front
• H0 p .25
• Ha p gt .25
• Test statistic z
• p-value Pr(Zgtz)
• How does this depend on n?
• Test of Significance Calculator

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Which Tire?

• n z-statistic p-value
• 50 1.14 .127
• 100 1.62 .053
• 150 1.98 .024
• 400 3.23 .001
• 1000 5.11 .000

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Morals Formal Statistical Inference

• Fun simple data collection
• Effect of sample size
• hard to establish result with small samples
• Never accept null hypothesis

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Activity 16 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.

72
Kissing the Right Way

• Is 1/2 a plausible value for p, the probability a
  kissing couple turns right?
• Coin Tossing applet
• Is 2/3 a plausible value for p, the probability a
  kissing couple turns right?
• Is the observed result in the tail of the what
  if distribution?

73
Kissing the Right Way

• Determine the plausible values for p, the
  probability a kissing couple turns right
• The values that produce an approximate p-value
  greater than .05 are not rejected and are
  therefore considered plausible values of the
  parameter. The interval of plausible values is
  sometimes called a confidence interval for the
  parameter.

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Kissing the Right Way

• How does this compare to estimate margin of
  error?
• Or the even simpler approximation?

75
Morals Kissing the Right Way

• Interval estimation as (more?) important as
  significance
• Confidence interval as set of plausible (not
  rejected) values
• Interpretation of margin-of-error

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Activity 17 Reeses Pieces Revisited

• Calculate 95 confidence interval for p from your
  sample proportion of orange
• Does everyone have same interval?
• Does every interval necessarily capture p?
• What proportion of class intervals would you
  expect?
• Simulating Confidence Intervals applet
• What percentage of intervals succeed?
• Change confidence level, sample size

77
Morals Reeses Pieces Revisited

• Interpretation of confidence level
• In terms of long-run results from taking many
  samples
• Effects of confidence level, sample size on
  confidence interval

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Example 18 Dolphin Therapy

• Subjects who suffer from mild to moderate
  depression were flown to Honduras, randomly
  assigned to a treatment

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Dolphin Therapy

• Is dolphin therapy more effective than control?
• Core question of inference
• Is such an extreme difference unlikely to occur
  by chance (random assignment) alone (if there
  were no treatment effect)?

80
Some approaches

• Could calculate test statistic, p-value from
  approximate sampling distribution (z, chi-square)
• But its approximate
• But conditions might not hold
• But how does this relate to what significance
  means?
• Could conduct Fishers Exact Test
• But theres a lot of mathematical start-up
  required
• But thats still not closely tied to what
  significance means
• Even though this is a randomization test

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80
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3S Approach

• Simulate random assignment process many times,
  see how often such an extreme result occurs
• Assume no treatment effect (null model)
• Re-randomize 30 subjects to two groups (using
  cards)
• Assuming 13 improvers, 17 non-improvers
  regardless
• Determine number of improvers in dolphin group
• Or, equivalently, difference in improvement
  proportions
• Repeat large number of times (turn to computer)
• Ask whether observed result is in tail of what if
  distribution
• Indicating saw a surprising result under null
  model
• Providing evidence that dolphin therapy is more
  effective

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Analysis

• http//www.rossmanchance.com/applets/
• Dolphin Study applet

82
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Conclusion

• Experimental result is statistically significant
• And what is the logic behind that?
• Observed result very unlikely to occur by chance
  (random assignment) alone (if dolphin therapy was
  not effective)

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Morals

• Re-emphasize meaning of significance and p-value
• Use of randomness in study
• Focus on statistical process, scope of conclusions

85
Activity 19 Sleep Deprivation

• Does sleep deprivation have harmful effects on
  cognitive functioning three days later?
• 21 subjects random assignment
• Core question of inference
• Is such an extreme difference unlikely to occur
  by chance (random assignment) alone (if there
  were no treatment effect)?

85
85
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Sleep Deprivation

• Simulate randomization process many times under
  null model, see how often such an extreme result
  (difference in group medians or means) occurs
• Start with tactile simulation using index cards
• Write each score on a card
• Shuffle the cards
• Randomly deal out 11 for deprived group, 10 for
  unrestricted group
• Calculate difference in group medians (or means)
• Repeat many times (Randomization Tests applet)

86
86
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Sleep Deprivation

• Conclusion Fairly strong evidence that sleep
  deprivation produces lower improvements, on
  average, even three days later
• Justification Experimental results as extreme as
  those in the actual study would be quite unlikely
  to occur by chance alone, if there were no effect
  of the sleep deprivation

88
Exact randomization distribution

• Exact p-value 2533/352716 .0072 (for difference
  in means)

89
Morals Randomizations Tests

• Emphasizes core logic of inference
• Takes advantage of modern computing power
• Easy to generalize to other statistics

90
Activity 6 Revisited (Memorizing Letters)

• Conduct randomization test to assess strength of
  evidence in support of research hypothesis
• Enter data into applet
• Summarize conclusion and reasoning process behind
  it
• Does non-significant result indicate that
  grouping of letters has no effect?

91
Activity 20 Cat Households

• 47,000 American households (2007)
• 32.4 owned a pet cat
• or the other way around!
• test statistic z-4.29
• p-value virtually zero
• 99 CI for p (.31844, .32956)

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Morals Limits of statistical significance

• Statistical significance is not practical
  significance
• Especially with large sample sizes
• Accompany significant tests with confidence
  intervals whenever possible

93
Activity 21 Female Senators

• 17 women, 83 men in 2010
• 95 CI for p
• .170 .074
• (.096, .244)

94
Morals Limitations of Inference

• Always consider sampling procedure
• Randomness is key assumption
• Garbage in, garbage out
• Inference is not always appropriate!
• Sample population here

95
Activity 22 Game Show Prices

• Sample of 208 prizes from The Price is Right
• Examine a histogram
• 99 confidence interval for the mean
• Technical conditions?
• What percentage of the prizes fall in this
  interval?
• Why is this not close to 99?

96
Morals Cautions/Limitations

• Prediction intervals vs. confidence intervals
• Constant attention to what the it is

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Activity 23 Government Spending

• 2004 General Social Survey Is there an
  association between American adults opinion on
  federal government spending on the environment
  and political inclinations?

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Government Spending

• Descriptive analysis

Liberal Moderate Conservative Total
Too Much 1 17 32 50
About Right 27 80 91 198
Too Little 127 158 113 398
Total 155 255 236 646
99
Government Spending

• Inferential analysis 3S approach
• 1. Chi-square statistic
• 2. Simulate sampling distribution of chi-square
  test statistic under null hypothesis of no
  association
• Randomly mix up political inclinations, determine
  could have been table
• Repeat many times and examine what if
  distribution of chi-square values under null
  hypothesis

100
Government Spending

• 3. Strength of evidence
• Is observed chi-square value in tail of
  distribution?
• Summarize What conclusions should be drawn?
• Very statistically significant
• Not cause and effect
• Ok to generalize to adult Americans

101
Government Spending

• What about federal spending on the space program?

More or less evidence of association? Larger or
smaller p-value?
102
General Advice

• Emphasize the process of statistical
  investigations, from posing questions to
  collecting data to analyzing data to drawing
  inferences to communicating findings
• Use simulation, both tactile and
  technology-based, to explore concepts of
  inference and randomness
• Draw connections between how data are collected
  (e.g., random assignment, random sampling) and
  scope of conclusions to be drawn (e.g.,
  causation, generalizability)
• Use real data from genuine studies, as well as
  data collected on students themselves
• Present important studies (e.g., draft lottery)
  and frivolous ones (e.g., flat tires) and
  especially studies of issues that are directly
  relevant to students (e.g., sleep deprivation)

103
General Advice (cont.)

• Lead students to discover and tell you
  important principles (e.g., association does not
  imply causation)
• Keep in mind the research question when analyzing
  data
• Graphical displays can be very useful
• Summary statistics (measures of center and
  spread) are helpful but dont tell whole story
  consider entire distribution
• Develop graph-sense, number-sense by always
  thinking about context
• Use technology to reduce the burden of rote
  calculations, both for analyzing data and
  exploring concepts
• Emphasize cautions and limitations with regard to
  inference procedures

104
Implementation Suggestions

• Take control of the course
• Collect data from students
• Encourage predictions from students
• Allow students to discover/tell you findings
• Precede technology simulations with tactile
• Promote collaborative learning
• Provide lots of feedback
• Follow activities with related assessments
• Intermix lectures with activities
• Dont underestimate ability of activities to
  teach materials
• Have fun!

105
Suggestion 1

• Take control of the course
• Not control in usual sense of standing at front
  dispensing information
• But still need to establish structure, inspire
  confidence that activities, self-discovery will
  work
• Be pro-active in approaching students
• Dont wait for students to ask questions of you
• Ask them to defend their answers
• Be encouraging
• Instructor as facilitator/manager

106
Suggestion 2

• Collect data from students
• Leads them to personally identify with data,
  analysis gives them ownership
• Collect anonymously
• Can do out-of-class
• E.g., matching variables to graphs

107
Suggestion 3

• Encourage predictions from students
• Fine (better) to guess wrong, but important to
  take stake in some position
• Directly confront common misconceptions
• Have to convince them they are wrong (e.g.,
  Gettysburg address) before they will change their
  way of thinking
• E.g., AIDS Testing

108
Suggestion 4

• Allow students to discover, tell you findings
• E.g., Televisions and life expectancy
• I hear, I forget. I see, I remember. I do, I
  understand. -- Chinese proverb

109
Suggestion 5

• Precede technology simulations with tactile/
  concrete/hands-on simulations
• Enables students to understand process being
  simulated
• Prevents technology from coming across as
  mysterious black box
• E.g., Gettysburg Address (actual before applet)

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Suggestion 6

• Promote collaborative learning
• Students can learn from each other
• Better yet from arguing with each other
• Students bring different background knowledge
• E.g., Matching variables to graphs

111
Suggestion 7

• Provide lots of feedback
• Danger of discovering wrong things
• Provide access to model answers after the fact
• Could write answers on board
• Could lead discussion/debriefing afterward

112
Suggestion 8

• Follow activities with related assessments
• Or could be perceived as fun and games only
• Require summary paragraphs in their own words
• Clarify early (e.g., quizzes) that they will be
  responsible for the knowledge
• Assessments encourage students to grasp concept
• Can also help them to understand concept
• E.g., fill in the blank p-value interpretation

113
Suggestion 9

• Inter-mix lectures with activities
• One approach Lecture on a topic after students
  have performed activity
• Students better able to process, learn from
  lecture having grappled with issues themselves
  first
• Another approach Engage in activities toward end
  of class period
• Often hard to re-capture students attention
  afterward
• Need frequent variety

114
Suggestion 10

• Do not under-estimate ability of activities to
  teach material
• No dichotomy between content and activities
• Some activities address many ideas
• E.g. Gettysburg Address activity
• Population vs. sample, parameter vs. statistic
• Bias, variability, precision
• Random sampling, effect of sample/population size
• Sampling variability, sampling distribution,
  Central Limit Theorem (consequences and
  applicability)

115
Suggestion 11

• Have fun!

116
Assessment Advice

• Two sample final exams
• Carefully match the course goals
• Be cognizant of any review materials you have
  given the students
• Use real data and genuine studies
• Provide students with guidance for how long they
  should spend per problem
• Use multiple parts to one context but aim for
  independent parts (if a student cannot answer
  part (a) they may still be able to answer part
  (b))
• Use open-ended questions requiring written
  explanation
• Aim for at least 50 conceptual questions rather
  than pure calculation questions
• (Occasionally) Expect students to think,
  integrate, apply beyond what they have learned.
• Sample guidelines for student projects

117
Promoting Student Progress

• Document and enhance student learning
• Element of instruction
• Interactive feedback loop
• Diagnostic with indicators for change
• Throughout the course
• To student and instructor
• Encourage self-evaluation
• Multiple indicators

118
Student Projects

• Best way to demonstrate to students the practice
  of statistics
• Experience the fine points of research
• Experience the messiness of data
• From beginning to end
• Formulation and Explanation
• Constant Reference
• statweb.calpoly.edu/bchance/stat217/projects.html

119
Resources

• www.causeweb.org

120
Resources

• GAISE reports

121
Resources

• TeachingWithData.org

122
Resources

• Inter-University Consortium for Political and
  Social Research (ICPSR)

123
Resources

• www.rossmanchance.com/applets/
• http//statweb.calpoly.edu/csi/

124
Resources

• https//app.gen.umn.edu/artist/

125
Resources

• http//lib.stat.cmu.edu/DASL/
• www.amstat.org/publications/jse/
• /jse_data_archive.html

126
Background Readings

• Guidelines for teaching introductory statistics
• Reflections on what distinguishes statistical
  content and statistical thinking
• Educational research findings and suggestions
  related to teaching statistics
• Collections of resources and ideas for teaching
  statistics
• Suggestions and resources for assessing student
  learning in statistics

127
Thanks very much!

• Questions, comments?
• bchance_at_calpoly.edu
• arossman_at_calpoly.edu

128
My Syllabus Briefly

• W1 Collecting Data
• W2 Graphical/Numerical
• W3 Normal Project 1
• W4 Exam 1 Project 2
• W5 Probability
• W6 Sampling Distributions
• W7 Inference
• W8 Inference

129
My Syllabus Briefly

• W9 Two Samples
• W10 Exam II Project 3
• W11 Two variables
• W12 Inference for Regression
• W13 Two-way Tables Project 4
• W14 ANOVA
• W15 Presentations

130
Non-simulation approach

• Exact randomization distribution
• Hypergeometric distribution
• Fishers Exact Test
• p-value
• .0127