Intuitions about Chance Devices
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Describe how your predicted bar graphs would change if there were 20
coins/dice/cards instead of 10.
Suppose that you receive about 10 text messages per day and about half of all text
messages come from your mother. Describe how you could use each of these
devices (coins, dice, cards) to simulate the number of text messages that you
receive from your mother per day.
In a family with 4 children:
• What is the most likely split of boys/girls?
• How often would you expect to see all boys or all girls?
• Which do you think is more likely: a 2/2 split or a 3/1 split in either
direction?
• Describe how you could use each of these devices (coins, dice, cards) to
simulate this random process and answer the questions above.
Checking Your Intuitions
• Compare your plots with those of several classmates.
– Are your plots identical?
– Comment on their similarities and differences.
– Based on examining several of these plots, graph what you would expect
to see if you could simulate 10,000 repetitions (rather than 100) of the
random process.
• Reconsider the questions about a family with 4 children (repeated below).
Conduct a simulation analysis to investigate these questions.
– What is the most likely split of boys/girls?
– How often would you see all boys or all girls?
– Which is more likely: a 2/2 split or a 3/1 split in either direction?
Can You Beat Randomness II
• Based on pooling class results, approximate probability of winning with each
strategy.
• Describe how you could (in principle anyway) produce more accurate
approximations of these probabilities.
• If light actually is green 90% of time and red 10% of time, what would be the best
strategy? Explain.
• Under what conditions (about actual probability of green/red lights) would human
and rat strategies be equally good/bad? Explain.
One Son
• Many people expect a “one son” policy to affect both family size and average
number of girls per family. Give an intuitive, non-mathematical explanation for
why this turns out not to be the case.
• Now consider a policy that allows (actually, requires) couples to keep having
children until they have one of each sex. Describe how to conduct a simulation to
investigate the effects of such a policy on family size and boy/girl ratio.
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Conduct a simulation analysis to investigate the effects of this policy. Write a
report in which you describe how you conducted the simulation and then
summarize your findings.
Cereal Box Simulation
• Would you say that your results display considerable consistency or much
variability? Explain.
• Suppose that you buy boxes all at once without inspecting them for prizes as you
go.
– Would buying 25 or even 50 boxes guarantee that you would obtain a
complete set of all 6 prizes? Explain.
– How many boxes should you buy in order to feel 90% confident that you
have enough boxes to obtain a complete set of all 6 prizes? Explain how
you determine your answer based on simulation results.
– If you’re willing to feel only 70% confident of obtaining a complete set,
how many boxes should you buy?
• Suppose that there were only 3 prizes in the set instead of 6.
– How would you expect your simulation results to change?
– Conduct a simulation and comment on how your results change. Be sure
to address variability as well as center.
• Repeat for a set of 10 prizes rather than 6 or 3.
• Devise and conduct a simulation analysis to investigate a different problem
– Allan has a 1 in 9 chance of winning when he plays a game of Solitaire,
and Beth has a 1 in 6 chance of winning when she plays. Allan plays first,
and then Beth, and then they keep taking turns. How likely is that Beth
wins before Allan does?
Matching Dogs to Owners
• Devise and conduct a simulation analysis to investigate a different problem
– 3 students drop their cell phones and pick them up completely at random
• How likely is it that everybody picks up their own phone?
• How likely is it that nobody picks up their own phone?
• What is the average number of students who would pick up their
own phone?
• How would the answers to these questions change if the number of
students and phones were 4, or 6, or 10?
Helper or Hinderer
• Suppose that the observed result had been 10 of 16 choosing the helper toy.
– Explain why you could assess the significance of this result using the same
simulation analysis that you already conducted.
– Based on your simulation analysis, what is the approximate p-value?
– What conclusion would you draw? Explain the reasoning process behind
this conclusion.
• Repeat, if the observed result had been 13 of 16 choosing the helper toy.
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Suppose that the study had involved only 8 infants rather than 16, and that 7 of 8
had chosen the helper toy
– Explain how you would need to change your set-up of the simulation
analysis
– Would you expect this result constitute more, less, or the same amount of
evidence that infants really prefer the helper toy, as compared to the 14 of
16 result?
– Conduct the simulation analysis and comment on whether your
expectation was correct
A recent study found that 8 of 12 kissing couples leaned their heads to the right.
– Conduct a simulation analysis to investigate whether this provides strong
evidence that more than half of all kissing couples lean their heads to the
right.
– Write a report describing how you conduct the simulation and also
summarizing your conclusion. Be sure to explain the reasoning process
behind your conclusion.
Sleep Deprivation
• Investigate effect of sample size
– Keep data same for both groups but double sample size
– Make prediction for change (if any) on p-value
– Conduct simulation to confirm/refute intuition
• Investigate effect of within-group variability
– Keep means, sample sizes same for both groups but increase within-group
variability
– Make prediction for change (if any) on p-value
– Conduct simulation to confirm/refute intuition
• Apply same reasoning process to observational study
– Comparing two groups with quantitative response
– Age discrimination case (Martin vs. Westvaco)
• Simulation analysis proceeds in identical manner
But scope of conclusions changes
Dolphin Therapy
• Investigate effect of sample size
– Keep success proportions same for both groups but double sample size
– Make prediction for change (if any) on p-value
– Conduct simulation to confirm/refute intuition
• Apply same reasoning process, simulation analysis to different study
– “Is yawning contagious?” from MythBusters
– Difference between groups turns out not to be statistically significant
• Not even close, despite what MythBusters concluded
• Some students struggle with idea of “not surprising”
• Apply same reasoning process, simulation analysis to observational study
– Comparing two groups with yes/no response
– Murder trial (Kristen Gilbert)
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Simulation analysis proceeds in identical manner
But scope of conclusions again changes
Cell Phone Distractions
• Apply wrong procedure (as if randomized design) to investigate how p-value,
conclusion change
• Apply procedure to observational data with paired design
– E.g., marriage ages
DRAFT OF LEARNING GOALS FOR UNIT 2 ACTIVITIES
Memory Study
• Collect data from (simple) randomized experiment
– Comparison, randomization, blindness
• Understand chance (null) model for comparing two groups
• Reinforce idea/process of statistical significance
– Same reasoning as with helper/hinderer
• Use knowledge of simulating chance devices to simulate randomization test
Sleep Deprivation
• Simulate randomization test for real data from genuine study
– Learn hazards of pulling all-nighters!
• Transition from TinkerPlots (visual, create components yourself) to R (powerful,
more generalizeable)
• Realize that randomization approach applies equally well to other measures (e.g.,
comparing medians, trimmed means)
Dolphin Therapy
• Simulate randomization test for real data from genuine study with yes/no response
– Identical reasoning process applies
– Identical simulation process applies
• With 0/1 responses
Difference in means = difference in success proportions
Cell Phone Distraction
• Recognize value of paired design
– Account for variability across subjects
• Apply randomization test to data from paired design
Randomization now occurs for each subject