Prob Models

1
Prob Models

• Probability is the study of random processes
• Toss a coin
• Take a test
• Grad QPR
• Distance of a thrown ball

2
Prob Models

• Each individual outcome may vary some
• Still reasonable to describe results
• We have in mind an underlying prob model for the
  outcome

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Prob Models

• Two important models
• 1. Binomial two distinct outcomes
• 2. Normal outcomes on a continuous scale

4
Prob Models

• In most prob models, there are things we do not
  know
• Free throw percentage
• Avg distance thrown
• Diff between SM121 and SM121A

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Statistics

• Statistics uses data to draw conclusions about
  our models
• Data is a sample from a population
• It can be too expensive to gather data on the
  entire population
• Or we may be trying to predict the future
• Always uncertainty about our conclusions

6
Guessing age

• Tenn. Recently passed a law that salespersons in
  liquor stores need to card all customers
• Salespersons estimates of customers ages will
  vary
• Construct a probability model

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Diff between guess and actual

• Model1 difference between estimated age and
  actual age
• Might be a normal model, esp if we consider age
  as continuous
• Want to know if salesperson is right on average
• Want to know how variable salesperson might be

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Guess correctly?

• Model2 Can salesperson correctly identify
  whether or not someone is of legal age?
• Binomial model 2 outcomes, either right or
  wrong
• Want to know probability of being right
• (Not sure how to account for diff between being
  right for someone over 21 and being right for
  someone under 21.)

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Types of studies

• Studies can be experiments or observational
  studies
• Experiments are planned in advance
• Control which subjects are which
• Observational studies are after the fact

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Experiment

• Experiment for guessing age
• Select photos of subjects of different ages and
  have salesperson guess their age

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Obs study

• Observational study
• In liquor store, have salesperson guess each
  customers age
• Then compare to ID

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Limits of observational

• Tricky to draw conclusions from observational
  studies
• Observe that many people who drive Cadillacs have
  gray hair
• Does driving a Cadillac cause gray hair?

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What can stat tell us?

• All of statistics is aimed at answering
  scientific questions
• How long does it take me to drive to work?
• Is my gas mileage worse when I use the AC?
• Does taking SM121A instead of SM121 improve
  students scores?
• Does abstinence only sex ed work?

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SM121A

• Example Does SM121A help?
• Placement into SM121A is based on the math
  placement test
• Suppose we define a range of borderline scores
  that could go either way
• We randomly assign some of these to SM121 and
  some to SM121A

15
Models

• Binomial consider how many of each group get at
  least a C
• Compare prob of C between the two groups
• Normal consider score on final exam
• Compare means of scores between the groups

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Uncertainty

• Any conclusions based on data will have
  uncertainty
• Two types
• 1. How sure are we that there is some difference?
• 2. How much difference might there be?

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Uncertainty

• We use results from our probability models to
  measure the uncertainty
• Suppose that 18 of 20 in SM121 get a C and 19 of
  25 in SM121A get a C
• If the probability of a C is the same for both
  groups, we still might get slightly different s
• Answer question 1 by If both groups were the
  same, what is the chance of getting what I
  observed?
• Which means we need to learn to calculate
  probabilities

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General Rule

• Whenever we use statistics, we should be able to
  identify the underlying probability model
• AND we should be able to relate our conclusions
  to something of interest in the scientific
  problem