Transcript Psyc 235: Introduction to Statistics

Psyc 235:
Introduction to Statistics
http://www.psych.uiuc.edu/~jrfinley/p235/
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To-Do
• ALEKS: aim for 18 hours spent by the
end of this week
• Jan 30th Target Date for Descriptive
Statistics
• Watch videos:
1. Picturing Distributions
2. Describing Distributions
3. Normal Distributions
Quiz 1
• NOT GRADED
• available starting 8am Thurs Jan 31st,
through Friday
• can do on ALEKS from home, etc
• No access to any other learning or
reviewing materials until either they finish
the quizzes or after Friday
• 3.5 hour time limit
Review:
(2 Steps Forward and 1 Step Back)
• Distribution
 For a given variable:
 the possible numerical values
 & the number of times they occur in the data
 Many ways to represent visually
Summarizing Distributions
• Descriptive Measures of Data
 Measures of C_nt__l T__d__cy
 Measures of D__p_rs__n
Central Tendency
• Mean, Median, Mode
 Mean vs Median & outliers
 (Bill Gates example)
 skewed distributions
Standard Deviation
• Conceptually:
 about how far, generally, each datum is from
the mean
 2 formulas??
Population vs Sample
• In Psychology:
 Population: hypothetical, unobservable
 not just all humans who ARE, but all humans who
COULD BE.
 must estimate mean, standard deviation, from:
 Sample is the only thing we ever have
Descriptive -> Inferential?
• How can we make inferences about a
population if we just have data from a
sample?
• How can we evaluate how good our
estimate is?
• “Do these sample data really reflect what’s
going on in the population, or are they
maybe just due to chance?”
PROBABILITY
• The tool that will allow us to bridge the gap
from descriptive to inferential
• we’ll start by using simple problems, in
which probability can be calculated by
merely COUNTING
Flipping a Coin
• Say I flip a coin...
 OMG Heads!!!!
 Do you care?
 Why Not?
• Sample Space:
 (draw on board)
 collection of all possible outcomes for a given
phenomenon
 coin toss: {H,T}
 mutually exclusive: either one happens, or the other
Flipping a Coin
• Probability(Heads)?
• So.... must the next one be Tails?
• No!
 Independent trials
 Random Phenomenon:
 can’t predict individual outcome
 can predict pattern in the LONG RUN
• Probability: relative # times something
happens in the long run
2 Coin Flips
• OMG 2 Heads!
 impressed yet?
• Sample space
 (draw on board)
 Prob(2 Heads): 1/4
 outcome: single observation
• OMG 2 of same!
 Prob(2 Heads OR 2 Tails):
 event: subset of the sample space made of 1 or more
possible outcomes
Larger Point
• OMG 30 Heads in a row!
 NOW maybe you’re finally interested...
• OMG drew 3 yellow cars!
 interesting? boring? can’t tell!
• Descriptive Stats: measuring & summarizing
outcomes
• Inferential Stats: to understand some outcome,
must consider it in context of all possible
outcomes that could’ve occurred (sample space)
Counting Rules
• Count up the possible outcomes
 that is: define the sample space
• 2 Main ways to do this:
 Permutations
 when order matters
 Combinations
 when order doesn’t matter
Permutation:
Ordered Arrangement
• Example used: Horse Race...
• MUTANT HORSE RACE!
Permutation:
Ordered Arrangement
“HorseFace McBusterWorthy wins 1st place!!”
...in a one-horse race!
# Horses (n)# Winning Places (r)
1
3
3
1
1
3
# Outcomes
Permutation:
Ordered Arrangement
• For n objects, when taking all of them (r=n),
there are n! possible permutations.
• 3 horses (n) & 3 winning places (r) -->
 3*2*1=6 possible outcomes
• For n objects taken r at a time:
n!
(n-r)!
• 7 horses & 3 winning places?...
Combination:
Unordered Arrangement
• Example used: Combo Plate!
QuickTime™ and a
decompressor
are needed to see this picture.
Combination:
Unordered Arrangement
• Mexican restaurant’s menu:
 taco, burrito, enchilada
• How many different 3-item combos can you get?
# Menu Items (n)
Combo Size (r)
3
3
# Outcomes
Combination:
Unordered Arrangement
• Mexican restaurant’s menu:
 taco, burrito, enchilada
 tamale, quesadilla, taquito, chimichanga
• How many different 3-item combos can you get?
# Menu Items (n)
Combo Size (r)
3
7
3
3
# Outcomes
Combination: Unordered
Arrangement
• For n objects, when taking all of them
(r=n), there is 1 combination
• For n objects taken r at a time:
n!
r!(n-r)!
Multiplication Principle
(a.k.a. Fundamental Counting Principle)
• For 2 independent phenomenon, how
many different ways are there for them to
happen together?
 # possible joint outcomes?
• Simply multiply the # possible outcomes
for the two individual phenomena
• Example: flip coin & roll die
• 2*6=12
Multiplication Principle
(a.k.a. Fundamental Counting Principle)
• Can be used with Permutations &/or
Combinations
• Ex: Lunch at the Racetrack
 7 horses racing
 7 items on the cafe menu
 I see the results of the race (1st, 2nd, 3rd)
and order a 3-item combo plate. How many
different ways can this happen?
Calculating Probabilities
• Counting rules (Permutation, Combination,
Multiplication):
 Define sample space (# possible outcomes)
• Probability of a specific outcome:
1
sample space
• Probability of an event?
 event: subset of sample space made of 1 or
more possible outcomes
Calculating Probabilities
• Sample Space:
 7 Micro Machines (3 yellow, 4 red)
• Outcome:
 draw the yellow corvette
 Probability = 1/7
• Event:
 draw any yellow car
 there are 3 outcomes that could satisfy this event:
yellow corvetter, yellow pickup, yellow taxi
 Probability = 3/7
Probability of Draws w/
Replacement
• Replacement: resetting the sample space
each time
 --> independent phenomena
 so use multiplication principle
• Ex: 3 draws with replacement
 Event: drawing a red car all 3 times
 Probability: 4/7 * 4/7 * 4/7
= 64/343 = 0.187 =18.7%
Probability of Draws w/o
Replacement
1. Use counting rules to define sample
space
2. Use counting rules to figure out how
many possible outcomes satisfy the
event
3. divide #2 by #1.
Probability of Draws w/o
Replacement
• Ex: Drawing 3 cars w/o replacement
 Event: drawing 2 red & 1 yellow
 (don’t care about order)
 --> use Combinations
 Define Sample space:
 Count outcomes that satisfy event
 treat red & yellow as independent
 use combinations, then multiplication principle
 Divide
Recap
• Today:
 Probability is the tool we’ll use to make
inferences about a population, from a sample
 Counting rules: define sample space for
simple phenomena
 Intro to calculating probability
• Next time:
 Probability rules, more about events, Venn
diagrams
Remember
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Quiz 1 starting Thursday
Office hours Thursday
Lab
Put your ALEKS hours in!!