Transcript Psyc 235: Introduction to Statistics

Psyc 235:
Introduction to Statistics
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Review:
?s about Descriptive Statistics
• Displaying Data
• Central Tendency: Mean, Median Mode
• Dispersion: Variance (sample, population)
Review: Probability
• Counting and Sample Space
 All possible outcomes n!
• Permutations (order matters)
 For n objects taken r at a time:
n!
(n-r)!
• Combinations (order doesn’t matter)
 For n objects taken r at a time:
n!
r!(n-r)!
Review Example
Say a little league coach has 14
kids and wants to know how many
distinct groups of 9 outfielders he can make.
• What do we do?
 Combination
14!
9!5!
What if the little league coach wants to know how many
unique ways he can arrange the kids into each of the 9
field positions?
• What do we do?
 Permutation
14!
5!
Review: Probability
• Multiplication Principle (Fundamental
Counting Principle)
 For 2 independent phenomenon, multiply the
# possible outcomes for the two individual
phenomena
Example: Multiplication
Principle
Say our coach has
2 pitchers, 1 catcher,
5 outfielders,
and 6 infielders.
Can you help me set up
how many unique teams
he has now (keeping in mind
those positions)?
Ok? So what does that have to
do with statistics?
• Once we know all the possible outcomes, we
can compute what the likelihood is of randomly
selecting one (or a set of potential) outcomes out
of all of those possible outcomes.
• Eventually, we’ll want to do this the other way-given the set of outcomes that we randomly
selected from an unknown group, what can we
conclude about the characteristics of all possible
outcomes from that group.
The Assumption of
Independence
• Independence: the outcomes of one
simple experiment are not linked to the
outcome of another
• Height and weight are NOT independent
• Eye color and IQ are independent
Important Terms/Concepts
• Simple experiment: some process that leads to
one possible outcome being selected from a set
of possibilities
• Sample space: the set of possible outcomes in a
simple experiment
• Sample points: members of the sample space
• Event: Any subset of the sample space
• Elementary Event: An event that contains a
single sample point
How would these terms relate to rolling a die?
Probabilities in Simple
Experiments
• Assume that each roll of the die (or elementary
event) is equally likely.
 Note: How do we know equally likely?
 Good assumption
 Relative frequency
 Subjective estimation based on some criteria
• What is the probability of any one roll of the die?
• What is the probability of getting three 6s in
three rolls?
Probability of Draw with
Replacement
• What is replacement?:
• All sample points in the sample space are
available each time you do a simple
experiment.
• Like rolling a die.
• When you want to know the likelihood of
obtaining some specific set of results, use
multiplication principle.
Example: The proverbial
Statistician’s Urn.
Probability of Draws without
Replacement
• Instead of considering each simple
experiment in isolation, think about the
events.
• How many events satisfy your conditions?
• How many events are in the sample
space?
• P(E)= # of events satisfy condition/total #
of events
Example: The proverbial
Statistician’s Urn.
Brief Psychology Experiment.
• Linda is 31 years old, single, outspoken, and
very bright. She majored in philosophy. As a
student, she was deeply concerned with issues
of discrimination and social justice, and also
participated in anti-nuclear demonstrations.
Which is more likely?
• 1.Linda is a bank teller.
• 2.Linda is a bank teller and is active in the
feminist movement.
So, now we’ve moved from talking about a
single group to multiple groups.
• Is there a better way to represent what
we’re talking about now?
• Yes!
• Venn Diagrams.
• Let’s draw one that represents what we
had in our grecian urn…
Venn Diagrams & Some Terms
• Universal Set (W): the largest set (sample
space)
• Mutually Exclusive Events: events that
never occur together
• Sure Event: event that always occurs
• Impossible Event: event that never occurs
(empty set)
• Exhaustive Events: If events are equal to
the universal Set
Let’s make our own…
• Super bowl Supporters:
• Giants, Patriots, Both, Neither
Venn Diagrams and Set Theory
• Now, we can count how many sample points fall
in each category.
• How do we talk about combinations of
categories (e.g. how many people favored a
team(s) in the super bowl?)
• Union: combination of two sets
 C= A U B
• Intersection: contains only objects common to A
and B
 D=AB
Set Theory and Probability
• Once we know number of total events and
number of events in each category, we
can calculate the probability of obtaining a
result in any one category.
• P(A)= A/W
• P(B)= B/W
• P(A  B) = A  B/W
• P(A U B) = AUB/W
Set Theory
• Set Compliment A = 1 - A
• Subsets
 Proper Subset (all objects in subset are not
only objects in set) C  A
 Subset (all objects in subset could be only
objects in set C  A
Set Theory Calculations
• P(A  B) = P(A) * P(B)
• P(A U B) = P(A) + P(B)- P(A  B)
Can you tell me…
•
•
•
•
•
•
A
AW
AU
AUW
AA
AUA
=
=A
=A
=W
=
=W
Some other examples
• A occurs or B does not (or both)
• A occurs without B or B without A
• A does not occur or B does not occur (or
both)
• A occurs without B or A and B both occur
• B occurs but A does not
• B occurs without A or A and B occur
Back to the Psych Experiment
Which is more likely?
• 1.Linda is a bank teller.
• 2.Linda is a bank teller and is active in the
feminist movement.
If/Then: Logic Fallacies
Venn Diagrams can help!
To-Do
• ALEKS: aim for about 24 hours spent by
the end of this week (including class time)
• Descriptive Statistics should already be
complete
• Start work on Probability Sections
• Make sure you are up to date on videos