Transcript 235_lecture6_080219

Psyc 235:
Introduction to Statistics
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(and check out the
lunar eclipse on
Thursday from about
9-10)
Graded Assessment
AL1: Feb 25th & BL1: Feb 27th
• In Psych 289: anytime between 9-5
• Can sign up for guaranteed spot:
 9am, 11:30am, 2pm, 4:30pm
 Sign up in lab or Office Hours (Thurs Rm 25)
• Bring ID and notes.
Graded Assessment
• ALEKS will be unavailable:
 AL1: 8am Mon - 11:59pm Wed
 BL1: 8am Wed - 11:59pm Fri
• Conflict/Makeup exams:
 must be within that window
 let us know ASAP (as in TODAY)
Quiz on this Thursday
• Use this quiz as practice exam for the
assessment.
• Get your notes ready beforehand.
• Complete like an assessment
• Make note of trouble areas, additional notes you
would like, etc.
• Can do the quiz in office hours and then ask
Jason questions
Questions?
Intersection & Union
• Intersection:
 P(A  B) = P(A)*P(B)
 (If mutually exclusive = 0)
• Union:
 P(A U B) =P(A)+P(B)- P(AB)
• Compliment:
 p(A)=1-p(A)
Independent vs. Dependent
Events
• Independent Events: unrelated events that
intersect at chance levels given relative
probabilities of each event
• Dependent Events: events that are related in
some way Concepts of union and intersection are the same
 However, P(AB)  P(A)*P(B)
• Do you think mutually exclusive events are
dependent or independent?
Conditional Probability
p(B|A) =
p(BA)
p(A)
Conceptually this means:
A
B
Baye’s Theorem
p(B|A) =
p(A|B)p(B)
p(A|B)p(B) + p(A|B)p(B)
• Can we break this down a little to
understand it better?
• p(A|B)*p(B)=p(AB)
A
• p(A|B)*p(B) + p(A|B)*p(B)
= p(AB) + p(AB)
= p(A)
p(B|A) =
•So, this is just:
p(BA)
p(A)
B
Law of Total Probabilities
A
B
• p(A) = p(AB) + p(AB)
• p(A) = p(A|B)p(B) + p(A|B)p(B)
_
B
Random Variables
• Where are we?
• In set theory, we were talking about
theoretical variables that only took on two
values: either a 0 or 1. They were in the
group or not.
• Now we’re going to talk about variables
that can take on multiple values.
Random Variables
• But wait, didn’t we already talk about
variables that had multiple values?
• When we were talking about central
tendancy and dispersion, we were talking
about specific distributions of data…now
we’re going to start discussing theoretical
distributions.
Data World vs. Theory World
• Theory World: Idealization of reality
(idealization of what you might expect from
a simple experiment)
 Theoretical probability distribution
• Data World: data that results from an
actual simple experiment
 Frequency distribution
But First…
• Before we get into random variables, we
need to spend a little bit of time thinking
about:
 the kinds of values variables can take on
 what those values mean
 how we can combine them
4 Standard Scales
• Categorical (Nominal) Scale
 Numbers serve only as labels
 Only relevant info is frequency
• Ordinal Scale
 Things that are ranked
 Numbers give you order of items, but not distance
between/relation between
• Interval Scale
 Scale with arbitrary 0 point and arbitrary units
 However, units give you proportional relationship between
values
• Ratio Scale
 Scale has an absolute 0 point
 Intervals between units is constant
What kind of scale is this?
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Temperature
Grades
Number Scale
Terror Alert Scale
Class Rank
What are other scales you are familiar with?
Discrete vs. Continuous
Random Variables
• Discrete
 Finite number of outcomes
 (x = sum of dice)
 Countable infinite number of outcomes
 Numbers from 1 to infinity
• Continuous
 Uncountably Infinite
 (x=number of flips to get a head)
 (Convergent series: the sum of 1-infinity approaches
some value)
Probability Density Distributions
• Discrete: draw on board
 Probability mass function
• Continuous
 (x= spot where pointer lands)
 Probability mass funtion
Next:
• Now that we know more about random
variables, we can apply everything that
we’ve learned so far.
• Graphing and displaying data
• Central tendency & dispersion
• Transformations of mean and variance
• Contingency Tables
Central Tendency in Random
Variables
• E(x)=∑(X*p(x))
• Var(x)=∑((X-E(x))2*p(x))
Properties of Expectation
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E(a)=a
E(aX)=a*E(X)
E(X+Y)=E(X)+E(Y)
E(X+a)=E(X)+a
E(XY)=E(X) * E(Y)
Properties of Variance
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Var(aX)=a2Var(X)
Var(X+a)=Var(X)
Var(X-a)=Var(X)
Var(X+Y)= Var(X) + Var(Y)
Var(X2)=E(X)+Var(X)2
Contingency Tables for 2
random variables
A(yes)
not A (no)
B (yes)
p(a)
p(b)
p(a+b)
not B(no)
p(c)
p(d)
p(c+d)
p(a+c)
p(b+d)
1
• A is facilitative of B when p(B|A)>P(B)
• A is inhibitory for B when p(B|A)<P(B)
Remember
• 1st Exam Feb 25/27
 Sign up for exam timeslots in lab Wed or
Office Hours Thurs
 (or also first-come-first-served on exam day)
• Quiz on Thursday