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MGMT 276: Statistical Inference in Management
Spring 2015
Schedule of readings
Before our next exam (March 24th)
We’ll be jumping
around some…we will
start with chapter 7
Lind (5 – 11)
Chapter 5: Survey of Probability Concepts
Chapter 6: Discrete Probability Distributions
Chapter 7: Continuous Probability Distributions
Chapter 8: Sampling Methods and CLT
Chapter 9: Estimation and Confidence Interval
Chapter 10: One sample Tests of Hypothesis
Chapter 11: Two sample Tests of Hypothesis
Plous (10, 11, 12 & 14)
Chapter 10: The Representativeness Heuristic
Chapter 11: The Availability Heuristic
Chapter 12: Probability and Risk
Chapter 14: The Perception of Randomness
Homework due – Tuesday (March 3rd)
On class website:
Please print and complete Homework Assignment 9
Chapter 5 Approaches to probabilities and
Chapter 7 Interpreting probabilities using the normal curve
Calculating z-score, raw scores and areas (probabilities)
under normal curve
Use this as your
study guide
By the end of lecture today
2/26/15
Counting ‘standard deviationses’ – z scores
Connecting raw scores, z scores and probability
Connecting probability, proportion and area of curve
Percentiles
Approaches to probability:
Empirical, Subjective and Classical
Normal distribution
Raw scores
z-scores
Have z
Find raw score
Formula
probabilities
Z
Scores
z table
Have z
Find area
Have area
Find z
Have raw score
Find z
Raw
Scores
Area &
Probability
Always draw a picture!
Homework worksheet
Homework worksheet
.6800
1
Homework worksheet
.9500
2
Homework worksheet
.9970
3
Homework worksheet
.5000
4
Homework worksheet
33-30 z = 1.5
z=
2
Go to
table
.4332
5
z=
33-30 z = 1.5
2
Go to
table
.4332
Add area
Lower half
.4332 + .5000 = .9332
6
Homework worksheet
Go to
33-30
z=
.4332
= 1.5
table
2
Subtract
from .5000
.5000 - .4332 = .0668
7
z=
29-30
2
= -.5
Go to
.1915
table
Add to upper
Half of curve .5000 - .1915 = .6915
8
25-30
2
31-30
=
2
=
= -2.5
=.5
Go to table
Go to table
.4938
.1915
.4938 + .1915 = .6853
9
z=
Go to
27-30
= -1.5 table
2
.4332
Subtract
From .5000
.5000 - .4332 = .0668
10
z=
25-30
2
= -2.5
Go to
table
.4938
Add lower
Half of curve .5000 + .4938 = .9938
11
z=
Go to
32-30
= 1.0 table
2
.3413
Subtract
from .5000
.5000 - .3413 = .1587
12
50th percentile = median
30
13
28
32
14
77th percentile
Find area
of interest
.7700 - .5000 = .2700
x = mean + z σ = 30 + (.74)(2) = 31.48
Find nearest z = .74
15
13th percentile
Find area
of interest
.5000 - .1300 = .3700
x = mean + z σ = 30 + (-1.13)(2) = 27.74
Find nearest z = -1.13
16
Please use the following distribution with
a mean of 200 and a standard deviation of 40.
.6800
17
.9500
18
.9970
19
=
230-200
= .75
40
Go to
table
.2734
20
190-200 = -.25 Go to
z=
table
40
.0987
Subtract
from .5000
.5000 - .0987 = .4013
21
180-200 = -.5 Go to
z=
table
40
.1915
Add to upper
.5000 + .1915 = .6915
Half of curve
22
236-200 = 0.9
z=
40
Go to
table
.3159
Subtract
from .5000
.5000 - .3159 = .1841
23
192 - 200
40
= 222 - 200
40
z=
z
= -.2
=.55
Go to table
Go to table
.0793
.2088
.0793 + .2088 = .2881
24
z=
275-200 = 1.875
40
Go to
table
.4693
or
.4699
Add area
Lower half
.4693 + .5000 = .9693
.4699 + .5000 = .9699
25
295-200 z = 2.375
z=
40
Go to
table
.4911
or
.4913
.5000 - .4911 = .0089
Add area
Lower half .5000 - .4913 = .0087
26
Add to upper
130-200 = -1.75 Go to
.5000 + .4599 = .9599
z=
.4599
table
Half
of
curve
40
27
Subtract
130-200 = -1.75 Go to
z=
.4599
table
from .5000
40
.5000 - .4599 = .0401
28
99th percentile
Find area
of interest
.9900 - .5000 = .4900
x = mean + z σ = 200 + (2.33)(40) = 293.2
Find nearest z = 2.33
29
33rd percentile
Find area
of interest
.5000 - .3300 = .1700
x = mean + z σ = 200 + (-.44)(40) = 182.4
Find nearest z = -.44
30
40th percentile
Find area
of interest
.5000 - .4000 = .1000
x = mean + z σ = 200 + (-.25)(40) = 190
Find nearest z = -.25
31
67th percentile
Find area
of interest
.6700 - .5000 = .1700
x = mean + z σ = 200 + (.44)(40) = 217.6
Find nearest z = .44
32
.
.8276
.1056
.2029
.1915
.4332
44 - 50
4
.3944
= -1.5
z of 1.5 = area of .4332
55 - 50
4
= +1.25
z of 1.25 = area of .3944
.4332 +.3944 = .8276
.3944
.3944
55 - 50
4
= +1.25
1.25 = area of .3944
.5000 - .3944 = .1056
52 - 50
4
= +.5
z of .5 = area of .1915
55 - 50
4
= +1.25
z of 1.25 = area of .3944
.3944 -.1915 = .2029
What is probability
1. Empirical probability: relative frequency approach
Number of observed outcomes
Number of observations
Probability of getting into an educational program
Number of people they let in
Number of applicants
400
600
66% chance of
getting admitted
Probability of getting a rotten apple
Number of rotten apples
Number of apples
5
100
5% chance of
getting a rotten
apple
What is probability
1. Empirical probability: relative frequency approach
“There is a 20% chance
“More than 30% of the
10% of people who buy a
that a new stock
results from major
Number
of observed
house with
no pool build
offered in outcomes
an initial
search engines for the
one. What is the
public offering (IPO)
Number
observations
keyword phrase “ring
likelihood that
Bob will?of will
reach or exceed
tone” are fake
its target price on
Probability of hitting
the corvette pages created by
the first day.”
spammers.”
Number of carts that hit corvette
Number of carts rolled
182
200
= .91
91% chance of
hitting a corvette
2. Classic probability: a priori probabilities based on logic
rather than on data or experience.
All options are equally likely (deductive rather than inductive).
Likelihood get Chosen at
Lottery question right
random to be
on multiple team captain
choice test
Number of outcomes of specific event
Number of all possible events
In throwing a die what is the probability of getting a “2”
Number of sides with a 2
Number of sides
1
16% chance of
getting a two
= 6
In tossing a coin what is probability of getting a tail
Number of sides with a 1
Number of sides
1
=
2
50% chance
of getting a tail
3. Subjective probability: based on someone’s personal
judgment (often an expert), and often used when empirical
and classic approaches are not available.
Likelihood get a
60% chance
Likelihood
”B” in the class
that Patriots
that company
will play at
will invent
Super
Bowl
There
Verizon
new
type ofis a 5% chance that
battery with Sprint
merge
will
Bob says he is 90% sure he could swim across the river
Approach
Example
Empirical
There is a 2 percent chance of
twins in a randomly-chosen birth
Classical
There is a 50 % probability of
heads on a coin flip.
Subjective
There is a 5% chance that
Verizon will merge with Sprint
The probability of an event is the relative likelihood that
the event will occur.
The probability of event A [denoted P(A)], must lie
within the interval from 0 to 1:
0 < P(A) < 1
If P(A) = 0, then the
event cannot occur.
If P(A) = 1, then the event
is certain to occur.
The probabilities of all simple events must sum to 1
P(S) = P(E1) + P(E2) + … + P(En) = 1
For example, if the following number of purchases were made by
credit card:
32%
P(credit card) = .32
debit card:
20% Probability
P(debit card) = .20
cash:
35%
P(cash) = .35
check:
13%
P(check) = .13
Sum = 100%
Sum = 1.0
What is the complement of the probability of an event
The probability of event A = P(A).
The probability of the complement of the event A’ = P(A’)
• A’ is called “A prime”
Complement of A just means probability of “not A”
• P(A) + P(A’) = 100%
• P(A) = 100% - P(A’)
• P(A’) = 100% - P(A)
Probability of
getting a rotten apple
5% chance of “rotten apple”
95% chance of “not rotten apple”
100% chance of rotten or not
Probability of getting
into an educational program
66% chance of “admitted”
34% chance of “not admitted”
100% chance of admitted or not
Two mutually exclusive characteristics: if the occurrence of any
one of them automatically implies the non-occurrence of the
remaining characteristic
Two events are mutually exclusive if they cannot occur at the
same time (i.e. they have no outcomes in common).
Two propositions that logically cannot both be true.
Warranty
No
Warranty
For example, a car repair is
either covered by the
warranty (A) or not (B).
http://www.thedailyshow.com/video/index.jhtml?videoId=188474&title=an-arab-family-man
Collectively Exhaustive Events
Events are collectively exhaustive if their union is
the entire sample space S.
Two mutually exclusive, collectively exhaustive
events are dichotomous (or binary) events.
Warranty
No
Warranty
For example, a car repair is
either covered by the
warranty (A) or not (B).
Satirical take on being “mutually exclusive”
Warranty
Recently a public figure in the heat of the
moment inadvertently made a statement that
reflected extreme stereotyping that many would
Arab
find highly offensive. It is within this context that
comical satirists have used the concept of
being “mutually exclusive” to have fun with the
statement.
Transcript:
Speaker 1:
“He’s an Arab”
Speaker 2:
“No ma’am, no ma’am.
He’s a decent, family
man, citizen…”
http://www.thedailyshow.com/video/index.jhtml?videoId=188474&title=an-arab-family-man
No
Warranty
Decent ,
family man
Union versus Intersection
Union of two events means
Event A or Event B will happen
∩
P(A
B)
Intersection of two events means
Event A and Event B will happen
Also called a “joint probability”
P(A ∩ B)
The union of two events: all outcomes in the
sample space S that are contained either in event
A or in event B or both (denoted A  B or “A or B”).
 may be read as “or” since one or the
other or both events may occur.
5A-55
The union of two events: all outcomes contained
either in event A or in event B or both
(denoted A  B or “A or B”).
What is probability of drawing a red card or a queen?
what is Q  R?
It is the possibility of drawing
either a queen (4 ways)
or a red card (26 ways)
or both (2 ways).
Probability of
picking a Queen
Probability of
picking a Red
26/52
4/52
2/52
P(Q) = 4/52
(4 queens in a deck)
P(R) = 26/52
(26 red cards in a deck)
P(Q  R) = 2/52
Probability of
picking both
R and Q
P(Q  R) = P(Q) + P(R) – P(Q  R)
= 4/52 + 26/52 – 2/52
= 28/52 = .5385 or 53.85%
(2 red queens in a deck)
When you add the P(A)
and P(B) together, you
count the P(A and B) twice.
So, you have to subtract
P(A  B) to avoid overstating the probability.
Union versus Intersection
Union of two events means
Event A or Event B will happen
∩
P(A
B)
Intersection of two events means
Event A and Event B will happen
Also called a “joint probability”
P(A ∩ B)
The intersection of two events: all outcomes
contained in both event A and event B
(denoted A  B or “A and B”)
What is probability of drawing red queen?
what is Q  R?
It is the possibility of drawing
both a queen and a red card
(2 ways).
If two events are mutually exclusive (or disjoint) their intersection is
a null set (and we can use the “Special Law of Addition”)
P(A ∩ B) = 0
Intersection of two events means
Event A and Event B will happen
Examples:
If A = Poodles
mutually
exclusive
If B = Labradors
Poodles and Labs:
Mutually Exclusive
(assuming purebred)
If two events are mutually exclusive (or disjoint) their intersection is
a null set (and we can use the “Special Law of Addition”)
Intersection of two events means
Event A and Event B will happen
P(A ∩ B) = 0
Dog Pound
Examples:
If A = Poodles
(let’s say 10% of dogs are poodles)
If B = Labradors
(let’s say 15% of dogs are labs)
What’s the probability of picking a
poodle or a lab at random from pound?
P(A B) = P(A) +P(B)
P(poodle or lab) = P(poodle) + P(lab)
∩
P(poodle or lab) = (.10) + (.15) = (.25)
Poodles and Labs:
Mutually Exclusive
(assuming purebred)
Conditional Probabilities
Probability that A has occurred given that B has occurred
P(A | B) =
P(A ∩ B)
P(B)
Denoted P(A | B):
The vertical line “ | ” is read as “given.”
The sample space is restricted to
B, an event that has occurred.
A  B is the part of B that is also
in A.
The ratio of the relative size of
A  B to B is P(A | B).
Conditional Probabilities
Probability that A has occurred given that B has occurred
Of the population aged 16 – 21 and not in college:
P(U) = .1350
Unemployed
13.5%
No high school diploma
29.05%
Unemployed with no high school diploma
5.32%
P(ND) = .2905
P(UND) = .0532
What is the conditional probability that a member of this
population is unemployed, given that the person has no diploma?
P(A | B) =
P(A ∩ B)
P(B)
=
.0532
= .1831
.2905
or 18.31%
Conditional Probabilities
Probability that A has occurred given that B has occurred
Of the population aged 16 – 21 and not in college:
P(U) = .1350
Unemployed
13.5%
No high school diploma
29.05%
Unemployed with no high school diploma
5.32%
P(ND) = .2905
P(UND) = .0532
What is the conditional probability that a member of this
population is unemployed, given that the person has no diploma?
P(A | B) =
P(A ∩ B)
P(B)
=
.0532
= .1831
.2905
or 18.31%