Download STA 291-021 Summer 2007

Lecture 6
Dustin Lueker
 (mu)
 (sigma)

population mean
population standard deviation
2
(sigma-squared)
population variance
x or xi (x-i) observation
x (x-bar)
s
2
s

sample mean
sample standard deviation
sample variance
summation symbol
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
Sample
◦ Variance
s2 
2
(
x
i

x
)

n 1
◦ Standard Deviation

Population
◦ Variance
2 
s
2
(
x
i

x
)

n 1
2
(
x
i


)

◦ Standard Deviation
N

2
(
x
i


)

N
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1.
2.
3.
4.
5.
Calculate the mean
For each observation, calculate the deviation
For each observation, calculate the squared
deviation
Add up all the squared deviations
Divide the result by (n-1)
Or N if you are finding the population variance
(To get the standard deviation, take the square root of the result)
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If the data is approximately symmetric and
bell-shaped then
◦ About 68% of the observations are within one
standard deviation from the mean
◦ About 95% of the observations are within two
standard deviations from the mean
◦ About 99.7% of the observations are within three
standard deviations from the mean
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Standardized measure of variation
◦ Idea
 A standard deviation of 10 may indicate great
variability or small variability, depending on the
magnitude of the observations in the data set

CV = Ratio of standard deviation divided by
mean
◦ Population and sample version
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Which sample has higher relative variability?
(a higher coefficient of variation)
◦ Sample A
 mean = 62
 standard deviation = 12
 CV =
◦ Sample B
 mean = 31
 standard deviation = 7
 CV =
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Experiment

Random (or Chance) Experiment
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Outcome

Sample Space

Event

Simple Event
◦ Any activity from which an outcome, measurement, or other
such result is obtained
◦ An experiment with the property that the outcome cannot
be predicted with certainty
◦ Any possible result of an experiment
◦ Collection of all possible outcomes of an experiment
◦ A specific collection of outcomes
◦ An event consisting of exactly one outcome
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

Let A denote an even
Complement of an event A
◦ Denoted by AC, all the outcomes in the sample
space S that do not belong to the even A
◦ P(AC)=1-P(A)

Example
◦ If someone completes 64% of his passes, then what
percentage is incomplete?
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
Let A and B denote two events
Union of A and B
◦ A∪B
◦ All the outcomes in S that belong to at least one of
A or B

Intersection of A and B
◦ A∩B
◦ All the outcomes in S that belong to both A and B
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Let A and B be two events in a sample space S
◦ P(A∪B)=P(A)+P(B)-P(A∩B)
 At State U, all first-year students must take chemistry
and math. Suppose 15% fail chemistry, 12% fail math,
and 5% fail both. Suppose a first-year student is
selected at random, what is the probability that the
student failed at least one course?
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

Let A and B denote two events
A and B are Disjoint (mutually exclusive)
events if there are no outcomes common to
both A and B
◦ A∩B=Ø
 Ø = empty set or null set

Let A and B be two disjoint (mutually
exclusive) events in a sample space S
◦ P(A∪B)=P(A)+P(B)
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The probability of an event occurring is
nothing more than a value between 0 and 1
◦ 0 implies the event will never occur
◦ 1 implies the event will always occur

How do we go about figuring out
probabilities?
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
Can be difficult
Different approaches to assigning probabilities to
events
◦ Subjective
◦ Objective
 Equally likely outcomes (classical approach)
 Relative frequency
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Relies on a person to make a judgment on
how likely an event is to occur
◦ Events of interest are usually events that cannot be
replicated easily or cannot be modeled with the
equally likely outcomes approach
 As such, these values will most likely vary from person
to person

The only rule for a subjective probability is
that the probability of the event must be a
value in the interval [0,1]
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The equally likely approach usually relies on
symmetry to assign probabilities to events
◦ As such, previous research or experiments are not
needed to determine the probabilities
 Suppose that an experiment has only n outcomes
 The equally likely approach to probability assigns a
probability of 1/n to each of the outcomes
 Further, if an event A is made up of m outcomes then
P(A) = m/n
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Selecting a simple random sample of 2
individuals
◦ Each pair has an equal probability of being selected

Rolling a fair die
◦ Probability of rolling a “4” is 1/6
 This does not mean that whenever you roll the die 6
times, you always get exactly one “4”
◦ Probability of rolling an even number
 2,4, & 6 are all even so we have 3 possibly outcomes in
the event we want to examine
 Thus the probability of rolling an even number is
3/6 = 1/2
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Borrows from calculus’ concept of the limit
a
P( A)  lim
n  n
◦ We cannot repeat an experiment infinitely many
times so instead we use a ‘large’ n
 Process
 Repeat an experiment n times
 Record the number of times an event A occurs, denote this
value by a
 Calculate the value of a/n
a
P( A) 
n
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
“large” n?
◦ Law of Large Numbers
 As the number of repetitions of a random experiment
increases, the chance that the relative frequency of
occurrence for an event will differ from the true
probability of the even by more than any small number
approaches 0
 Doing a large number of repetitions allows us to
accurately approximate the true probabilities using the
results of our repetitions
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