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Sample ◦ Variance ◦ Standard Deviation Population ◦ Variance ◦ Standard Deviation 3STA 291 Fall 2009 Lecture 7
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STA 291Fall 2009
Lecture 7Dustin Lueker
Symbols
2
2
(mu) (sigma)
(sigma-squared) or (x-i) (x-bar)
i
population meanpopulation standard deviation
population variancex x observationx sample means
s
2
sample standard deviation
s sample varianceummation symbol
STA 291 Fall 2009 Lecture 7
Variance and Standard Deviation Sample
◦ Variance
◦ Standard Deviation
Population◦ Variance
◦ Standard Deviation
3
22 ( )
1ix x
sn
2( )1
ix xs
n
22 ( )ix
N
2( )ix
N
STA 291 Fall 2009 Lecture 7
4
Variance Step By Step1. Calculate the mean2. For each observation, calculate the
deviation3. For each observation, calculate the squared
deviation4. Add up all the squared deviations5. 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)
STA 291 Fall 2009 Lecture 7
Empirical Rule 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
5STA 291 Fall 2009 Lecture 7
Empirical Rule
STA 291 Fall 2009 Lecture 7 6
Probability Terminology Experiment
◦ Any activity from which an outcome, measurement, or other such result is obtained
Random (or Chance) Experiment◦ An experiment with the property that the outcome cannot
be predicted with certainty Outcome
◦ Any possible result of an experiment Sample Space
◦ Collection of all possible outcomes of an experiment Event
◦ A specific collection of outcomes Simple Event
◦ An event consisting of exactly one outcome
7STA 291 Fall 2009 Lecture 7
Experiment, Sample Space, Event
STA 291 Fall 2009 Lecture 7 8
Examples:Experiment1. Flip a coin2. Flip a coin 3 times3. Roll a die4. Draw a SRS of size
50 from a population
Sample Space
1.2.3.4.
Event1.2.3.4.
Complement Let A denote an event Complement of an event A
◦ Denoted by AC, all the outcomes in the sample space S that do not belong to the event A
◦ P(AC)=1-P(A)
Example◦ If someone completes 64% of his passes, then
what percentage is incomplete?
9STA 291 Fall 2009 Lecture 7
SA
Union and Intersection 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
10STA 291 Fall 2009 Lecture 7
Additive Law of Probability Let A and B be two events in a sample
space S◦ P(A∪B)=P(A)+P(B)-P(A∩B)
11STA 291 Fall 2009 Lecture 7
S
A B
Additive Law of Probability 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?
12STA 291 Fall 2009 Lecture 7
Disjoint Events (Mutually Exclusive) 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 events in a sample space S◦ P(A∪B)=P(A)+P(B)
13STA 291 Fall 2009 Lecture 7
S
A B
Assigning Probabilities to Events 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?
14STA 291 Fall 2009 Lecture 7
Assigning Probabilities to Events Can be difficult Different approaches to assigning probabilities to
events◦ Subjective◦ Objective
Equally likely outcomes (classical approach) Relative frequency
15STA 291 Fall 2009 Lecture 7
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]
Subjective Probability Approach
STA 291 Fall 2009 Lecture 7 16
Equally Likely (Laplace) 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 thenP(A) = m/n
STA 291 Fall 2009 Lecture 7 17
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
Examples
18STA 291 Fall 2009 Lecture 7
Borrows from calculus’ concept of the limit
◦ 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
Relative Frequency (von Mises)
19
naAP
n lim)(
naAP )(
STA 291 Fall 2009 Lecture 7
“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
Relative Frequency Approach
20STA 291 Fall 2009 Lecture 7
Probabilities of Events Let A be the event A = {o1, o2, …, ok},
where o1, o2, …, ok are k different outcomes
Suppose the first digit of a license plate is randomly selected between 0 and 9◦ What is the probability that the digit 3?
◦ What is the probability that the digit is less than 4?
21
1 2( ) ( ) ( ) ( )kP A P o P o P o
STA 291 Fall 2009 Lecture 7
Conditional Probability
◦ Note: P(A|B) is read as “the probability that A occurs given that B has occurred”
22
( )( | ) , provided ( ) 0( )
P A BP A B P BP B
STA 291 Fall 2009 Lecture 7
Independence If events A and B are independent, then the
events have no influence on each other◦ P(A) is unaffected by whether or not B has
occurred◦ Mathematically, if A is independent of B
P(A|B)=P(A)
Multiplication rule for independent events A and B◦ P(A∩B)=P(A)P(B)
23STA 291 Fall 2009 Lecture 7
Example Flip a coin twice, what is the probability of
observing two heads?
Flip a coin twice, what is the probability of observing a head then a tail? A tail then a head? One head and one tail?
A 78% free throw shooter is fouled while shooting a three pointer, what is the probability he makes all 3 free throws? None?
24STA 291 Fall 2009 Lecture 7