Lecture 7 Dustin Lueker. 2STA 291 Fall 2009 Lecture 7

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


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)


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


Experiment, Sample Space, Event


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?


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


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)


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?


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)


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?


Assigning Probabilities to Events Can be difficult Different approaches to assigning probabilities to

events◦ Subjective◦ Objective

Equally likely outcomes (classical approach) Relative frequency


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


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


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


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 )(


“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


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


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


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)


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?


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Lecture 7 Dustin Lueker. 2STA 291 Fall 2009 Lecture 7