Review

Detecting Outliers

Review

Detecting Outliers– Standard Deviation– Percentiles/Box Plots

– Suspected and Highly Suspected Outliers

Review

Detecting Outliers– Standard Deviation– Percentiles/Box Plots

– Suspected and Highly Suspected Outliers

Review

Detecting Outliers– Standard Deviation

• Chebyshev’s Rule• Emperical Rule• Which points are within k standard deviations?• Z-scores• Suspected and Highly Suspected Outliers

Review

Detecting Outliers– Percentiles/Box Plots

• Find Percentiles

• Find Qu, M, QL, IQR. – *** Use the method I showed you, not your calculator***

• Building a box plot– Calculate the Upper/Lower Inner and Outer Fences– *** Use the method I showed you, not your calculator***– Include a menu and show all your work

• Suspected and Highly Suspected Outliers

Examples

Big Picture

Detecting Outliers

Big Picture (Outliers)

Typically we know a lot of historical data about what we are trying to test. From that data we estimate what the population center (the mean) and population standard deviation are. We can:

1) make predictions (within a certain percentage chance) about future events.

2) collect new data and check to see if that would be an outlier in the old data.

Probability

Probability

An experiment is any process that allows researchers to obtain observations.

An event is any collection of results or outcomes of an experiment.

A simple event is an outcome or an event that cannot be broken down any further.

Example

Rolling a die is an experiment. It has 6 different possible outcomes

An example of an event is rolling a 5.

Rolling a 5 is a simple event. It cannot be broken down any further.

Example

Rolling a die is an experiment. It has 6 different possible outcomes.

Another example of an event is rolling an odd number.

This event can be broken down into three simple events: Rolling a 1, rolling a 3 and rolling a 5.

Sample Space

The sample space for an experiment consists of all simple events.

Example: When we roll on die the sample space is: 1, 2, 3, 4, 5, 6

Sample Space

Example: When we roll on die the sample space is: 1, 2, 3, 4, 5, 6

Example: When we roll two dice the sample space is:

Sample SpaceExample: When we roll two dice the sample space is all possible pairs of rolls

1,1 1,2 1,3 1,4 1,5 1,6

2,1 2,2 2,3 2,4 2,5 2,6

3,1 3,2 3,3 3,4 3,5 3,6

4,1 4,2 4,3 4,4 4,5 4,6

5,1 5,2 5,3 5,4 5,5 5,6

6,1 6,2 6,3 6,4 6,5 6,6

Sample Space

Sample Space

Event

Simple Events (all the red dots)

We often represent the sample space with a Venn Diagram.

Sample Space

Sample Space

Event

Simple Events

Usually the simple events are not included in our diagram

Sample Space

Here is a Venn Diagram depicting two events which overlap, or intersect.

Assigning Probabilities

Sample Space

Event

Simple Events (all the red dots)

Assigning Probabilities

Each Simple event has a probability associated with it.

trialsofnumber

successes possible ofnumber )( sP

Assigning Probabilities

Each Simple event has a probability associated with it.

This is really the relative frequency of the simple event.

occurcan points sample theall waysofnumber

occurcan s waysofnumber )( sP

Assigning Probabilities

Each Simple event has a probability associated with it.

This is really the relative frequency of the simple event.

To find the probability of an event, add up the probabilities of the simple events inside of it.

Example

A cage contains 7 black mice, 4 brown mice and 1 white mouse. A mouse is selected at random from the cage. What is the probability it is either a black mouse or a white mouse?

Frequency

Black 7

Brown 4

White 1

Total 12

Example

A cage contains 7 black mice, 4 brown mice and 1 white mouse. A mouse is selected at random from the cage. What is the probability it is either a black mouse or a white mouse?

Frequency Probability

Black 7 7/12

Brown 4 4/12

White 1 1/12

Total 12 1

Example

A cage contains 7 black mice, 4 brown mice and 1 white mouse. A mouse is selected at random from the cage. What is the probability it is either a black mouse or a white mouse?

P (Black or White) =

Example

A cage contains 7 black mice, 4 brown mice and 1 white mouse. A mouse is selected at random from the cage. What is the probability it is either a black mouse or a white mouse?

P (Black or White) = P(Black) +P(White)

Example

A cage contains 7 black mice, 4 brown mice and 1 white mouse. A mouse is selected at random from the cage. What is the probability it is either a black mouse or a white mouse?

P (Black or White) = P(Black) +P(White)

= 7/12 + 1/12 = 8/12

Sample SpaceExample: Roll two dice. What is the probability of rolling a 9?

1,1 1,2 1,3 1,4 1,5 1,6

2,1 2,2 2,3 2,4 2,5 2,6

3,1 3,2 3,3 3,4 3,5 3,6

4,1 4,2 4,3 4,4 4,5 4,6

5,1 5,2 5,3 5,4 5,5 5,6

6,1 6,2 6,3 6,4 6,5 6,6

Sample Space

Example: Roll two dice. What is the probability of rolling a 9?

)3,6()4,5()5,4()6,3()9( PPPPP

Sample Space

Example: Roll two dice. What is the probability of rolling a 9?

9

1

36

4

36

1

36

1

36

1

36

1

)3,6()4,5()5,4()6,3()9(

PPPPP

Properties of Probability

1)()()()( (2)

then},,,{

events simple of up made is space sample theIf

1)(0 (1)

satisfies sevent simple a ofy Probabilit The

321

321

n

n

sPsPsPsP

ssss

sP

Union

The union of events A and B is the event that A or B (or both) occur.

Union

The union of events A and B is the event that A or B (or both) occur.

AB

A or B

Intersection

The intersection of events A and B is the event that both A and B occur.

Intersection

The intersection of events A and B is the event that both A and B occur.

A B

A and B

Compliment

The compliment of an event A is the event that A does not occur.

Compliment

The compliment of an event A is the event that A does not occur.

AC

Compliment

The compliment of an event A is the event that A does not occur.

We use AC to denote the compliment of A.

P(AC)= 1 - P(A)

Compliment

The compliment of an event A is the event that A does not occur.

We use AC to denote the compliment of A.

P(A)= 1 - P(AC)

ExampleFor an experiment of randomly selecting one card from

a deck of 52 cards, letA=event the card selected is the King of Hearts

B=event the card selected is a King

C=event the card selected is a Heart

D=event the card selected is a face card.

Find:

a) P(DC) b) P(B and C)

c) P(B or C) d) P(C and D)

e) P(A or B) f) P(B)

ExampleFor an experiment of randomly selecting one card from

a deck of 52 cards, letA=event the card selected is the King of Hearts

B=event the card selected is a King

C=event the card selected is a Heart

D=event the card selected is a face card.

Find:

a) P(DC) =40/52 b) P(B and C)= 1/52

c) P(B or C)=16/52 d) P(C and D)=3/52

e) P(A or B)=4/52 f) P(B)=4/52

Unions and Intersections

Unions and Intersections are related by the following formulas

P(A and B)= P(A) + P(B) - P(A or B)

P(A or B)= P(A) + P(B) - P(A and B)

Mutually Exclusive

Two events are mutually exclusive if

P (A and B) = 0.

Mutually Exclusive

Two events are mutually exclusive if

P (A and B) = 0.

Suppose P (E) = .3, P (F) = .5, and E and F are mutually exclusive. Find:

P(E and F)= P(E or F)=

P(EC)= P(FC)=

P((E or F) C)= P((E and F) C)=

Mutually Exclusive

Two events are mutually exclusive if

P (A and B) = 0.

Suppose P (E) = .3, P (F) = .5, and E and F are mutually exclusive. Find:

P(E and F) = 0 P(E or F) = 0.8

P(EC) = 0.7 P(FC) = 0.5

P((E or F) C)=0.2 P((E and F) C)=1

Example

In buying a new computer (tower, monitor, keyboard and mouse) studies show that 4% have problems with their mouse and 2% have problems with their monitor and 0.2% have problems with both before the expirations of their manufactured warranty.a) Find the probability that a computer set purchased has one of the two problemsb) Neither c) Just a monitor problem

Example

In buying a new computer (tower, monitor, keyboard and mouse) studies show that 4% have problems with their mouse and 2% have problems with their monitor and 0.2% have problems with both before the expirations of their manufactured warranty.a) Find the probability that a computer set purchased has one of the two problems. (5.8%) b) Neitherc) Just a monitor problem

Example

In buying a new computer (tower, monitor, keyboard and mouse) studies show that 4% have problems with their mouse and 2% have problems with their monitor and 0.2% have problems with both before the expirations of their manufactured warranty.

a) Find the probability that a computer set purchased has one of the two problems. (5.8%)

b) Neither (94.2%)

c) Just a monitor problem (1.8%)

Example

In buying a new computer (tower, monitor, keyboard and mouse) studies show that 4% have problems with their mouse and 2% have problems with their monitor and 0.2% have problems with both before the expirations of their manufactured warranty.

a) Find the probability that a computer set purchased has one of the two problems. (5.8%)

b) Neither (94.2%)

c) Just a monitor problem (1.8%)

Example

Suppose P (E) = 0.4, P (F) = 0.3, and

P(E or F)=0.6. Find:

P(E and F) P(EC or FC)

P(EC and FC) P(EC and F)

Example

Suppose P (E) = 0.4, P (F) = 0.3, and

P(E or F)=0.6. Find:

P(E and F) =0.1 P(EC or FC)

P(EC and FC) P(EC and F)

Example

Suppose P (E) = 0.4, P (F) = 0.3, and

P(E or F)=0.6. Find:

P(E and F) =0.1 P(EC or FC)=0.9

P(EC and FC) P(EC and F)

Example

Suppose P (E) = 0.4, P (F) = 0.3, and

P(E or F)=0.6. Find:

P(E and F) =0.1 P(EC or FC)=0.9

P(EC and FC)=0.4 P(EC and F)

Example

Suppose P (E) = 0.4, P (F) = 0.3, and

P(E or F)=0.6. Find:

P(E and F) =0.1 P(EC or FC)=0.9

P(EC and FC)=0.4 P(EC and F)= 0.2

Example

Suppose P (E) = 0.4, P (F) = 0.3, and

P(E or F)=0.6. Find:

P(E and F) =0.1 P(EC or FC)=0.9

P(EC and FC)=0.4 P(EC and F)

Review

Probabilities– Definitions of experiment, event, simple event,

sample space, probabilities, intersection, union compliment

– Finding Probabilities– Drawing Venn Diagrams – If A and B are two events then

P(A or B) = P(A) + P(B) - P(A and B),

P(not A) = 1 - P(A). – Two events A and B are mutually exclusive if P(A and

B) = 0.

57

Homework

• Finish reading Chapter 3.1-3.7

• Assignment 1 due Thursday

• Quiz next Tuesday on Chapters 1 and 2

• Problems on next slide

58

Problems

• 3.15, 3.20,

• 3.44, 3.45, 3.54