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 Q u, M, Q L, 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

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 Space Example: When we roll two dice the sample space is all possible pairs of rolls 1,1 1,2 1,31,4 1,5 1,6 2,1 2,2 2,32,4 2,5 2,6 3,13,2 3,33,4 3,5 3,6 4,14,2 4,34,4 4,5 4,6 5,15,2 5,35,4 5,5 5,6 6,16,2 6,36,4 6,5 6,6

Sample Space Event Simple Events (all the red dots) We often represent the sample space with a Venn Diagram.

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.

Assigning Probabilities Each Simple event has a probability associated with it. This is really the relative frequency of the simple event.

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 Black7 Brown4 White1 Total12

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? FrequencyProbability Black77/12 Brown44/12 White11/12 Total121

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 Space Example: Roll two dice. What is the probability of rolling a 9? 1,1 1,2 1,31,4 1,5 1,6 2,1 2,2 2,32,4 2,5 2,6 3,13,2 3,33,4 3,5 3,6 4,14,2 4,34,4 4,5 4,6 5,15,2 5,35,4 5,5 5,6 6,16,2 6,36,4 6,5 6,6

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

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

Properties of Probability

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. A B 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. ACAC

Compliment The compliment of an event A is the event that A does not occur. We use A C to denote the compliment of A. P(A C )= 1 - P(A)

Compliment The compliment of an event A is the event that A does not occur. We use A C to denote the compliment of A. P(A)= 1 - P(A C )

Example For an experiment of randomly selecting one card from a deck of 52 cards, let A=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(D C ) b) P(B and C) c)P(B or C) d) P(C and D) e)P(A or B)f) P(B)

Example For an experiment of randomly selecting one card from a deck of 52 cards, let A=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(D C ) =40/52 b) P(B and C)= 1/52 c)P(B or C)=16/52d) P(C and D)=3/52 e)P(A or B)=4/52f) 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(E C )= P(F C )= 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(E C ) = 0.7 P(F C ) = 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 problems b) 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) 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) 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(E C or F C ) P(E C and F C )P(E C and F)

Example Suppose P (E) = 0.4, P (F) = 0.3, and P(E or F)=0.6. Find: P(E and F) =0.1P(E C or F C ) P(E C and F C )P(E C and F)

Example Suppose P (E) = 0.4, P (F) = 0.3, and P(E or F)=0.6. Find: P(E and F) =0.1P(E C or F C )=0.9 P(E C and F C )P(E C and F)

Example Suppose P (E) = 0.4, P (F) = 0.3, and P(E or F)=0.6. Find: P(E and F) =0.1P(E C or F C )=0.9 P(E C and F C )=0.4P(E C and F)

Example Suppose P (E) = 0.4, P (F) = 0.3, and P(E or F)=0.6. Find: P(E and F) =0.1P(E C or F C )=0.9 P(E C and F C )=0.4P(E C 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) =0.1P(E C or F C )=0.9 P(E C and F C )=0.4 P(E C 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 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