Chapter 2 - Sections 2.3 And 2
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CHAPTER 2
One – way Table
One-way or frequency tables represent the simplest method for analyzing categorical (nominal) data. Nominal Variables are variables which take on one of a set of discrete values, such as Gender={Male, Female}.
Example: Net New Workers, 1985-2000 Category Percent -------------------------------------White women 42 White men 15 Nonwhite women 14 Nonwhite men 7 Immigrant women 9 Immigrant men 13
Obs. In general, the information in a table is mutually exclusive.
Two – way Table
When only two variables are cross tabulated, we call the resulting table a two-way table.
Women Men Total --------------------------------------------------------------------White | 42% 15% 57% Nonwhite | 14% 7% 21% Immigrant | 9% 13% 22% Total | 65% 35% 100%
Question: Does adding across rows or columns give numbers that are interpretable?
Venn Diagram
One useful way of understanding the relations between sets is by using Venn diagrams.
Women
Women
White
Nonwhite
White
Venn Diagram
SET OF NOTATION
We have a set S, with elements S={1,2,3,4}. If A={1,2} and B={2,3,4}, then A and B are subsets of S. The element 2 is an element of A.
The union of A and B is the set consisting of all points is either A or B or in both. A U B = {1,2,3,4}. If C={3} then A U C ={1,2,3}
The intersection of A and B is the set consisting of all points that are both in A and B. A ∩ B =AB= {2}. Therefore A ∩ C = Ø, the empty set, the set consisting of no points. The complement of A with respect to S, is the set of all points in S that are not in A and is denoted Ā. Ā = {3,4}.
Venn Diagram
Two sets are said to be mutually exclusive or disjoint, if they have no points in common, as in A and C previously.
We can observe that A U Ā = S
Distributive Laws:
De Morgan’s Laws:
A ∩ (B U C) = (A ∩ B) U (A ∩ C) A U (B ∩ C) = (A U B) ∩ (A U C) AUB=A ∩ B A ∩B =AUB
Venn Diagram
Example:
Twenty electric motors are pulled from an assembly line and inspected for defects. Eleven motors are free of defects, eight have defects on the exterior finish and three have defects in their assembly and will not run. Let A denote the set of motors having assembly defects and F the set having defects on their finish. Using A and F, write a symbolic notation for the following. Then give the number of motors in each set.
A) the set of motors having both types of defects B) the set of motors having at least one type of defect C) the set of motors having no defects D) the set of motors having exactly one type of defect
A={set of motors having assembly defects} and F={set of motors having defects on their finish} a) Motors with both types of defects must be in A and F, therefore in A ∩ F. Since only 9 motors have defects, while A contains 3 and F contains 8 motors, 2 motors must be in A ∩ F.
b) The motors having at least one type of defect must have either an assembly or a finish defect. Therefore, this is written as A U F. Since we have 20 motors and 11 are free of defects, the answer here is 9. c) Motors having no defects are 11. This can be written as ¯AUF . d) Motors with exactly one type of defect are in either A or either B, not in both. This is written as A U F – A ∩ F or ¯A ∩ F U ¯F ∩ A . The answer is 7 motors.
HOMEWORK
2.13
2.4 Definition of Probability
A sample space S is a set that includes all possible outcomes for a random selection from a specified population, listed in a mutually exclusive (elements do not overlap) and exhaustive manner (the list contains all possible outcomes).
For a die toss: S={1,2,3,4,5,6}
For a coin toss: S={H,T}
S={x | x > 2} – all real numbers x, such that x >2.
An event is any subset of a sample space.
When tossing a die: E ={4} E={1,2,3}
2.4 Definition of Probability
Using a Venn diagram display the following events and sample space:
In the die-tossing experiment let us define:
A – to be “an odd number” B – to be “ an even number” C – to be “a number less than 4” E – to be “observe a 6” S – to be “the sample space foe tossing a die”
Every time a die is tossed one of the integers 1,2,3,4,5,6 must occur, so the total probability associated with the sample space must be 1.
How many times should a “4” occur, if the six outcomes occur with the same likelihood?
2.4 Definition of Probability
Definition: Suppose a random number from a specified population has associated with it a sample space S. A probability is a numerically valued function that assigns a number P(A) to every event A so that the following axioms hold:
1. P(A) ≥ 0 2. P(S) =1 3. If A1, A2 … is a sequence of mutually exclusive events (that is Ai Aj = Ø for any i≠ j), then
P(Ui=1∞ Ai) = ∑i=1∞ P(Ai)
If A and B are two mutually exclusive events, then P(A U B) = P(A) + P(B).
If A is a subset of B, then P(A) ≤ P(B). Explain!
2.4 Definition of Probability
If a die is balanced, we could toss it a few times to see if the upper faces all seem equally likely to occur.
We could simply assume this would happen and assign probability 1/6 to each of the six elements in S as follows: P(Ei)=1/6, i=1,2…6.
Calculate the following:
P (of getting an even number) P (of getting an odd number) P (of getting a number less than 5)
2.4 Definition of Probability
In 1956 the Bureau of Labor Statistics provided the following information: 32% are married men, 37% are single, 16% are married women and the rest are widowed or divorced. In 1982, the situation is as follows: 25% are married men, 42% are single, 18% are married women and the rest are widowed or divorced.
Represent graphically the two situations. If we choose a person from 1956, what is the probability of selecting A) a married woman B) a single person C) a married person D) a person that is not married
Answer the same questions for a person from 1982.
Answer
a) P(married woman) = 0.16 b) P (single)= 0.37 c) P(married) = P(married woman) + P( married man) = 0.16+0.32=0.48
HOMEWORK
2.18
One – way Table
One-way or frequency tables represent the simplest method for analyzing categorical (nominal) data. Nominal Variables are variables which take on one of a set of discrete values, such as Gender={Male, Female}.
Example: Net New Workers, 1985-2000 Category Percent -------------------------------------White women 42 White men 15 Nonwhite women 14 Nonwhite men 7 Immigrant women 9 Immigrant men 13
Obs. In general, the information in a table is mutually exclusive.
Two – way Table
When only two variables are cross tabulated, we call the resulting table a two-way table.
Women Men Total --------------------------------------------------------------------White | 42% 15% 57% Nonwhite | 14% 7% 21% Immigrant | 9% 13% 22% Total | 65% 35% 100%
Question: Does adding across rows or columns give numbers that are interpretable?
Venn Diagram
One useful way of understanding the relations between sets is by using Venn diagrams.
Women
Women
White
Nonwhite
White
Venn Diagram
SET OF NOTATION
We have a set S, with elements S={1,2,3,4}. If A={1,2} and B={2,3,4}, then A and B are subsets of S. The element 2 is an element of A.
The union of A and B is the set consisting of all points is either A or B or in both. A U B = {1,2,3,4}. If C={3} then A U C ={1,2,3}
The intersection of A and B is the set consisting of all points that are both in A and B. A ∩ B =AB= {2}. Therefore A ∩ C = Ø, the empty set, the set consisting of no points. The complement of A with respect to S, is the set of all points in S that are not in A and is denoted Ā. Ā = {3,4}.
Venn Diagram
Two sets are said to be mutually exclusive or disjoint, if they have no points in common, as in A and C previously.
We can observe that A U Ā = S
Distributive Laws:
De Morgan’s Laws:
A ∩ (B U C) = (A ∩ B) U (A ∩ C) A U (B ∩ C) = (A U B) ∩ (A U C) AUB=A ∩ B A ∩B =AUB
Venn Diagram
Example:
Twenty electric motors are pulled from an assembly line and inspected for defects. Eleven motors are free of defects, eight have defects on the exterior finish and three have defects in their assembly and will not run. Let A denote the set of motors having assembly defects and F the set having defects on their finish. Using A and F, write a symbolic notation for the following. Then give the number of motors in each set.
A) the set of motors having both types of defects B) the set of motors having at least one type of defect C) the set of motors having no defects D) the set of motors having exactly one type of defect
A={set of motors having assembly defects} and F={set of motors having defects on their finish} a) Motors with both types of defects must be in A and F, therefore in A ∩ F. Since only 9 motors have defects, while A contains 3 and F contains 8 motors, 2 motors must be in A ∩ F.
b) The motors having at least one type of defect must have either an assembly or a finish defect. Therefore, this is written as A U F. Since we have 20 motors and 11 are free of defects, the answer here is 9. c) Motors having no defects are 11. This can be written as ¯AUF . d) Motors with exactly one type of defect are in either A or either B, not in both. This is written as A U F – A ∩ F or ¯A ∩ F U ¯F ∩ A . The answer is 7 motors.
HOMEWORK
2.13
2.4 Definition of Probability
A sample space S is a set that includes all possible outcomes for a random selection from a specified population, listed in a mutually exclusive (elements do not overlap) and exhaustive manner (the list contains all possible outcomes).
For a die toss: S={1,2,3,4,5,6}
For a coin toss: S={H,T}
S={x | x > 2} – all real numbers x, such that x >2.
An event is any subset of a sample space.
When tossing a die: E ={4} E={1,2,3}
2.4 Definition of Probability
Using a Venn diagram display the following events and sample space:
In the die-tossing experiment let us define:
A – to be “an odd number” B – to be “ an even number” C – to be “a number less than 4” E – to be “observe a 6” S – to be “the sample space foe tossing a die”
Every time a die is tossed one of the integers 1,2,3,4,5,6 must occur, so the total probability associated with the sample space must be 1.
How many times should a “4” occur, if the six outcomes occur with the same likelihood?
2.4 Definition of Probability
Definition: Suppose a random number from a specified population has associated with it a sample space S. A probability is a numerically valued function that assigns a number P(A) to every event A so that the following axioms hold:
1. P(A) ≥ 0 2. P(S) =1 3. If A1, A2 … is a sequence of mutually exclusive events (that is Ai Aj = Ø for any i≠ j), then
P(Ui=1∞ Ai) = ∑i=1∞ P(Ai)
If A and B are two mutually exclusive events, then P(A U B) = P(A) + P(B).
If A is a subset of B, then P(A) ≤ P(B). Explain!
2.4 Definition of Probability
If a die is balanced, we could toss it a few times to see if the upper faces all seem equally likely to occur.
We could simply assume this would happen and assign probability 1/6 to each of the six elements in S as follows: P(Ei)=1/6, i=1,2…6.
Calculate the following:
P (of getting an even number) P (of getting an odd number) P (of getting a number less than 5)
2.4 Definition of Probability
In 1956 the Bureau of Labor Statistics provided the following information: 32% are married men, 37% are single, 16% are married women and the rest are widowed or divorced. In 1982, the situation is as follows: 25% are married men, 42% are single, 18% are married women and the rest are widowed or divorced.
Represent graphically the two situations. If we choose a person from 1956, what is the probability of selecting A) a married woman B) a single person C) a married person D) a person that is not married
Answer the same questions for a person from 1982.
Answer
a) P(married woman) = 0.16 b) P (single)= 0.37 c) P(married) = P(married woman) + P( married man) = 0.16+0.32=0.48
HOMEWORK
2.18
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