THINK, VALUE, COMMUNICATE, ACT

1
THINK, VALUE, COMMUNICATE, ACT
Class Pay attention Ask questions
Before class Read assignment
Tutor
After class Do assigned homework before next class
Additional resources
Group study
WWW
MSC
2
PROBLEM SOLVING

• List everything that you know about the problem.

• Identify the question or unknown.

• List any underlying formulae.

• Draw a picture.

• Solve the problem.
• (Dont take short cuts or skip any steps).

3
DESCRIBING SETS

• Verbal Definition
• Set of integers between one and 10
• integers between one and 10

• Roster or Listing
• 2,3,4, . . . ,9

• Set Builder Notation
• x 1 lt x lt 10 ? x ? I

4
2 SET OPERATIONS - REGIONS
5
NUMBER SETS
U UNIVERSAL SET
IRRATIONAL NUMBERS
R REAL NUMBERS
INTEGERS
WHOLE NUMBERS
NATURAL NUMBERS
RATIONAL NUMBERS
6
SET OPERATIONS UNION AND INTERSECTION
U UNIVERSAL SET
(A?B)
B
A
A?B
A?B
7
2 SET OPERATIONS - REGIONS
8
3 SET OPERATIONS - REGIONS
U UNIVERSAL SET
A
B
I
III
II
V
IV
VI
VII
VIII
C
9
LOGIC SYMBOLS
? Negation
? Conjunction
? Disjunction
? Conditional statement
? Biconditional statement
? Equivalent Statements
10
TRUTH TABLE TWO SIMPLE STATEMENTS
11
TRUTH TABLE SHORT CUTS
Conjunction (and) - Is true only when both are
true.
Disjunction (or) - Is false only when both are
false.
Conditional Statement (if then) - Is false only
when
the antecedent is true and the consequent is
false.
Biconditional Statement (if and only if) - Is
true only when
the antecedent and the consequent have the same
truth value.
12
TRUTH TABLE THREE SIMPLE STATEMENTS
13
VARIATIONS OF THE CONDITIONAL STATEMENT
Equivalent
Equivalent
Contra- positive ?q ? ?p T F T T
Conditional p ? q T F T T
Converse q ? p T T F T
Inverse ?p ? ?q T T F T
14
LINES AND LINE SEGMENTS
15
LINES IN PLANES
D
B
A
B
D
A
C
Parallel Lines
C
Intersecting Lines
B
A
D
C
Skew Lines
16
CLASSIFICATION OF ANGLES
17
EQUAL ANGLES
18
CLASSIFICATION OF TRIANGLES
BY ANGLES
BY SIDES
19
LABELING THE PARTS OF TRIANGLES
20
TYPES OF QUADRILATERALS
21
CIRCLES
Circumference
Diameter
Radius
22
PERIMETER AND AREA OF TRIANGLES
Perimeter p s1 s2 s3
Area A ½ bh
23
PERIMETER AND AREA
Triangle P s1 s2 s3 A ½ bh
Rectangle P s1 s2 s3 s4 A bh
Square P 4 s A s 2
Trapezoid P s1 s2 s3 s4 A b1 b2 h 2
Parallelogram P s1 s2 s3 s4 A bh
Circle P C 2 ? r A ? r 2
24
CONGRUANT TRIANGLES
? ABC
? DEF
PROPERTIES
Side-Angle-Side (SAS)
cf BE ad
Angle-Side-Angle (ASA)
BE ad CF
Side-Side-Side (SSS)
ad be cf
25
SOLID POLYGONS AND THEIR VOLUMES
26
COUNTING TECHNIQUES
27
PROBABILITY DEFINITIONS
Probability mathematical estimate of the
likelihood that some event will occur.
Empirical Probability
frequency of occurrence of an event as
determined by an experiment.
Theoretical Probability
frequency of occurrence of an event as
determined mathematically
28
PROBABILITY DEFINITIONS
Experiment the examination estimate of the
likelihood that some event will occur.
Event subset of outcomes of an experiment.
Outcome result of an experiment.
Sample Space set of all possible outcomes.
Equal Likelihood each event in the sample space
has the same probability of occurring.
29
PROBABILTY EQUATIONS
Empirical Probability P(E) number of times
event E occurred . number of
times experiment performed
Theoretical Probability P(E) number of
favorable outcomes . total number of
possible outcomes
30
BASIC PROPERTIES OF PROBILITIES
1. The probability of an event is always between
0 and 1 inclusive. 0 lt P(E) lt 1
2. The probability of an event that cannot occur
(impossible event) is 0.
3. The probability of an event that must occur
(certain event) is 1.
4. The sum of the probabilities of all possible
outcomes of an experiment is 1. SP(E) 1
5. The probability of an event not occurring is
1 minus the probability of its occurring.
P(E) 1 P(E) or P(E) 1 P(E)
31
LAWS OF PROBABILITIESOR
P(A or B) P(A ? B) P(A) P(B) P(A ? B)
P(A or B) A and B Mutually Exclusive P(A ? B)
P(A) P(B)
32
PROBABILITY VENN DIAGRAM
U UNIVERSAL SET
P(A?B)
P(B)
P(A)
P(A?B)
P(A?B)
33
LAWS OF PROBABILITIESAND
P(A given B) P(AB)P(A ? B) P(B)
P(A and B) P(A ? B) P(A) P(BA)
P(A and B) A and B Independent P(A ? B) P(A)
P(B)
34
EXPECTED VALUE
Expected Value The mean (average) value of
weighted events. Weighted Events - Events with
different probabilities of occurrence.
35
EXPECTED VALUEEXAMPLE 1
S ggg, ggb, gbg, bgg, gbb, bgb, bbg, bbb
36
EXPECTED VALUEEXAMPLE 2
Cost to Play 3.00
37
EXPECTED VALUEEXAMPLE 5
Cost to Play 1.00
38
FAIR PRICE EXAMPLE 4
Cost to Play 1.00
Fair Price Expected Value Cost to Play
-0.40 1.00 0.60
39
EXPECTED VALUE - EXAMPLE 6
40
STATISTICS
41
CLASSIFYING STATISTICAL STUDIES
Descriptive Statistics
Inferential Statistics
42
POPULATION AND SAMPLE
43
DATA
44
TERMINOLOGY
Frequency The number of identical
observations.   Frequency distribution A listing
of all observations along with their
frequencies.   Relative frequency The ratio of
the frequency of an observation to the total
number of observations.   Relative-frequency
distribution A listing of all observations along
with their relative frequencies.
45
FREQUENCY DISTRIBUTION TABLE
Number of Siblings
46
GROUPING DATA
Classes Categories for grouping data
Class Width Interval for each class. Widths
should all be the same.
Lower Class Limit The smallest value that can
go into a class.
Upper Class Limit The largest discrete value
that can go into a class or, for continuous data,
The lower class limit of the next higher class
Modal Class Class with the greatest frequency.
Midpoint of a Class (Class Mark) Point half
way between the upper and lower limit.
47
TABLE FOR GROUPED DATA
Hours Spent Studying Weekly
48
GRAPHS AND CHARTS
49
GRAPHED DATA
50
CONSTRUCTING A PIE CHART
Student Monthly Expenses
51
PIE CHART
Student Monthly Expenses
52
MEASURES OF CENTRAL TENDENCY
Mean of a data set - The sum of the observations
divided by the number of observations.
Weighted Mean of a data set The mean of
observations multiplied by their respective
weights (frequency).
Median of a data set - The observation exactly in
the middle of an ordered list.
Mode of a data set - The observation that occurs
the most frequently.
Midrange Value half way between the highest and
lowest value
53
MEASURES OF CENTRAL TENDANCY
Ages of Students
19 20 22 25 26 28
1 3 8 7 4 2 n25
.04 .12 .32 .28 .16 .08 1.00
0.76 2.40 7.04 7.00 4.16 2.24 23.60
22.00
25.00
54
SKEWED DISTRIBUTIONS
55
MEASURES OF DISPERSION
Range of a data set The difference between the
highest and lowest observation..
Standard Deviation of a data set A statistical
measure of how far observations are from the mean.
Variance of a data set The square of the Standard
Deviation.
56
SAMPLE MEASURES OF VARIATION
57
MEASURES OF VARIATION
32 41 47 53 57
-14 - 5 1 7 11 0 (always)
196 25 1 49 121 392
Mean 46
230/546
58
MEASURES OF VARIATIONWEIGHTED DISTRIBUTIONS
2 3 4 5
-1.36 -0.36 0.64 1.64 Not 0
5 8 10 2 n25
10 24 40 10
1.8496 0.1296 0.4096 2.6896
9.2480 1.0368 4.0960 5.3792 19.7600
84/253.36
Mean 3.36
59
NOTATION USED FOR A SAMPLE AND FOR A POPULATION
Standard Deviation s ?
60
COEFFICIENT OF VARIATION
RELATIVE DISPERSION
Sample Coefficient
Population Coefficient
61
Z-SCORE
The number of standard deviations to the left (-)
or right () of the mean
62
QUARTILES
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
?
0 2 4 6 8 10 14 18 22
26 30 34 38 42 44
Q1
Q2
Q3
Q4
63
COMMON DISTRIBUTION SHAPES
64
PROPERTIES OF THE NORMAL DISTRIBUTION
65
GRAPH OF GENERIC NORMAL DISTRIBUTION
66
PROPERTIES OF THE STANDARD NORMAL CURVE
All of the properties of the Normal Curve
plus Mean 0 Standard Deviation 1
67
AREA UNDER THE STANDARD NORMAL CURVE
A
? z