+
Chapter 3
The Normal
Distributions
3.4
3.5
3.6
3.7
3.8
The 68 95 99.7 Rule
The Standard Normal Distribution
Finding Normal Proportions
Using the Standard Normal Table
Finding a Value Given a Proportion
+
3.4
The 68 95 99.7 Rule
nIn
the Normal distributi
+
Chapter 2
Describing
Distributions
with Numbers
2.1 Measuring Center: The Mean
2.2 Measuring Center: The Median
2.3 Comparing the Mean and the Median
+
Measuring the Center: the Mean
n
The most common measure of center is the arithmetic
average, or mean
+
Chapter 2
Describing
Distributions
with Numbers
2.4 Measuring Variability: the Quartiles
2.5 The Five-Number Summary and Boxplots
+
Quick Review
n
What characteristics do you need to describe when asked to
interpret a histogram?
G
S
O
C
S
-Gaps
-Shape
+
Chapter 5
Regression
5.1 Regression Lines
5.2 The Least-Squares Regression Line
5.3 Using Technology
5.4 Facts about Least-Squares Regression
+
Regression Lines
A regression line is
a line that describes the overall relationship between two
quantitativ
+
Chapter 1
Picturing
Distributions
with Graphs
1.3
1.4
1.5
1.6
Quantitative Variables: Histograms
Interpreting Histograms
Quantitative Variables: Stemplots
Time Plots
+
Histograms
n
Similar to a bar graph, but displays the distribution of a
quantitative
+
Chapter 2
Describing
Distributions with
Numbers
2.6 Spotting Suspected Outliers and Modified Boxplots
2.7 Measuring Variability: the Standard Deviation
2.8 Choosing Measures of Center and Variability
+
The Interquartile Range
n The
interquartile range (
+
Chapter 4
Scatterplots and
Correlation
4.4 Adding Categorical Variables to Scatterplots
4.5 Measuring Linear Association: Correlation
4.6 Facts about Correlation
+
Categorical Variables & Scatterplots
While
scatterplots are used to display the
relation
+
Chapter 1
Picturing
Distributions
with Graphs
1.1 Individuals and Variables
1.2 Categorical Variables: Pie Charts and Bar Graphs
+
Definitions
nIndividual
n The person or object described in a set of data.
nVariable
n Any characteristic of an individual
+
Chapter 3
The Normal
Distributions
3.1 Density Curves
3.2 Describing Density Curves
3.3 Normal Distributions
+
Density Curves
3.1
+
Density Curves
n A
density curve is a curve that:
Is always on or above a horizontal axis.
n Has an area of exactly 1 und
+
Chapter 4
Scatterplots and
Correlation
4.1 Explanatory and Response Variables
4.2 Displaying Relationships: Scatterplots
4.3 Interpreting Scatterplots
+
A Change of Focus
Distribution of 1 Quantitative Variable
Relationship Between 2 Quantitative Varia
TEST CODE
FORM TP 2015037
MAY/JUNE 2015
CARIBBEAN
EXAMINATIONS
CARIBBEAN
01254020
COUNCIL
SECONDARY EDUCATION
EXAMINATION
[email protected]
ADDITIONAL MATHEMATICS
Paper 02 - General Proficiency
2 hours 40 minutes
(
05 MAY 2015 (p.m.)
READ THE FOLLOWING
)
INSTR
1a(i)
f ( x ) = x 3 + 1; g ( x ) = x + 5
2b
g ( f ( x ) = ( x 3 + 1) + 5
= x3 + 6
(ii)Sincetherangeoff(x)is 0 x 3 andisa
subsetofRealNumbers,then 0 x 3 isthe
domainofg(f(x).Therefore:
Range of g ( f ( x) :
g ( f ( x) = x 3 + 6
= 0 +6
=6
g ( f ( x) = x 3 +
2(c) Arithmetic Sequence Problem
2(b)Range of values of x.
2x 5
>0
3x + 1
2x 5
> 0 ( 3x + 1)
( 3x + 1)
3x + 1
2x 5 > 0
By analysis we find that a1 = 30 and common difference d = 5.
to find the amount paid at the 24th month.
we use the Explicit Formula fo
1.
[1]
(a) Express cos 2x in terms of sin x .
(b)
(i) Hence show that 3 sin
+ 3 sin x
x - 2 cos 2x = Zsin?
- 1 for all values of x.
[2]
(ii) Solve the equation 3 sin x - cos 2x
2.
sin2x
-=
l-cos2x
(a) Prove the identity
(b) Hence solve the equation
3.
()a
3.3 Lone-divider method
fair division of a continuous asset among n players
inductive proof: start from the smallest cases(Divider-Chooser
works)
The Lone-Divider method will always produce a fair division of
the assets among the n players
S1
S2
S3
D
33
Chapter 3: The math of sharing
Fair division
A number of players want to divide some assets fairly.
Continuous asset: money, water, cake (can be divided) (3.2
-> 3.4)
Discrete asset: car, house (3.5 3.6)
A value function
Each player can define how val
Judgement and Decision Making
Math 114-01 Syllabus
Fall 2016
Instructor : Ryan Pellico
Contact Information :
Office Hours: MWF 1:00 - 3:00 pm,
Tues 9:30 - 10:30 am,
or by appointment
Office:
MECC 273
Phone:
860-297-2297
Email:
[email protected]
Le
11/4/16
3.2 Divider-Chooser method
Fair division of a continuous asset bw 2 people
The 2 players randomly decide who is the divider and who is
the chooser
Divider devide into 2 shares, chooser chooses 1
-> always fair share
Exam 2 Review:
(will be on
Hamiltons Method:
1.Give each its lower quota
o Ideal case: SQ1 +SQ2 +SQ3 = total # seats
o But usually LQ SQ
o -> LQ1 + LQ2 + LQ3 total seats
2. If there is a surplus of seats, give them out 1 at a time
starting with the state with the largest residue
3.5
The Method of Sealed Bids
Procedure:
Each player makes a sealed bid, giving an estimate of the
dollar value of each item. Each players fair share is computed
by dividing the total of the players bids by the # of players.
Each item goes to the highes
A fair share is 1/N of the total value, where N is the number
of players
o If 2 people are dividing a cake, they should each get a
piece worth at least 50% to them
o If 3 people,., at least 33.33% to them
A fair division method is an algorithm for assign
11/16/16
How do we opposition seats to states based on population?
Apportion went
# seats you get /total # seats = population of your state/total
popu
Standard divisor (there is only one)
= total population/total #seats = #people reported by each
seat
Divider-Chooser
Dividing a sub that has turkey and ham
o
75%
Turkey
Deb
75
Connor
50
Ham
25
50
Deb divides so that s1 & s2 are each 50% to her
S1
50
33.3
25%
Deb
Connor
Where does she cut?
How does Connor value s1 & s2?
X is a fraction of turkey piece t
Math 114-01
Exam 2 Review
1. A certain club consists of 6 freshmen, 4 sophomores, 3 seniors, and 2 faculty.
(a) How many coalitions can be formed with 2 freshmen and 2 sophomores?
(b) How many coalitions can be formed with 2 seniors and 2 faculty?
(c) How
10/7/16
cfw_S1,S2,S3,S4,S5,S6,F
There are
coalitions of cfw_S,S,F
There are
coaltions of cfw_S1,S,F
If there are n ways to do task 1 & if there are m ways to do task
2, there are m*n ways to do both.
A deck of cards: 4 suits & 13 ranks
10/14/16
Exam 2: Oc
9/30/16
2.2 Banzhaf power
H1,H2,OB,NH,LB,GC
[ 59: 31, 31, 28,21, 2, 2]
1. Find all winning coalitions
2. Find critical players for each winning coalition
3. Compute total critical count T = B1 + B2 +.+ Bn
4. Compute Banzhaf power index i = Bi/T , T =
10/3/16
[q: 10,9,5,2,1]
Find all q for which P1 and P2 have veto power (and no one else)
If P2 has veto power, so does P1
V = 27
14 q 27
P2 has veto power if the rest combined cant meet the quota
V w2 < q 18 < q (which is not not true)
If P2 has veto,
9/21/2016
Exam 1 on Monday, in class
The plurality with elimination methods can violate the monotonicity
criterion.
The plurality method satisfies the monotonicity criterion.
If youre the plurality winner and youre moved up in some
preference ballot(s),