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POMPTON LAKES PUBLIC SCHOOLS
ADVANCED PLACEMENT STATISTICS
June 2012
COURSE OF STUDY
Submitted By
The Mathematic Department
Dr. Paul Amoroso, Superintendent
Mr. Vincent Przybylinski, Principal
Mr. Anthony Mattera, Vice Principal
Frances J. Macdonald, Mathematics Supervisor K-12
BOARD MEMBERS
Mr. Jose A. Arroyo, Mrs. Catherine Brolsma, Mr. Shawn Dougherty,
Mrs. Nancy Lohse-Schwartz, Mr. Garry Luciani, Mr. Carl Padula,
Mr. Tom Salus, Mrs. Stephanie Shaw, Mr. Timothy Troast, Jr.
AP Statistics
I.
2
RATIONALE
This year long course in Statistics is not only a college preparatory course, but
also a course that could earn college credit dependent on future college
attendance.
II.
DESCRIPTION
This year long course, Advanced Placement (AP) Statistics, offers a
course description and examination in statistics to students who wish to
complete studies in secondary school equivalent to a one-semester, introductory,
non-calculus based, and college course in statistics. In colleges and universities,
the number of students who take a statistics course is almost as large as the
number of students who take a calculus course. At least one statistics course is
typically required for majors such as engineering, psychology, sociology, health
sciences and business. The purpose of the AP course in statistics is to
introduce students to the major concepts and tools for collecting, analyzing and
drawing conclusions from data. Students are exposed to four broad conceptual
themes:
1. Exploring Data: Observing patterns and departures from patterns
2. Planning a Study: Deciding what and how to measure
3. Anticipating Patterns: Producing models using probability and
simulation
4. Statistical Inference: Confirming models
Students who successfully complete the course and examination may receive
credit and/or advanced placement for a one-semester introductory college
statistics course. The AP Statistics course is an excellent option for any student
who has successfully completed a second-year course in Algebra, regardless of
the student=s intended college major. Much of the content of an introductory
statistics course does not require any symbolic manipulation beyond the level of
first-year Algebra; however, most students who have studied only first-year
algebra do not possess sufficient mathematical maturity and quantitative
reasoning ability to complete an introductory statistics course successfully.
Consequently, Algebra II is the prerequisite course.
III.
THE CORE CURRICULUM CONTENT STANDARDS
4.1
4.2
4.3
All students will develop the ability to pose and solve mathematical
problems in mathematics, other disciplines, and everyday
experiences.
All students will communicate mathematically through written, oral,
symbolic, and visual forms of expression.
All students will connect mathematics to other learning by
understanding the interrelationships of mathematical ideas and the
roles that mathematics and mathematical modeling play in other
disciplines and in life.
A.P. Statistics
3
4.4
All students will develop reasoning ability and will become selfreliant, independent mathematical thinkers.
4.5
All students will regularly and routinely use calculators, computers,
manipulatives and other tools to enhance mathematical thinking,
understanding and power.
4.6
All students will develop number sense and an ability to represent
numbers in a variety of forms and use numbers in diverse
situations.
4.7
All students will develop spatial sense and an ability to represent
geometric properties and relationships to solve problems in
mathematics and in everyday life.
4.8
All students will understand, select, and apply various methods of
performing numerical operations.
4.9
All students will develop an understanding of and will use
measurement to describe and analyze phenomena.
4.10 All students will use a variety of estimation strategies and recognize
situations in which estimation is appropriate.
4.11 All students will develop an understanding of patterns,
relationships, and functions and will use them to represent and
explain real world phenomena.
4.12 All students will develop an understanding of statistics and
probability and will use them to describe sets of data, model
situations, and support appropriate inferences and arguments.
4.13 All students will develop algebraic concepts and processes and will
use them to represent and analyze relationships among variable
quantities and to solve problems.
4.14 All students will apply the concepts and methods of discrete
mathematics to model and explore a variety of practical situations.
4.15 All students will develop an understanding of the conceptual
building blocks of calculus and will use them to model and analyze
natural phenomena.
4.16 All students will demonstrate high levels of mathematical thought
through experiences which extend beyond traditional computation,
algebra, and geometry.
IV.
STANDARD 9.1 (Career and Technical Education)
All students will develop career awareness and planning, employment skills, and
foundational knowledge necessary for success in the workplace.
Strands and Cumulative Progress Indicators
Building knowledge and skills gained in preceding grades, by the end of Grade
12, students will:
A.
Career Awareness/Preparation
1.
Re-evaluate personal interests, ability, and skills through various
measures including self assessments.
2.
Evaluate academic and career skills needed in various career
A.P. Statistics
3.
4.
5.
B.
4
clusters.
Analyze factors that can impact an individual’s career
Review and update their career plan and include plan in portfolio.
Research current advances in technology that apply to a sector
occupational career cluster.
Employment skills
1.
Assess personal qualities that are needed to obtain and retain a job
related to career clusters.
2.
Communicate and comprehend written and verbal thoughts, ideas,
directions and information relative educational and occupational
settings.
3.
Select and utilize appropriate technology in the design and
implementation of teacher-approved projects relevant to
occupational and /or higher educational settings
4.
Evaluate the following academic and career skills as they relate to
home, school, community, and employment.
Communication
Punctuality
Time management
Organization
Decision making
Goal setting
Resources allocation
Fair and equitable competition
Safety
Employment application
Teamwork
5.
Demonstrate teamwork and leadership skills that include student
participation in real world applications of career and technical
educational skills.
All students electing further study in career and technical education will
also: participate in a structural learning experience that demonstrates
interpersonal communication, teamwork, and leadership skills.
V.
UNITS
A
EXPLORING DATA: OBSERVING PATTERNS AND DEPARTURE
FROM PATTERNS
TIME LINE 10 days
CCCS 4.1,4.2,4.3,4.4,4.5,4.6,4.8,4.10,4.11,4.12,4.13,4.14
1.
Objectives
a.
Interpret graphical displays of distributions of univariate data
b.
Summarize distributions of univariate data
A.P. Statistics
2.
3.
5
c.
Compare distributions of univariate data
d.
Explore bivariate data
e.
Examine categorical data using frequency tables
Content
a.
Dot-plot, stem-plot, box-plot, histograms and cumulative
frequency plot.
b.
Center and spread of data
c.
Clusters and gaps in data
d.
Outlines
e.
Shape
f.
Measures of center: median, mean
g.
Measures of spread: range, inter-quartile range, standard
deviation
h.
Effects of changing units on summary measures
i.
Patterns in scatter-plots
j.
Correlation and linearity
k.
Least squares regression line
I.
Residual plots
m.
Logarithmic and power transformations
n.
Marginal and joint frequencies for two-way tables
o.
Conditional relative frequencies
Assessments
1.
Consider the following murder rates (per 100,000 people).
AL
AK
AZ
AR
CA
CO
CT
DE
FL
GA
HI
ID
IL
13.3
12.9
9.4
9.1
11.7
7.3
4.2
6.7
11.0
14.4
6.7
5.4
9.9
IN
IA
KS
SY
LA
ME
MD
MA
MI
MN
MS
MO
MT
6.2
2.6
5.7
8.9
15.8
2.7
8.2
3.7
10.6
2.0
12.6
10.4
4.8
NE
NV
NH
NM
NY
NC
ND
OH
OK
OR
PA
RI
2.9
15.5
1.4
10.2
10.3
10.8
1.2
6.9
8.5
5.0
6.2
SC
SD
TN
TX
UT
VT
VA
WA
WV
WI
WY
11.5
1.9
9.4
14.2
3.7
3.3
8.8
4.6
6.8
2.5
7.1
What does a histogram show about the shape of this distribution?
A.P. Statistics
2.
6
According to the 1992 NAEP Trial State Assessment the average
mathematics proficiency scores in eighth grade for 41 states were
as follows:
AL
AZ
AR
CA
CO
CT
DE
FL
GA
HI
ID
251
265
255
260
272
273
262
259
259
257
274
IN
IA
KY
LA
ME
MD
MA
MI
MN
MS
269
283
261
249
278
264
272
267
282
246
MO
NE
NH
NJ
NM
NY
NC
ND
OH
OK
270
277
278
271
259
266
258
283
267
267
PA
RI
SC
TN
TX
UT
VA
WV
WI
WY
271
265
260
258
264
274
267
258
277
274
What does a stem-plot show about the shape of this distribution?
3.
According to a 1988 New York Time article, the ten car
models with the highest theft rates were as follows:
Vehicle Model
Pontiac Firebird
Chevrolet Camaro
Chevrolet Monte Carlo
Toyota MR2
Buick Regal
Mitsubishi Starion
Ferrari Mondial
Mitsubishi Mirage
Pontiac Fiero
Oldsmobile Cutlass
Thefts per 1000 cars
30.14
26.02
20.28
19.25
14.70
14.70
13.60
12.80
12.68
11.73
What are the mean, range, and standard deviation of these theft
rates? Explain how each of these values changes if each theft rate
increases by 1.5 and if each increases by 15%.
B.
PLANNING A STUDY: DECIDING WHAT AND HOW TO MEASURE
TIME LINE 8 days
CCCS 4.1,4.2,4.3,4.4,4.5,4.6,4.8,4.10,4.11,4.12,4.13,4.14
1.
Objects
a.
Distinguish various methods of data collections
b.
Plan and conduct surveys
A.P. Statistics
2.
3.
7
c.
Plan and conduct experiments
Content
a.
Census
b.
Sample survey
c.
Experiment
d.
Observational study
e.
Random sampling
f.
Sampling error
g.
Bias in surveys
h.
Stratifying to reduce variations
i.
Experiments versus observational studies versus surveys: ie
treatments
j.
Confounding, control groups, placebo effects, blinding
k.
Randomized paired comparison design: Block design
including matched pairs design
l.
Replication, blocking and generalization of results
m.
Generalization of results and types of conclusions which can
be drawn from observational studies, experiments and
surveys.
Assessment
1. A questionnaire is being designed to determine whether most
people are or are not in favor of legislation protecting the habitat
of the spotted owl. Give two examples of poorly worded
questions, one biased toward each response.
2. To obtain a sample of 25 students from among the 500
students present in school one day, a surveyor decides to pick
every twentieth student waiting on line to attend a required
assembly in the gym.
a) Explain why this procedure will not result in a simple random
sample of the students present that day.
b) Describe a procedure that will result in a simple random
sample of the students present that day.
C.
ANTICIPATING PATTERNS: PRODUCING MODELS USING
PROBABILITY AND SIMULATION
TIME LINE 12 days
CCCS 4.1,4.2,4.3,4.4,4.5,4.6,4.8,4.10,4.11,4.12,4.13,4.14
1.
Objectives:
a.
Understand probability as relative frequency
b.
Combine independent random variables
c.
Use the normal distribution as a model for measurements
d.
Work with the properties of the normal distribution: ie:
Addition rule: multiplication rule; conditional probability;
A.P. Statistics
2.
3.
8
independence
e.
Simulate sampling distribution
Content
a.
Notion of independent versus dependence
b.
Mean and standard deviation for sums and differences of
independent random variables
c.
Properties of the normal distribution
d.
Tables of the normal distribution
e.
The normal distribution as a model for measurement
f.
Sampling distribution of a sample proportion
g.
Sampling distribution of a sample mean
h.
Central limit theorem
i.
Sampling distribution of a difference between two
independent sample proportions
j.
Sampling distribution of a difference between two
independent sample means
k.
Simulation of sampling distribution
l.
“Law of Large Numbers” concept
m.
Discrete random variables and their probability distribution,
including binomial and geometric.
n.
Simulations of random behavior and probability distributions.
o.
Mean (expected value) and standard deviation of a random
variable.
p.
Linear transformations of a random variable
q.
Notion of independence versus dependence.
r.
Mean and Standard deviation for sums and differences of
independent random variables.
Assessment
1. Suppose you will be taking exams in English, statistics, and
chemistry
tomorrow, and from past experience you know that
for each exam you have a 50% chance of receiving an A. List
all eight possibilities for receiving A’s on the various exams.
Letting X represent the number of A’s you will receive, show the
distribution of X, that is, the values and associated probabilities.
2. A sample of applicants for a management position yields the
following numbers with regard to age and experience:
Years of experience
0-5
6-10
>10
Less than 50
years old
80
125
20
More than 50
years old
10
75
50
A.P. Statistics
9
a) What is the probability that an applicant is less than 50 years
old? Has more than 10 years’ experience? Is more than 50
years old and has five or fewer years’ experience?
b) What is the probability that an applicant is less than 50 years
old given that she has between 6 and 10 years experience?
c) Are the two events “less than 50 years old” and “more than 10
years’ experience” independent events? How about the two
events “more than 50 years old” and “between 6 and 10 years’
experience”? Explain.
D.
STATISTICAL INFERENCE: CONFIRMING MODELS
TIME LINE 10 days
CCCS 4.1,4.2,4.3,4.4,4.5,4.6,4.8,4.10,4.11,4.12,4.13,4.14
1.
Objectives
a.
Use statistical inference to select appropriate models
b.
Establish confidence intervals
c.
Work with a variety of Tests of Significance
d.
Identify special cases of normally distributed data
2.
Content:
a.
Meaning of Confidence interval; estimating population
parameters and margins of error.
b.
Confidence interval for a proportion
c.
Confidence interval for a mean
d.
Confidence interval for a difference between two proportions
or two means
e.
Logic of Significance testing: Type I and Type II errors
f.
Null and alternative hypotheses
g.
P-values; concept of power
h.
One- and two-sided tests
i.
Large sample tests for a proportion, a mean, and the
difference between two proportions or two means
j.
Chi-square tests for goodness of fit, homogeneity of
proportions and independence; Chi-square distribution
k.
T-distributions
l.
T-procedures: single sample; two sample
m.
Test for the slope of a least-square regression line.
n.
Properties of point estimators, including unbiased ness and
variability.
o.
Point estimators
3.
Assessment
1.
Suppose that the heights of college basketball players are
normally distributed with a mean of 74 inches and a
standard deviation of 4 inches.
a)
What percentage of players are over 7 feet?
A.P. Statistics
b)
c)
d)
10
What is the probability that at least one of ten randomly
selected players is over 7 feet?
What is the probability that the mean height in an SRS of
size 10 is over 6 feet?
If an outlier is defined to be any value more than 1.5
interquartile ranges above the third quartile or below the first
quartile, what percentage of heights of players are outliers?
2.
The mathematics department at a state university notes that
the SAT math scores of high school seniors applying for
admission into their program are normally distributed with a
mean of 610 and standard deviation of 50.
a)
What is the probability that a randomly chosen applicant to
the department has an SAT math score above 700?
What is the shape, mean, and standard deviation of the
sampling distribution of the mean of a sample of 40 randomly
selected applicants?
What is the probability that the mean SAT math score in an
SRA of 40 applicants is above 625?
Would your answers to (a), (b), or (c) be affected if the
original population of SAT math scores were highly skewed
instead of normal? Explain.
b)
c)
d)
VI.
EVALUATIONS
A.
Tests
B.
Quizzes
C.
Project/Research
D.
Homework
E.
Class participation
F.
Assessments which require students to use calculator functions
appropriate to statistics
G.
Semester Exams
VII.
BENCHMARKS
A.
(Semester I Exam)
1.
Dot-plot, stem-plot, box-plot, histograms and cumulative
frequency plot.
2.
Center and spread of data
3.
Clusters and gaps in data
4.
Outlines
5.
Shape
6.
Measures of center: median, mean
7.
Measures of spread: range, inter-quartile range, standard
deviation
A.P. Statistics
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
B.
11
Effects of changing units on summary measures
Patterns in scatter-plots
Correlation and linearity
Least squares regression line
Residual plots
Logarithmic and power transformations
Marginal and joint frequencies for two-way tables
Conditional relative frequencies
Census
Sample survey
Experiment
Observational study
Random sampling
Sampling error
Bias in surveys
Stratifying to reduce variations
Experiments versus observational studies versus surveys: ie
treatments
Confounding, control groups, placebo effects, blinding
Randomized paired comparison design: Block design
including matched pairs design
Replication, blocking and generalization of results
Generalization of results and types of conclusions which can
be drawn from observational studies, experiments and
surveys.
(Semester II Exam)
1.
Notion of independent versus dependence
2.
Mean and standard deviation for sums and differences of
independent random variables
3.
Properties of the normal distribution
4.
Tables of the normal distribution
5.
The normal distribution as a model for measurement
6
Sampling distribution of a sample proportion
7.
Sampling distribution of a sample mean
8.
Central Limit Theorem
9.
Sampling distribution of a difference between two independent
sample proportions
10.
Sampling distribution of a difference between two independent
sample means
11.
Simulation of sampling distribution
12.
“Law of Large Numbers” concept
13.
Discrete random variables and their probability distribution,
including binomial and geometric.
14.
Simulations of random behavior and probability distributions.
A.P. Statistics
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30
31.
32.
33.
VIII.
12
Mean (expected value) and standard deviation of a random
variable.
Linear transformations of a random variable
Notion of independence versus dependence.
Mean and Standard deviation for sums and differences of
independent random variables.
Meaning of Confidence interval; estimating population
parameters and margins of error.
Confidence interval for a proportion
Confidence interval for a mean
Confidence interval for a difference between two proportions or two
means
Logic of Significance testing: Type I and Type II errors
Null and alternative hypotheses
P-values; concept of power
One- and two-sided tests
Large sample tests for a proportion, a mean, and the difference
between two proportions or two means
Chi-square tests for goodness of fit, homogeneity of proportions
and independence; Chi-square distribution
T-distributions
T-procedures: single sample; two sample
Test for the slope of a least-square regression line.
Properties of point estimators, including unbiased ness and
variability.
Point estimators
AFFIRMATIVE ACTION
A-1 minorities and females incorporated in the plans
A-2 human relations concepts being taught
A-3 teaching plans to change ethnic and racial stereotypes
IX.
BIBLIOGRAPHY
Moore, D., The Basic Practice of Statistics ,W. H. Freeman, New York, NY 1994
Moore, D. and McCabe, G., Introduction to the Practice of Statistics, Second
Edition, W. H. Freeman, New York, NY 1993
Rossman, A., Workshop Statistics, Springer-Verlag, New York, NY 1995
Siegel, A. Morgan, Statistics and Data Analysis, C. Second Edition, John Wiley
& Sons, New York, NY 1996