GLEN RIDGE PUBLIC SCHOOLS
Curriculum Guide
Course Title: PROBABILITY
AND STATISTICS
Subject: Mathematics
Grade Level: 12
Department/School: Mathematics/High School
Duration: Full Year
Number of Credits: 5
Prerequisite: Algebra
II
Elective: Elective
Author: Catherine McCarthy
Date Submitted: Summer 2007
COURSE DESCRIPTION
Probability and Statistics is a full year course intended
for students who have successfully completed Algebra II with a minimum grade of
85%. It is an introductory, non-calculus
based course in statistics. The purpose
of the course is to introduce and develop strategies for collecting, organizing,
analyzing, and drawing conclusions from data.
Students are exposed to four broad conceptual themes:
- Exploring
Data: Observing patterns
and departures from patterns
- Planning
a Study: Deciding
what and how to measure
- Anticipating
Patterns: Producing
models using probability and simulation
- Statistical
Inference: Confirming
models.
COURSE GOALS:
students are expected to learn to:
- Produce
convincing oral and written statistical arguments, using appropriate
terminology, in a variety of applied settings
- Determine
when and how to use technology to aid them in solving statistical problems
- Use
graphical and numerical techniques to study patterns and departures from
patterns in the exploration and analysis of data
- Collect
and produce data according to statistically well-designed methods in order
to determine valid conjectures and conclusions
- Use
methods of probability analysis to anticipate and conjecture what a
distribution of data should look like using appropriate model structures
- Use
statistical inference to guide them in selecting appropriate models and
thereby draw conclusions from data
- Become
informed and critical consumers when presented with statistical results.
Teaching materials for this course include a primary
textbook, Elementary Statistics, Picturing the World, by Ron Larsen and
Betsy Farber, classroom lectures, power point presentations, lab
assignments and activities. Students are
expected to provide their own TI-83/TI-83+/TI-84 graphing calculator for use in
class and at home. Students will use
their graphing calculators extensively throughout the course and the teacher
will use a graphing calculator with an overhead display as well as a projection
unit for class demonstrations.
Throughout the primary text, students are provided with instruction and
practice on the statistical capabilities of their calculator. Students also have access to the computer
software Fathom, and are further
provided examples of computer printouts from other software packages such as Minitab, Data Desk, etc.
GLEN RIDGE PUBLIC SCHOOLS
MATHEMATICS
Mathematics
and Computer Science are an integral part of our lives. Students must be actively involved in their
mathematics education with problem solving being an essential part of the
curriculum. The mathematics and computer
science curricula should emphasize thinking skills through a balance of
computation, intuition, common sense, logic, analysis and technology. Students will be engaged and challenged in a
student-centered learning environment that is developmentally appropriate. Students will communicate mathematical ideas
effectively by applying hands-on manipulatives, basic computational skills,
mathematical models, and technology in order to solve practical problems.
New Jersey Mathematics Standards
The Mathematics Standards
consist of five statements, which describe what is essential to excellent
mathematics education, and present a view of mathematics teaching and learning
that integrates the processes of mathematical activity, the content of
mathematics, and the learning environment in the classroom. The following standards were adopted by the
New Jersey State Board of Education.
This course
will cover the following Core Curriculum Standards:
Descriptive Statement: Numbers and arithmetic operations are what most of the
general public think about when they think of mathematics; and, even though
other areas like geometry, algebra, and data analysis have become increasingly
important in recent years, numbers and operations remain at the heart of
mathematical teaching and learning. Facility with numbers, the ability to
choose the appropriate types of numbers and the appropriate operations for a
given situation, and the ability to perform those operations as well as to
estimate their results, are all skills that are essential for modern day life.
Number Sense
Numerical Operations
Estimation
Descriptive
Statement: Spatial
sense is an intuitive feel for shape and space. Geometry and measurement both
involve describing the shapes we see all around us in art, nature, and the
things we make. Spatial sense, geometric modeling, and measurement can
help us to describe and interpret our physical environment and to solve
problems.
Geometric Properties
Transforming Shapes
Coordinate Geometry
Units of Measurement
Measuring Geometric Objects
Descriptive Statement: Algebra is a symbolic language used to
express mathematical relationships. Students need to understand how
quantities are related to one another, and how algebra can be used to concisely
express and analyze those relationships. Modern technology provides tools
for supplementing the traditional focus on algebraic procedures, such as
solving equations, with a more visual perspective, with graphs of equations
displayed on a screen. Students can then focus on understanding the
relationship between the equation and the graph, and on what the graph
represents in a real-life situation.
Patterns
Functions and Relationships
Modeling
Procedures
Descriptive
Statement: Data
analysis, probability, and discrete mathematics are important interrelated
areas of applied mathematics. Each provides students with powerful
mathematical perspectives on everyday phenomena and with important examples of
how mathematics is used in the modern world. Two important areas of
discrete mathematics are addressed in this standard;
a third area, iteration and recursion, is addressed in Standard 4.3 (Patterns
and Algebra).
Data Analysis
Probability
Discrete Mathematics
– Systematic Listing and Counting
Discrete Mathematics
– Vertex-edge Graphs and Algorithms
Descriptive Statement: The mathematical processes described here highlight ways of
acquiring and using the content knowledge and skills delineated in the first
four mathematics standards.
Problem Solving Reasoning
Communication Representations
Connections Technology
Curriculum Description
UNIT I: WHAT
IS STATISTICS
CCCS: 4.1, 4.3, 4.4,
4.5
Objectives:
The students will be able to:
1. Recognize
whether a study is an experiment, a survey, or an observational study that is
not a survey.
2. Determine
the best method for producing data to answer a specific question: experiment, survey, or other observational
study.
3. Locate
available data on the Internet to help answer a question of interest.
4. Identify
the individuals and variables in a set of data.
5. Classify
each variable as categorical or quantitative.
6. Identify
the units in which each quantitative variable is measured.
7. Answer
key questions – who, what, when, where, how, and by whom? – about a given set
of data.
8. Construct a bar graph of the distribution of a
categorical variable.
9. Interpret bar graphs.
10. Construct a dotplot of the distribution of a
quantitative variable.
11. Describe
patterns observed in a dotplot.
Approximate duration:
5 days
Activities:
-
Move-up day – How Many Licks to Get to the
Middle of a Tootsie Pop?
UNIT II: EXPLORING DATA
CCCS: 4.1, 4.3, 4.4, 4.5
Objectives:
The students will be able to:
1. Construct
stem and leaf plots of the distribution of a quantitative variable.
2. Construct
a histogram of the distribution of a quantitative variable.
3. Construct
and interpret an ogive of a set of quantitative variable.
4. Observe
the overall pattern and deviations from the pattern.
5. Characterize
the shape of a stem and leaf plot, dotplot, or histogram.
6. Determine
numerical measures of center and spread for a given distribution: mean, standard deviation, five number
summary.
7. Determine
which measures of center and spread are more appropriate for a given
distribution.
8. Recognize
and determine outliers.
9. Understand
the effects of outliers.
10. Construct
time plot of data.
11. Recognize
strong trends or other patterns in a time plot.
12. Calculate
the mean of a set of observations.
13. Determine
the median of a set of observations.
14. Understand
that the median is more resistant than the mean.
15. Recognize
the effects of the skewness of a data distribution on the mean.
16. Define and
calculate five-number summary, IQR, and outliers.
17. Construct a
boxplot with and without calculator.
18. Use a
calculator or software to calculate the standard deviation for a set of
observations.
19. Use TI-83
menu to determine univariate data statistics.
20. Determine
the effect of a linear transformation on measures of center and spread.
21. Calculate
new measures of center and spread on transformed data.
22. Construct
side-by-side bar graphs to compare distributions of categorical data.
23. Construct
back-to-back stem and leaf plots and side-by-side boxplots to compare
distributions of quantitative variables.
24. Write
narrative comparisons of the shape, center, spread, and outliers for two or
more quantitative distributions.
Approximate duration:
20 days
UNIT III: DESCRIBING LOCATION IN A DISTRIBUTION
CCCS: 4.1, 4.3, 4.4,
4.5
Objectives:
The students will be able to:
1. Determine
the standardized value (z score) of an observation.
2. Interpret
z-scores in context.
3. Use
percentiles to locate individual values within distributions of data.
4. Apply
Chebyshev’s inequality to a given distribution of data.
5. Understand
the concept that areas under a density curve represent proportions of all
observations and that the total area under a density curve is 1.
6. Approximately
locate the median and the mean on a density curve.
7. Understand
and recognize that the mean and median both lie at the center of a symmetric
density curve.
8. Recognize
the effect on the relationship between the mean and the median of a skewed
density curve.
9. Recognize
the shape of the
10. Estimate
the mean and standard deviation from a density curve.
11. Develop and
apply the Empirical Rule to state what percent of the observations from a
Normal distribution within 1, 2, or 3 standard deviations away from the mean.
12. Use the
standard Normal distribution to determine the proportion of values in a
specified range.
13. Use
z-scores to standardize non-standard Normal distributions.
14. Calculate
probabilities using the Normal distribution using either table or calculator.
15. Determine a
z-score from a percentile.
16. Given that
a variable has a Normal distribution with stated mean and standard deviation,
use table and calculator to find the value of the observation having a stated
proportion of values to the left or to the right of it.
17. Solve
problems involving the Normal distribution.
18. Be familiar
with Normal Distribution functions of the TI-83.
19. Construct
and interpret
Approximate duration: 10 days
Activities:
-
Video: AGAINST ALL ODDS PROGRAM # 4 & 5
UNIT IV: EXAMINING RELATIONSHIPS
CCCS: 4.1, 4.3, 4.4, 4.5
Objectives:
The students will be able to:
1. Recognize
whether a variable is categorical or quantitative.
2. Identify
the explanatory and response variables in situations where one variable
explains or influences another.
3. Construct
a scatterplot to display the relationship between two quantitative variables
with and without the use of a calculator.
4. Describe
the direction, form, and strength of the overall pattern of a scatterplot.
5. Recognize
a positive or negative association and linear pattern in a scatterplot.
6. Recognize
outliers in a scatterplot.
7. Use
a calculator to find the correlation coefficient between two quantitative
variables.
8. Understand
the relationship between the value of the correlation coefficient and
linearity.
9. Using
a calculator, find the least-squares regression line and use it to make
predictions.
10. Explain the
meaning in context of the variables used in the equation for the least-squares
regression line.
11. Determine
the slope and intercept of the least-squares regression line from the means and
standard deviations of x and y and their
correlation.
12. Recognize
the difference between interpolation and extrapolation.
13. Recognize
the dangers of extrapolation.
14. Determine
the equation of the least-squares regression line from reading an appropriate
computer output.
15. Analyze the
best regression equation to use for a data set.
16. Calculate
the residuals and plot them against the explanatory variable.
17. Use the Residual
plot function of the calculator.
18. Recognize
any unusual patterns in a residual plot.
19. Interpret
the residual plot in order to analyze whether an equation is a good fit.
20. Use the
coefficient of determination to describe how much of the variation in one variable
can be accounted for by a straight-line relationship with the other variable.
21. Recognize
outliers and potentially influential observations from a scatterplot with the
regression line drawn on it.
Approximate
duration: 20 days
Activities:
-
GUESS CELEBRITY AGES
-
GUESS THE CORRELATION
-
MATCHING SCATTERPLOTS WITH THEIR DESCRIPTION
-
LEAST SQUARES LINE APPLET
-
Short Project: WHAT’S YOUR BEST OFFER
UNIT VI:
PRODUCING DATA
CCCS: 4.1, 4.3, 4.4, 4.5
Objectives:
The students will be able to:
1. Identify
a population, sample, parameter, or statistic.
2. Recognize
bias due to voluntary response sampling and other inferior sampling methods.
3. Define
and identify a simple random sample from a population.
4. Recognize,
compare and utilize sampling methods.
5. Recognize
the presence of undercoverage, response and nonresponse bias in sample surveys.
6. Differentiate
between experiments and observational studies.
7. Recognize
bias due to confounding of explanatory variables with lurking variables in
either an observational study or an experiment.
8. Define
and identify the factors, treatments, response variables, and experimental
units or subjects, in an observational study and in an experiment.
9. Outline
the design of a completely randomized experiment including elements of
randomization, control and replication.
10. Define and
recognize the placebo effect.
11. Define,
recognize and apply principle of double-blind experimental design techniques.
12. Recognize
and apply a block design in an appropriate experimental setting.
13. Define and recognize
the appropriate setting for a matched-pair experimental design.
14. Explain why
a randomized comparative experiment can give a good evidence for cause and
effect relationships.
Approximate duration:
20 days
Activities:
-
RANDOM RECTANGLES
-
CLUSTER SAMPLING
-
GALLOP POLL WEBQUEST
-
Videos: AGAINST ALL ODDS: Program # 14,12,13
-
Short Project:
Students will conduct a survey on a topic of interest to them.
UNIT VII: PROBABILITY AND SIMULATION: THE STUDY OF RANDOMNESS
CCCS: 4.1, 4.3, 4.4,
4.5
Objectives:
The students will be able to:
1. Recognize
that many random phenomena can be investigated by means of a carefully designed
simulation.
2. Design
and run a simulation using either a random number table or the random number
generator function of a calculator.
3. Describe
the sample space of a random phenomenon.
4. Apply
appropriate counting techniques to determine the finite number of outcomes of
an event.
5. Determine
simple probabilities.
6. Define
and apply the Law of Large Numbers.
7. Know
the probability rules and be able to apply them to determine probabilities of
defined events.
8. Determine
a valid probability distribution.
9. Distinguish
between events that are disjoint, complementary, or independent.
10. Determine
the probabilities of the union and/or intersection of events.
11. Understand and apply the rules fo