Download Probability and Statistics Curriculum - Overview

Probability and Statistics Curriculum - Overview
Probability and Statistics is a full year study designed primarily as a preparation course for college, technical school or junior college. The key
components in probability are probability terms, the concept of the probability of an event, predicting and determining probabilities, expected value,
the relationship between theoretical and experimental probabilities, and compound events. In statistics, the key components are data collection,
organization, representation, sampling, central tendency, variance and correlation, and analysis and inference.
Probability and Statistics are the mathematics used to understand chance and to collect, organize, describe, and analyze numerical data. From weather
reports to sophisticated studies of genetics, from election results to product preference survey, probability and statistical language and concepts are
increasingly present in the media and in everyday conversations. Students need this mathematics to help them judge the correctness of an argument
supported by seemingly persuasive data.
Course topics will include the study of introduction to statistics, summarizing and graphing data, statistics for describing, exploring, and
comparing data, probability, discrete probability distributions, normal probability distributions, estimates and sample sizes, hypothesis testing,
inferences from two samples, and correlation and regression. Graphing calculators, Excel, GeoGebra and real life applications are used
throughout the course to develop conceptual understanding and analysis of data.
By the end of the course students will be sensible, critical users of probability and statistics, able to apply the processes and principles developed in
this course to real-world problems. Students should not think that those people who did not win the lottery yesterday have a greater chance of
winning today! They should not believe an argument merely because various statistics are offered. Rather, they should be able to judge whether the
statistics are meaningful and are being used appropriately.
Curriculum – Standards Probability & Statistics
Unit 1
Unit 2
Unit 3
Unit 4
S-IC3
S-ID1
S-ID2
S-ID3
S-ID5
S-CP1
S-CP2
S-CP3
S-CP4
S-CP5
S-CP6
S-CP7
S-CP8
S-CP9
S-MD1
S-MD2
S-MD3
S-MD4
S-MD5
S-ID4
S-MD1
S-ID6
S-IC1
S-IC3
S-IC4
S-MD7
S-IC4
S-IC5
S-IC6
S-ID6
S-ID7
S-ID8
S-ID9
PCTI MATHEMATICS DEPARTMENT
Probability and Statistics
UNIT 1
• Introduction to Statistics
• Summarizing and Graphing Data
• Statistics for Describing, Exploring, and Comparing Data
TECHNOLOGY STANDARDS
Use a graphing calculator/geogebra/ videos to:
• Graphs
https://www.youtube.com/results?search_query=pareto+chart+statistics
https://www.youtube.com/watch?v=qHW3bo5SH0g
https://www.youtube.com/watch?v=qHW3bo5SH0g
https://www.khanacademy.org/math/pre-algebra/applying-math-reasoningtopic/reading_data/v/u08-l1-t2-we3-stem-and-leaf-plots
https://www.youtube.com/results?search_query=pie+chart
https://www.youtube.com/results?search_query=pareto+chart+statistics
http://tube.geogebra.org/search/results/uid/3AFIIbWBcm
https://www.youtube.com/results?search_query=graph+historam+with+geogebra
• Descriptive Statistics
https://www.khanacademy.org/math/probability/descriptive-statistics
https://www.youtube.com/results?search_query=find+mean+median+mode+range+with+ti+84
https://www.youtube.com/results?search_query=find+mean+median+mode+range+with+ti+nspire
https://www.youtube.com/results?search_query=find+boxplot+with+ti+nspire
KEY VOCABULARY
Data
Statistics
Population
Census
Sample
Parameter
Quantitative
Categorical
Discrete data
Continuous (numerical)
data
Nominal level of
measurement
Ordinal level of
measurement
Interval level of
measurement
Voluntary response
sample
Frequency
distribution
Lower class
limits
Upper class
limits
Class boundaries
Class midpoints
Class width
Histogram
Relative
frequency
histogram
Frequency
polygon
Ogive
Dotplots
Stemplots
Bar graphs
Observational study
Experiment
Simple random sample
Random sample
Probability sample
Systematic sampling
Convenience sampling
Stratified sampling
Clustering sampling
Cross-sectional study
Retrospective study
Confounding
Sampling error
Outliers
Percentiles
Pareto charts
Pies charts
Scatterplots
Time-Series
Graph
Measure of
Center
Arithmetic mean
Median, Mode,
Midrange
Weighted mean
Skewed
Symmetric
Range
Standard
Deviation
Variance
Coefficient of
variation
Z score
Quartiles
Boxplot
#
TOPICS
(textbook reference; # of days for instruction)
I
II
#
STUDENT LEARNING OBJECTIVES
CCSS code
INTRODUCTION TO STATISTICS
(1.1-1.5 ; 10 days)
1 day
2 days
2 days
2 days
3 days
1
2
3
4
5
Review and Preview
Statistical Thinking
Types of Data
Critical Thinking
Collecting Sample Data
S-IC3
S-IC3
S-IC3
S-IC3
SUMMARIZING AND GRAPHIND DATA
(2.1-2.5 ; 15 days)
1 day
4 days
4 days
5 days
2 days
1
2
3
4
5
Review and Preview
Frequency Distributions
Histograms
Statistical Graphics
Critical Thinking: Bad Graphs
S-ID5
S-ID1, S-ID5
S-ID1, S-ID2
S-ID1, S-ID2
1
2
3
4
Review and Preview
Measures of Center
Measures of Variation
Measures of Relative Standing and Boxplots
S-ID2, S-ID3
S-ID2, S-ID3
S-ID1
lll STATISTICS FOR DESCRIBING, EXPLORING, AND
COMPARING DATA
(3.1-3.4 ; 15 days)
1 day
6 days
5 days
3 days
Selected Opportunities for Connections to
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Code #
S-IC3
S-ID1
S-ID2
S-ID3
S-ID5
Common Core State Standards
Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how
randomization relates to each.
Represent data with plots on the real number line (dot plots, histograms, and box plots).
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile
range, standard deviation) of two or more different data sets.
Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data
points (outliers).
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the
data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
PCTI MATHEMATICS DEPARTMENT
Probability and Statistics
UNIT 2
• Probability
• Discrete Probability Distributions
TECHNOLOGY STANDARDS
Use a graphing calculator/geogebra/ videos to:
• Find probability.
https://www.youtube.com/results?search_query=find+probability+addition+rule
https://www.youtube.com/results?search_query=find+probability+multiplication+rule
https://www.youtube.com/results?search_query=find+complements+and+conditional+probability
https://www.youtube.com/results?search_query=probability+through+simulation
https://www.youtube.com/results?search_query=counting+principle
https://www.youtube.com/results?search_query=permutation+and+combination
https://www.youtube.com/results?search_query=permutation+and+combination+using+calculator
https://www.youtube.com/results?search_query=permutation+and+combination+using+ti+nspire
KEY VOCABULARY
Event
Simple Event
Sample
Space
Complement
Actual Odds
Against
Actuals
Odds in
Favor
Payoff Odds
Compound
Event
• Find Probability distributions.
Disjoint
https://www.youtube.com/results?search_query=random+variables
Independent
https://www.youtube.com/results?search_query=binomial+probability+distribution
https://www.youtube.com/results?search_query=mean+variance+standard+deviation+binomial+probability+distribution Conditional
Probability
https://www.youtube.com/results?search_query=poisson+distribution
Simulation
http://www.geogebratube.org/search/results/uid/4CEjGqE0H7
Random
Variable
Probability
Distribution
Discrete
Random
Variable
Continuous
Random
Variable
Expected
Value
Binomial
Probability
Distribution
Poisson
Distribution
http://www.geogebratube.org/search/results/uid/EPS7dvERw2
#
TOPICS
(textbook reference; # of days for instruction)
IV PROBABILITY
(4.1-4.8 ; 20 days)
1 day
4 days
3 days
3 days
3 days
V
#
STUDENT LEARNING OBJECTIVES
1
2
3
4
5
Review and Preview
Basic Concepts of Probability
Addition Rule
Multiplication Rule: Basics
Multiplication Rule: Complements and Conditional
Probability
Probabilities through Simulations
Counting
3 days
3 days
6
7
DISCRETE PROBABILITY DISTRIBUTIONS
(5.1-5.5 ; 20 days)
1 day
6 days
5 days
5 days
1
2
3
4
Review and Preview
Random Variables
Binomial Probability Distributions
Mean, Variance, and Standard Deviation for the Binomial
Distribution
CCSS code
S-CP1-2, S-MD6,
S-CP1, S-CP7, S-MD4
S-CP2, S-CP6
S-CP3, S-CP6, S-CP8
S-CP4, S-CP5
S-CP9
S-MD1-5
S-MD1
S-MD2, S-ID2-3
Selected Opportunities for Connections to
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Code #
S-CP1
S-CP2
S-CP3
S-CP4
Common Core State Standards
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as
unions, intersections, or complements of other events (“or,” “and,” “not”).
Understand that two events A and B are independent if the probability of A and B occurring together is the product of their
probabilities, and use this characterization to determine if they are independent.
Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the
conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as
the probability of B.
Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified.
Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For
example, collect data from a random sample of students in your school on their favorite subject among math, science, and
English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in
tenth grade. Do the same for other subjects and compare the results.
S-CP5
S-CP6
S-CP7
S-CP8
S-CP9
S-MD1
S-MD2
S-MD3
S-MD4
S-MD5
Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung
cancer. Use the rules of probability to compute probabilities of compound
Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in
terms of the model.
Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the
answer in terms of the model.
(+) Use permutations and combinations to compute probabilities of compound events and solve problems.
(+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the
corresponding probability distribution using the same graphical displays as for data distributions.
(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
(+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be
calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers
obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected
grade under various grading schemes.
(+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned
empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the
United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100
randomly selected households?
(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at
a fastfood restaurant.
b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a lowdeductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
PCTI MATHEMATICS DEPARTMENT
Probability and Statistics
UNIT 3
• Normal Probability Distributions
• Estimates and Sample Sizes
TECHNOLOGY STANDARDS
Use a graphing calculator/geogebra/ videos to:
• Find normal probability Distributions.
https://www.youtube.com/results?search_query=stanard+normal+distribution
https://www.youtube.com/results?search_query=appliction+of+normal+distribution
https://www.youtube.com/results?search_query=central+limit+theorem
https://www.youtube.com/results?search_query=normal+as+an+approximation+to+binomial
https://www.youtube.com/results?search_query=assessing+normaility+in+statistics
http://www.geogebratube.org/search/results/uid/ZUIZkefdAN
• Estimate and find sample size for population parameters
https://www.youtube.com/results?search_query=estimate+population+proportion
https://www.youtube.com/results?search_query=estimate+population+mean
https://www.youtube.com/results?search_query=estimate+population+variance+statistics
KEY VOCABULARY
Normal Distribution
Uniform Distribution
Standard Normal
Distribution
Sampling Distribution
of a Statistics
Sampling Distribution
of the Mean
Sampling Distribution
of the Variance
Sampling Distribution
of the Proportion
Continuity Correction
Normal Quantile Plot
Point Estimate
Confidence
Interval
Confidence
Level
Critical Value
Margin of Error
Degrees of
Freedom
#
TOPICS
(textbook reference; # of days for instruction)
VI
NORMAL PROBABILITY DISTRIBUTIONS
(6.1-6.7 ; 25 days)
1 day
4 days
4 days
4 days
4 days
4 days
4 days
VII ESTIMATES AND SAMPLE SIZES
(7.1-7.5 ; 15 days)
1 day
5 days
3 days
3 days
3 days
#
STUDENT LEARNING OBJECTIVES
CCSS code
1
2
3
4
5
6
7
Review and Preview
The Standard Normal Distribution
Applications of Normal Distribution
Sampling Distributions and Estimators
The Central Limit Theorem
Normal as Approximation to Binomial
Determining Normality
S-ID4
S-ID4
S-ID4
S-ID4
S-ID4, S-MD1
S-ID6
1
2
Review and Preview
Estimating a Population Proportion.
3
4
5
Estimating a Population Mean: δ Known.
Estimating a Population Mean: δ Not Known
Estimating a Population Variance
S-IC1, S-IC3, S-IC4, SID4
S-IC1, S-IC3, S-IC4
S-IC1, S-IC3, S-IC4
S-IC1, S-IC3, S-IC4
Selected Opportunities for Connections to
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Code #
S-ID4
S-MD1
S-ID6
Common Core State Standards
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages.
Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to
estimate areas under the normal curve.
(+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the
corresponding probability distribution using the same graphical displays as for data distributions.
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or
choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
S-IC1
S-IC3
S-IC4
c. Fit a linear function for a scatter plot that suggests a linear association.
Understand statistics as a process for making inferences about population parameters based on a random sample from that
population.
Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how
randomization relates to each.
Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of
simulation models for random sampling.
PCTI MATHEMATICS DEPARTMENT
Probability and Statistics
UNIT 4
• Hypothesis Testing
• Correlation and Regression
•
• Inferences from two samples
TECHNOLOGY STANDARDS
Use a graphing calculator/geogebra/ videos to:
• Test a claim
https://www.youtube.com/results?search_query=hypothesis+testing+statistics
https://www.youtube.com/results?search_query=test+claim+about+proportion
https://www.youtube.com/results?search_query=test+claim+about+mean
https://www.youtube.com/results?search_query=test+claim+about+variation
http://www.geogebratube.org/search/results/uid/6Rm5JpOUmP
• Test a claim about two samples
https://www.youtube.com/results?search_query=inferences+about+two+proportions
https://www.youtube.com/results?search_query=inferences+about+two+means
https://www.youtube.com/results?search_query=comparing+variation+in+two+samples
•
Find correlation and regression
KEY VOCABULARY
Hypothesis
Hypothesis
Test
Independent
Dependent
Correlation
Linear Correlation Coefficient
Regression Equation
Regression Line
Marginal Change
Outlier
Influential Points
Residual
Least-squares property
Residual Plot
Total deviation
Explained Deviation
Unexplained Deviation
Coefficient of Determination
Prediction Interval
Multiple Regression Equation
https://www.youtube.com/results?search_query=correlation+and+regression+in+statistics
https://www.youtube.com/results?search_query=correlation+and+regression+ti+nspire
http://www.geogebratube.org/search/results/uid/8oQC2sCx6D
#
TOPICS
(textbook reference; # of days for instruction)
VIII HYPOTHESIS TESTING
(8.1-8.6 ; 15 days)
1 day
4 days
4 days
3 days
2 days
2 days
IX
INFERENCES FROM TWO SAMPLES
(9.1-9.5 ; 15 days)
1 day
5 days
3 days
3 days
3 days
Adjusted Coefficient of
Determination
#
STUDENT LEARNING OBJECTIVES
CCSS code
1
2
3
4
5
6
Review and Preview
Basics of Hypothesis Testing
Test a Claim about a Proportion
Test a Claim about a Mean: δ Known
Test a Claim about a Mean: δ Not Known
Test a Claim about Variation
S-MD7
S-IC4
S-IC4
S-IC4
S-IC4
1
2
3
4
5
Review and Preview
Inferences about Two Proportions
Inferences about Two Means; Independent Samples
Inferences from Matched Pairs
Comparing Variation in Two Samples
S-IC4, S-IC5, S-IC6
S-IC4, S-IC5, S-IC6
S-IC4, S-IC5, S-IC6
S-IC4, S-IC5, S-IC6
X
CORRELATION AND REGRESSION
(10.1-10.6 ; 10 days)
1 day
1 day
2 days
2 days
2 days
2 days
1
2
3
4
5
6
Review and Preview
Correlation
Regression
Variation and Prediction Intervals
Multiple Regressions
Modeling
Selected Opportunities for Connections to
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
S-ID6
S-ID6, S-ID8
S-ID9
S-ID8, S-ID9
S-ID7
Code #
S-MD7
Common Core State Standards
S-ID8
Analyze decisions and strategies using probability concepts ( eg. Product testing, medical testing, pulling a hockey goalies at the
end of a game ).
Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of
simulation models for random sampling.
Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between
parameters are significant.
Evaluate reports based on data.
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or
choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
Interpret the slope (rate of change) and the intercept (constant term)
of a linear model in the context of the data.
Compute (using technology) and interpret the correlation coefficient of a linear fit.
S-ID9
Distinguish between correlation and causation.
S-IC4
S-IC5
S-IC6
S-ID6
S-ID7