PROBABILITY & STATISTICS · probability and statistics to assist us in making informed decisions. The concepts learned ... How is probability used in everyday life?

FREEHOLD REGIONAL HIGH SCHOOL DISTRICT

OFFICE OF CURRICULUM AND INSTRUCTION

MATHEMATICS DEPARTMENT

PROBABILITY & STATISTICS

Grade Level: 11-12

Credits: 5

BOARD OF EDUCATION ADOPTION DATE:

AUGUST 30, 2010 SUPPORTING RESOURCES AVAILABLE IN DISTRICT RESOURCE SHARING

APPENDIX A: ACCOMMODATIONS AND MODIFICATIONS

APPENDIX B: ASSESSMENT EVIDENCE

APPENDIX C: INTERDISCIPLINARY CONNECTIONS

https://docs.google.com/document/d/1mEcdd5YsukZgMjnuU4GLQbRsGh-qzbT5eC0sUSt3M0o/edit?usp=sharing

Course Philosophy We live in a world where uncertainty is ever-present. In order to avoid common errors in human judgment and decision-making, we rely on probability and statistics to assist us in making informed decisions. The concepts learned in an introductory statistics course are utilized often in myriad disciplines and form the foundation for effective research. In an increasingly global world, the need for proper decision-making is of paramount importance. Individuals must understand how to analyze and justify claims in all facets of life. Statistics provides that basis and is used in everything from public policy planning to research in the behavioral sciences. Statistics is an essential component to a well-rounded education.

Course Description

In this course, students will learn the concepts that serve as the foundation for the study of probability and statistics. Students will see how fields outside of mathematics use statistics to analyze and interpret data to make informed decisions. With the assistance of technology such as the TI83/84 graphing calculator, they will apply these concepts in myriad ways to critically analyze and synthesize information. This course mirrors that of a college level introductory statistics course, and as such students who dedicate an appropriate amount of study time will be prepared to take a similar course in college.

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Freehold Regional High School District Curriculum Map

Probability & Statistics

Assessments

Relevant Standards1

Enduring Understandings

Essential Questions Diagnostic

(before) Formative (during)

Summative (after)

9.2 A1-5; B1-3; C1-2 4.4 A1-6

Statistics is necessary to make accurate decisions involving data.

How does one define statistics? Why is accurate decision making important? What are some implications of the inappropriate use of data?

4.4 A 1-6 Proper experimental

design is necessary to ensure non-biased results.

What considerations should be made when designing an experiment? What does it mean for results to be considered biased?

8.1 A1 4.4 A 4-5, B 2; C 4

Graphs produce visual displays of data in meaningful ways.

How do graphs enhance the display of data? How does one know which graph is appropriate to use for a given set of data?

4.4 A 5 Measuring the spread of data is essential for comparing data sets.

Why does one need to analyze the spread of data? In what situations might it be useful to compare the spread of data?

4.4 B 1-6; C 1-4 Probability describes the likelihood an event will occur.

How is probability used in everyday life? How does the study of probability integrate itself into the study of statistics?

4.4 A 5 The distribution of outcomes of many real life events can be approximated by the normal curve.

What is a normal curve? Why is an understanding of the normal curve essential to statistics? In what situations can the normal curve be applied to data?

Pretest Student Survey Oral Questions/ Discussion Anticipatory Set Questions

Journals Quizzes Chapter Test Written Assignments Oral Presentations Observations Participatory Rubrics Research Assignments

Portfolios Projects Mid Terms Final Exam

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Assessments Relevant

Standards1 Enduring

Understandings Essential Questions

Diagnostic (before)

Formative (during)

Summative (after)

4.4 A 2, 5; B 5-6 The larger the sample, the more accurate the data is when mapped onto a population.

What is a confidence interval? Why is it necessary to apply confidence intervals when attempting to generalize results of a sample to the population in the aggregate?

4.4 A 2-4; B 5 Claims must be rigorously tested against quantitative sets of standards.

What is hypothesis testing? What is the value in using hypothesis testing when trying to validate a claim?

8.1 A 1 4.4 A 4

Modeling an equation to data allows one to predict future behavior.

What is regression analysis? What are the benefits of using an equation to model data? How does one know how well an equation models a set of data?

4.4 A 2 To determine if two data sets affect each other, Chi-Square analysis is used.

Why might it be necessary to determine if there is a difference between two data sets? What does it mean for two data sets to be considered independent? Why is it important to understand hypothesis testing prior to using Chi-Square?

8.1A 1-3 ; B 1; D 2 9.1 B 1; E 1-2

Technology is integral to the study of statistics.

What types of technology is used for statistical analysis? In what ways can technology be useful when designing your own experiment?

Pretest Student Survey Oral Questions/ Discussion Anticipatory Set Questions

Journals Quizzes Chapter Test Written Assignments Oral Presentations Observations Participatory Rubrics Research Assignments

Portfolios Projects Mid Terms Final Exam

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Freehold Regional High School District

Course Proficiencies and Pacing

Probability & Statistics

Unit Title

Unit Understandings and Goals

Recommended Duration

Unit #1: What is Statistics? Statistics is necessary to make accurate decisions involving data. Proper experimental design is necessary to ensure non-biased results. 1. Students will examine the role of statistics in decision making processes. 2. Students will define the necessary steps to ensure proper experimental design and critique current published research for hidden biases.

2 weeks

Unit #2: Statistical Graphs Graphs produce visual displays of data in meaningful ways. 1. Students will decide which statistical graph best fits a given set of data. 2. Students will synthesize various types of graphs for given data sets.

2 weeks

Unit #3: Measures of Central Tendency

Measuring the spread of data is essential for comparing data sets. 1. Students will utilize various methods of central tendency to determine the spread of data. 2. Students will determine whether they are working with a sample or a population and utilize the

appropriate formulae to measure central tendencies.

3 weeks

Unit #4: Probability Probability describes the likelihood an event will occur. 1. Students will examine the role of probability in everyday life. 2. Students will utilize probability to determine the likelihood of outcomes in myriad situations.

5 weeks

Unit #5: The Normal Curve The distribution of outcomes of many real life events can be approximated by the normal curve. 1. Students will utilize the area under the Normal Curve to answer probabilistic questions. 2. Students will apply the Central Limit Theorem to determine the likelihood of the occurrence of

an event.

4 weeks

Unit #6: Confidence Intervals The larger the sample, the more accurate the data is when mapped onto a population. 1. Students will employ the use of confidence intervals to determine the likelihood of a population

mean being between a certain set of values. 2. Students will apply the concepts of confidence intervals to determine differences between two

population means.

4 weeks

Unit #7: Hypothesis Testing Claims must be rigorously tested against quantitative sets of standards. 1. Students will decide whether to reject or fail to reject a claim by using hypothesis testing. 2. Students will determine whether the data fits a mean or a proportion, and then utilize

appropriate testing methods.

5 weeks

Unit #8: Regression Analysis Modeling an equation to data allows one to predict future behavior. 1. Students will synthesize scatter diagrams to determine if linear correlation exists. 2. Students will determine how a population correlation coefficient fits a set of data using

hypothesis testing.

3 weeks

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Unit Title

Unit Understandings and Goals

Recommended Duration

Unit #9: Chi-Square Analysis To determine if two data sets affect each other, Chi-Square analysis is used. 1. Students will determine if independence exists between two sets of data by utilizing the Chi-

Square Test for Independence. 2. Students will determine if significant differences exists between two data sets by using Chi-

Square Goodness-of-Fit test.

3 weeks

Unit #10: Independent Projects Technology is integral to the study of statistics. Proper experimental design is necessary to ensure non-biased results. 1. Students will create a project utilizing many of the topics covered in the course. 2. Students will utilize technology, such as spreadsheets, graphing calculators, and graphing

software to analyze data and create their project.

3 weeks

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Freehold Regional High School District Probability & Statistics

Unit #1: What is Statistics? Enduring Understandings: Statistics is necessary to make accurate decisions involving data.

Proper experimental design is necessary to ensure non-biased results. Essential Questions: How does one define statistics? Why is accurate decision making important?

What are some implications of the inappropriate use of data? What considerations should be made when designing an experiment? What does it mean for results to be considered biased?

Unit Goals: Students will examine the role of statistics in decision making processes. Students will define the necessary steps to ensure proper experimental design and critique current published research for hidden biases.

Duration of Unit: 2 weeks NJCCCS: 8.1 A1, 9.2 A1-5; B1-3; C1-2; D1-5, 4.4 A1-6

Guiding / Topical Questions

Content, Themes, Concepts, and Skills

Instructional Resources

and Materials

Teaching Strategies

Assessment Strategies

Descriptive statistics involves methods of organizing, picturing, and summarizing information from data.

What are the two types of statistics?

Inferential statistics uses samples to make inferences about a population.

What is the difference between a sample and a population?

Define a sample and a population and show examples of each.

What are the different levels of measurement that can be applied to data?

Define the four levels of measurement: nominal, ordinal, interval, and ratio.

What is involved in ensuring proper experimental design?

List steps necessary to follow when creating an experiment.

How can one truly ensure a selection of random samples?

Show how to use a random number table and how to generate a random number on the graphing calculator.

What does it mean for an experiment to be biased?

Define bias and show examples of how bias may be introduced.

What is a double-blind experiment and how are they used?

Define a double-blind experiment and show how it is used.

What is a placebo effect? Define placebo effect with an example.

Current textbook and resource binders Internet Magazines Newspapers Videos Graphing Calculators

Lecture and class discussion Complete the chapter study guides Student investigation activities

Written tests and quizzes Worksheets Project assessments Notebook assessments Responses to discussion questions Journal assessments Student response systems for immediate feedback

Suggestions on how to differentiate in this unit: • Students will work individually, engage in cooperative learning, and utilize discovery learning on certain activities. • Through the use of lectures, the internet, and interactive whiteboards, students will be exposed to various teaching methods to appeal to visual, auditory, and kinesthetic learners. • Students will be given copies of data sets and other important notes.

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Unit #2: Statistical Graphs Enduring Understandings: Graphs produce visual displays of data in meaningful ways. Essential Questions: How do graphs enhance the display of data?

How does one know which graph is appropriate to use for a given set of data? Unit Goals: Students will decide which statistical graph best fits a given set of data.

Students will synthesize various types of graphs for given data sets. Duration of Unit: 2 weeks NJCCCS: 8.1 A1, 4.4 A 4-5; B 2; C 4

Guiding / Topical Questions Content, Themes, Concepts, and Skills

Instructional Resources and

Materials

Teaching Strategies


What is a bar graph?

Define and provide examples of bar graphs.

What are the differences between a bar graph and a Pareto chart?

Define the characteristics of a Pareto chart and compare and contrast to bar graphs.

Why is a correct scale important when creating graphs?

Explain the use of the break and show examples of graphs with inconsistent scales to emphasize importance of correct scales.

What is a circle graph and how are they created?

Define and create circle graphs.

What is a time-series graph? Create and define a time-series graph.

What is a histogram and how is it different from a bar graph?

Delineate the steps in creating a histogram while emphasizing the differences between it and a bar graph.

How is data ordered using a stem-and-leaf plot?

Define and create a stem-and-leaf plot.




Suggestions on how to differentiate in this unit: • Students will work individually, engage in cooperative learning, and utilize discovery learning on certain activities. • Through the use of lectures, the internet, and interactive whiteboards, students will be exposed to various teaching methods to appeal to visual, auditory, and

kinesthetic learners. • Students will be given copies of data sets and other important notes. • Students can create graphs on poster board to hang in the classroom.

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Unit #3: Measure of Central Tendency Enduring Understanding: Measuring the spread of data is essential for comparing data sets. Essential Questions: Why does one need to analyze the spread of data?

In what situations might it be useful to compare the spread of data? Unit Goals: Students will utilize various methods of central tendency to determine the spread of data.

Students will determine whether they are working with a sample or a population and utilize the appropriate formulae to measure central tendencies.

Duration of Unit: 3 weeks NJCCCS: 4.4 A 5

Guiding / Topical Questions Content, Themes, Concepts, and Skills Instructional

Resources and Materials

Teaching Strategies


How do mean, median, and mode relate to the concept of central tendency?

Define mean, median, and mode.

In what situations would a trimmed mean be preferable to a regular mean?

Define trimmed mean and explain its use when dealing with outliers.

How does one find an average when values have different weights?

Define weighted average.

What are some limiting factors when using mean, median, and mode to describe data?

Introduce and define the concepts of standard deviation and variance.

How does standard deviation relate to the spread of data?

Define Chebyshev’s Theorem and its use in determining percentages of data within a set number of standard deviations.

How does the median relate to percentiles?

Establish the median is the 50th percentile and then use box-and-whisker plots to visualize the remaining percentiles.





kinesthetic learners. • Students will be given copies of data sets and other important notes.

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Unit #4: Probability Enduring Understanding: Probability describes the likelihood an event will occur. Essential Questions: How is probability used in everyday life?

How does the study of probability integrate itself into the study of statistics? Unit Goals: Students will examine the role of probability in everyday life.

Students will utilize probability to determine the likelihood of outcomes in myriad situations. Duration of Unit: 5 weeks NJCCCS: 4.4 B 1-6; C 1-4



Teaching Strategies


How does one define probability?

Display the scale from 0 to 1 and define its relation to probability.

How do corporations and similar entities use probability to make decisions?

Define the Law of Large Numbers and its application to the insurance industry, casinos, and similar industries.

Why is it necessary to determine if two events are independent when calculating probabilities?

Define compound events and explain replacement versus non-replacement.

What is the difference between permutations and combinations?

Explain the importance of determining whether or not order is important.

What is the difference between a discrete and a continuous variable?

Define discrete and continuous variables.

What is a probability distribution?

Establish how discrete and continuous variables are applied to a probability distribution.

What is a binomial probability distribution?

Establish the conditions necessary for a probability distribution to be considered binomial.

How does a geometric probability distribution differ from a binomial distribution?

Show a geometric probability distribution builds on a binomial distribution by dealing with the occurrence of the first success.





kinesthetic learners. • Students will be given copies of data sets and other important notes. • Manipulatives such as dice and playing cards can be used to illustrate the fundamental concepts of probability.

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Unit #5: The Normal Curve Enduring Understanding: The distribution of outcomes of many real life events can be approximated by the normal curve. Essential Questions: What is a normal curve?

Why is an understanding of the normal curve essential to statistics? In what situations can the normal curve be applied to data?

Unit Goals: Students will utilize the area under the Normal Curve to answer probabilistic questions. Students will apply the Central Limit Theorem to determine the likelihood of the occurrence of an event.



Instructional Resources and Materials

Teaching Strategies


When does a normal curve exist?

Establish conditions for the normal curve.

What is the Empirical Rule?

Explain the connection between standard deviation and the normal curve.

What is a z-score?

Apply the formula for calculating z-scores and applying them to the normal curve.

How does one use the area under the normal curve to calculate probabilities?

Use the normal cumulative density calculator functions normalpdf and normalcdf.

What is the different between a parameter and a statistic?

Define both parameter and statistic.

What is the Central Limit Theorem?

Define the Central Limit Theorem.

How is the Central Limit Theorem applied to sampling distributions?

Establish conditions for the application of the Central Limit Theorem.






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Unit #6: Confidence Intervals Enduring Understanding: The larger the sample, the more accurate the data is when mapped onto a population. Essential Questions: What is a confidence interval?

Why is it necessary to apply confidence intervals when attempting to generalize results of a sample to the population in the aggregate? Unit Goals: Students will employ the use of confidence intervals to determine the likelihood of a population mean being between a certain set of values.

Students will apply the concepts of confidence intervals to determine differences between two population means. Duration of Unit: 4 weeks NJCCCS: 4.4 A 2, 5; B 5-6



Teaching Strategies


What is a confidence interval?

Define a confidence interval. Distinguish between a point estimate and an interval estimate.

How does one construct a confidence interval?

Utilize the graphing calculator to solve for a ZInterval.

How does one interpret the solution to a confidence interval?

Discuss various ways to write the solution in layman’s terms.

How does one compute a confidence interval for two population means?

Apply the appropriate formula using either the z statistic or the t statistic.

Why is it important to determine if the standard deviation is known or unknown?

Show the difference in the formulae given a known or unknown standard deviation.






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Unit #7: Hypothesis Testing Enduring Understanding: Claims must be rigorously tested against quantitative sets of standards. Essential Questions: What is hypothesis testing?

What is the value in using hypothesis testing when trying to validate a claim? Unit Goals: Students will decide whether to reject or fail to reject a claim by using hypothesis testing.

Students will determine whether the data fits a mean or a proportion, and then utilize appropriate testing methods. Duration of Unit: 5 weeks NJCCCS: 4.4 A 2-4; B 5



Teaching Strategies


What is hypothesis testing?

Define hypothesis testing and all applicable terms.

What are the types of hypothesis tests?

Define left-tailed, right-tailed, and two-tailed testing.

What are the two types of errors are possible when hypothesis testing?

Define Type I and Type II errors.

How does one test for differences between dependent samples?

Apply hypothesis testing for dependent samples.

How does one test for difference in means and proportions in independent samples?

Apply hypothesis testing for independent samples.

How does one apply the graphing calculator for hypothesis testing?

Use the Ztest, Ttest, , 1-PropZtest2-SampZTest, and 2-PropZtest functions on the graphing calculator.






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Unit #8: Regression Analysis Enduring Understanding: Modeling an equation to data allows one to predict future behavior. Essential Questions: What is regression analysis?

What are the benefits of using an equation to model data? How does one know how well an equation models a set of data?

Unit Goals: Students will synthesize scatter diagrams to determine if linear correlation exists. Students will determine how a population correlation coefficient fits a set of data using hypothesis testing.

Duration of Unit: 3 weeks NJCCCS: 8.1 A 1, 4.4 A 4



Materials

Teaching Strategies


How does one construct a scatter diagram?

Plot the explanatory and response variables on a coordinate plane.

How can one decide if data is linearly correlated?

Determine the correlation coefficient r.

What is a lurking variable?

Define a lurking variable and provide an example.

What is a least-squares regression line?

Apply the formula for a least-squares regression line.

What is the coefficient of determination?

Define the coefficient of determination.

How can one test a population correlation coefficient ρ (rho) from a sample correlation coefficient r?

Apply hypothesis testing techniques to test correlation coefficients.






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Unit #9: Chi-Square Analysis Enduring Understanding: To determine if two data sets affect each other, Chi-Square analysis is used. Essential Questions: Why might it be necessary to determine if there is a difference between two data sets?

What does it mean for two data sets to be considered independent? Why is it important to understand hypothesis testing prior to using Chi-Square?

Unit Goals: Students will determine if independence exists between two sets of data by utilizing the Chi-Square Test for Independence. Students will determine if significant differences exists between two data sets by using Chi-Square Goodness-of-Fit test.




Teaching Strategies Assessment Strategies

What is a Chi-Square Test for Independence?

Define a Chi-Square Test for Independence.

How are matrices used to test for independence?

Apply techniques to use Chi-Square Test for Independence.

What is a Chi-Square Goodness-of-Fit test?

Define a Chi-Square Goodness-of-Fit test.

What is the difference between an observed and expected value?

Define observed versus expected values.






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Unit #10: Independent Project Enduring Understandings: Technology is integral to the study of statistics.

Proper experimental design is necessary to ensure non-biased results. Essential Questions: What types of technology is used for statistical analysis?

In what ways can technology be useful when designing your own experiment? What considerations should be made when designing an experiment? What does it mean for results to be considered biased?

Unit Goals: Students will create a project utilizing many of the topics covered in the course. Students will utilize technology, such as spreadsheets, graphing calculators, and graphing software to analyze data and create their project.

Duration of Unit: 3 weeks NJCCCS: 4.4 A 1-6, 8.1A 1-3; B 1; D 2, 9.1 B 1; E 1-2



Materials

Teaching Strategies


How does one ensure proper experimental design?

Reinforce steps to proper experimental design.

How does one ensure non-biased questions and/or results?

Discuss hidden biases and review student generated questions prior to their being utilized in the experiment/survey.

How can one use spreadsheets to create meaningful displays of data?

Demonstrate how to use Excel or any similar program to create graphs from data.


Lecture and class discussion Student investigation activities

Project assessments Notebook assessments Responses to discussion questions Journal assessments Student response systems for immediate feedback



Documents

PROBABILITY & STATISTICS · probability and statistics to assist us in making informed decisions. The concepts learned ... How is probability used in everyday life?