COURSE TITLE

Statistics and Probability

LENGTH

Full Year

DEPARTMENT

STEM Department

SCHOOL

Rutherford High School

Primary Content

Mathematics

 Initial Board of Education Approval Date (Born on):  5/13/2024

Revisions:  

Embedded Content

 Career Readiness, Life Literacies and Key Skills

 Initial Board of Education Approval Date (Born on):  5/13/2024

Computer Science and Design Thinking

 Initial Board of Education Approval Date (Born on):  5/13/2024

Statistics and Probability

I.        Introduction/Overview/Philosophy

Statistics and Probability is designed for the college bound student who has demonstrated success in Algebra 2 and wishes to continue to explore a large range of topics with an emphasis on “real world” applications such as games of chance, random population, and actuarial science.  

Technology plays an important role in statistics and probability by making it possible to generate plots, regression functions, and correlation coefficients, and to simulate many possible outcomes in a short amount of time. Students will regularly apply the tools of technology including the graphing calculator and computer to solve problems.  They will be challenged through critical thinking exercises and participate in various group and individual activities that will enhance their mathematical reasoning ability and communication skills.  Students are expected to use the information and technology in various ways in real world applications.  

II.        Objectives

Course Outline:

1. Introductions to Statistics

1. Distinguish between a population and a sample, a parameter and a statistic, descriptive statistics and inferential statistics, qualitative data and quantitative data
2. Classify data with respect to the four levels of measurement
3. Collect data through observational study, performing an experiment, using a simulation, or using a survey
4. Design an experiment
5. Create a sample using random sampling, simple random sampling, stratified sampling, cluster sampling, and systematic sampling
6. Identify a biased sample

2. Descriptive Statistics

1. Construct a frequency distribution including limits, midpoints, relative frequencies, cumulative frequencies, and boundaries
2. Construct frequency histograms, frequency polygons, relative frequency histograms, and ogives
3. Graph quantitative data sets using stem-and-leaf plots and dot plots
4. Graph and interpret paired data sets using scatter plots and time series charts
5. Graph qualitative data sets using pie charts and Pareto charts
6. Find the mean, median, and mode of a population and a sample
7. Find a weighted mean of a data set and the mean of a frequency distribution
8. Describe the shape of a distribution as symmetric, uniform, or skewed and how to compare the mean and median for each
9. Find the range of a data set
10. Find the variance and standard deviation of a population and a sample
11. find the variance and standard deviation of a population and a sample
12. Use the Empirical Rule and Chebychev’s Theorem to interpret standard deviation
13. Approximate the sample standard deviation for grouped data
14. Find the quartiles and interquartile range of a data set
15. Draw a box-and-whisker plot
16. Interpret other fractiles such as percentiles
17. Find and interpret the standard score (z -score)

3. Discrete Probability Distributions

1. Distinguish between discrete random variables and continuous random variables
2. Determine if a distribution is a probability distribution
3. Construct a discrete probability distribution and its graph and find the mean, variance, and standard deviation of a discrete probability distribution
4. Find the expected value of a discrete probability distribution
5. Determine if a probability experiment is a binomial experiment
6. Find binomial probabilities using the binomial probability formula,a binomial probability table, and technology
7. Construct a binomial distribution and its graph and find the mean, variance, and standard deviation of a binomial probability distribution
8. Find probabilities using the geometric distribution and  the Poisson distribution

4. Normal Probability Distributions

1. Interpret graphs of normal probability distributions
2. Find areas under the standard normal curve
3. Find probabilities for normally distributed variables
4. Find a z -score given the area under the normal curve
5. Transform a z -score to an x -value
6. Find a specific data value of a normal distribution given the probability
7. Find sampling distributions and verify their properties
8. Apply the Central Limit Theorem to find the probability of a sample mean
9. Decide when a normal distribution can approximate a binomial distribution
10. Find the continuity correction
11. Use a normal distribution to approximate binomial probabilities

5. Probability

1. Identify the sample space of a probability experiment and how to identify simple events
2. Use the Fundamental Counting Principle to find the number of ways two or more events can occur
3. Distinguish among classical probability, empirical probability, and subjective probability
4. Find the probability of the complement of an event and how to find other probabilities using the Fundamental Counting Principle
5. Find conditional probabilities
6. Distinguish between independent and dependent events
7. Use the Multiplication Rule to find the probability of two events occurring in sequence
8. Determine if two events are mutually exclusive
9. Use the Addition Rule to find the probability of two events
10. Find the number of ways a group of objects can be arranged in order and the number of ways to choose several objects from a group without regard to order
11. Use counting principles to find probabilities

6. Confidence Intervals

1. Find a point estimate and a margin of error
2. Construct and interpret confidence intervals for the population mean
3. Determine the minimum sample size required when estimating µ
4. Interpret the t -distribution and use a t -distribution table
5. Construct confidence intervals when n < 30, the population is normally distributed, and is the standard deviation is unknown
6. Find a point estimate for a population proportion
7. Construct a confidence interval for a population proportion
8. Determine the minimum sample size required when estimating a population proportion
9. Interpret the chi-square distribution and use a chi-square distribution table
10. Use the chi-square distribution to construct a confidence interval for the variance and standard deviation

7. Hypothesis Testing with One Sample

1. State a null hypothesis and an alternative hypothesis
2. Identify type I and type II errors
3. Know whether to use a one-tailed or a two-tailed statistical test
4. Interpret a decision based on the results of a statistical test
5. Find P -values and use them to test a mean µ
6. Use p-values for a z -test
7. Find critical values and rejection regions in a normal distribution
8. Use rejection regions for a z -test
9. Find critical values in a t -distribution
10. Use the t -test to test a mean m
11. Use technology to find p-values and use them with a t -test to test a mean µ
12. Use the z -test to test a population proportion p
13. Find critical values for a chiSquare-test
14. Use the chiSquare -test to test a variance or a standard deviation

8. Hypothesis Testins with Two Samples

1. Decide whether two samples are independent or dependent
2. Perform a two-sample test for the difference between two means µ1 and µ2 using large independent samples
3. Perform a test for the difference between two population means µ1 and µ2 using small independent samples
4. Perform a test to test the mean of the differences for a population of paired data
5. Perform a test for the difference between two population proportions p1 and p2

9. Correlations and Regression

1. Construct a scatter plot
2. Perform a hypothesis test for a population correlation coefficient r
3. Find the equation of a regression line, y = mx + b
4. Predict y -values using a regression equation
5. find and interpret the coefficient of determination r2
6. find and interpret the standard error of estimate for a regression line
7. Construct and interpret a prediction interval for y, y-E<y<y+E

Student Outcomes:

After successfully completing this course, the student will:

• Summarize, represent, and interpret data on a single count or measurement variable
• Understand and evaluate random processes underlying statistical experiments
• Make inferences and justify conclusions from sample surveys, experiments and observational studies
• Understand the independence and conditional probability and use them to interpret data
• Use the rules of probability to compute probabilities of compound events in a uniform probability model
• Summarize, represent, and interpret data on a single count or measurement variable
• Summarize, represent, and interpret data on two categorical and quantitative variables
• Interpret functions that arise in applications in terms of the context

New Jersey Student Learning Standards

Career Readiness, Life Literacies, and Key Skills Practices

CRLLKSP 1   Act as a responsible and contributing community members and employee.

Students understand the obligations and responsibilities of being a member of a community, and they demonstrate this understanding every day through their interactions with others. They are conscientious of the impacts of their decisions on others and the environment around them. They think about the near-term and long-term consequences of their actions and seek to act in ways that contribute to the betterment of their teams, families, community and workplace. They are reliable and consistent in going beyond the minimum expectation and in participating in activities that serve the greater good.

CRLLKSP 2   Attend to financial well-being.

Students take regular action to contribute to their personal financial well-being, understanding that personal financial security provides the peace of mind required to contribute more fully to their own career success.

CRLLKSP 3   Consider the environmental, social and economic impacts of decisions.

Students understand the interrelated nature of their actions and regularly make decisions that positively impact and/or mitigate negative impact on other people, organization, and the environment. They are aware of and utilize new technologies, understandings, procedures, materials, and regulations affecting the nature of their work as it relates to the impact on the social condition, the environment and the profitability of the organization.

CRLLKSP 4   Demonstrate creativity and innovation.

Students regularly think of ideas that solve problems in new and different ways, and they contribute those ideas in a useful and productive manner to improve their organization. They can consider unconventional ideas and suggestions as solutions to issues, tasks or problems, and they discern which ideas and suggestions will add greatest value. They seek new methods, practices, and ideas from a variety of sources and seek to apply those ideas to their own workplace. They take action on their ideas and understand how to bring innovation to an organization.

CRLLKSP 5   Utilize critical thinking to make sense of problems and persevere in solving them.

Students readily recognize problems in the workplace, understand the nature of the problem, and devise effective plans to solve the problem. They are aware of problems when they occur and take action quickly to address the problem; they thoughtfully investigate the root cause of the problem prior to introducing solutions. They carefully consider the options to solve the problem. Once a solution is agreed upon, they follow through to ensure the problem is solved, whether through their own actions or the actions of others.

CRLLKSP 6   Model integrity, ethical leadership and effective management.

Students consistently act in ways that align personal and community-held ideals and principles while employing strategies to positively influence others in the workplace. They have a clear understanding of integrity and act on this understanding in every decision. They use a variety of means to positively impact the directions and actions of a team or organization, and they apply insights into human behavior to change others’ action, attitudes and/or beliefs. They recognize the near-term and long-term effects that management’s actions and attitudes can have on productivity, morals and organizational culture.

CRLLKSP 7   Plan education and career paths aligned to personal goals.

Students take personal ownership of their own education and career goals, and they regularly act on a plan to attain these goals. They understand their own career interests, preferences, goals, and requirements. They have perspective regarding the pathways available to them and the time, effort, experience and other requirements to pursue each, including a path of entrepreneurship. They recognize the value of each step in the education and experiential process, and they recognize that nearly all career paths require ongoing education and experience. They seek counselors, mentors, and other experts to assist in the planning and execution of career and personal goals.

CRLLKSP 8   Use technology to enhance productivity, increase collaboration and communicate effectively.

Students find and maximize the productive value of existing and new technology to accomplish workplace tasks and solve workplace problems. They are flexible and adaptive in acquiring new technology. They are proficient with ubiquitous technology applications. They understand the inherent risks-personal and organizational-of technology applications, and they take actions to prevent or mitigate these risks.

CRLLKSP 9   Work productively in teams while using cultural/global competence.

Students positively contribute to every team, whether formal or informal. They apply an awareness of cultural difference to avoid barriers to productive and positive interaction. They find ways to increase the engagement and contribution of all team members. They plan and facilitate effective team meetings.

Career Readiness, Life Literacies, and Key Skills 

9.1.12.CDM.6: Compute and assess the accumulating effect of interest paid over time when using a variety of sources of credit.

9.1.12.CDM.8: Compare and compute interest and compound interest and develop an amortization table using business tools.

9.1.12.PB.1: Explain the difference between saving and investing.

9.4.12.TL.1: Assess digital tools based on features such as accessibility options, capacities, and utility for accomplishing a specified task (e.g., W.11-12.6.).

Computer Science and Design Thinking

8.2.12.NT.1: Explain how different groups can contribute to the overall design of a product.

8.2.12.NT.2: Redesign an existing product to improve form or function.

8.1.12.IC.2: Test and refine computational artifacts to reduce bias and equity deficits.

Mathematics

S.CP.A.1. (+) 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”).

S.CP.A.2. (+) 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.

S.CP.A.3. (+)  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.

S.CP.A.4. (+) 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.

S.CP.A.5. (+) Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

S.CP.B.6. (+) 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.

S.CP.B.7. (+) 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.

S.CP.B.8. (+) 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.

S.CP.B.9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

S.IC.A.1.(+) Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

S.IC.A.2. (+)Decide if a specified model is consistent with results from a given data.generating process, e.g., using simulation.

S.IC.B.3.(+) Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

S.IC.B.4. (+) 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.

S.IC.B.5. (+) Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

S.IC.B.6. (+) Evaluate reports based on data.

S.ID.A.1. Represent data with plots on the real number line (dot plots, histograms, and box plots).

S.ID.B.5. 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.

S.ID.B.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

S.ID.C.7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S.ID.C.8. Compute (using technology) and interpret the correlation coefficient of a linear fit.

S.ID.C.9. Distinguish between correlation and causation.

S.MD.A.1. (+) 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.

S.MD.A.2. (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

S.MD.A.3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

S.MD.A.4. (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

S.MD.B.5a. Find the expected payoff for a game of chance.

S.MD.B.5b. Evaluate and compare strategies on the basis of expected values.

S.MD.B.6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

S.MD.B.7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

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.

English Language Arts

SL.PE.11–12.1. Initiate and participate effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with peers on grades 11–12 topics, texts, and issues, building on others’ ideas and expressing their own clearly and persuasively.

A. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring to evidence from texts and other research on the topic or issue to stimulate a thoughtful, well-reasoned exchange of ideas.

B. Collaborate with peers to promote civil, democratic discussions and decision-making, set clear goals and assessments (e.g., student developed rubrics), and establish individual roles as needed.

C. Propel conversations by posing and responding to questions that probe reasoning and evidence; ensure a hearing for a full range of positions on a topic or issue; clarify, verify, or challenge ideas and conclusions; and promote divergent and creative perspectives.

D. Respond thoughtfully to diverse perspectives; synthesize comments, claims, and evidence made on all sides of an issue; resolve contradictions when possible; and determine what additional information or research is required to deepen the investigation or complete the task.

III.         Proficiency Levels

Statistics and Probability is available to students who have successfully completed Algebra 2 or its equivalent.  

IV.        Methods of Assessment

Student Assessment

The teacher will provide a variety of assessments during the course of the year. The assessment may include but is not limited to: chapter and unit tests and quizzes, teacher observations, open-ended problems, cooperative work, and homework.

Curriculum/Teacher Assessment

The teacher will provide the subject area supervisor with suggestions for changes on an ongoing basis.

V.        Grouping

Statistics and Probability is a heterogeneously grouped junior/senior level course.  

VI.        Articulation/Scope & Sequence/Time Frame

Course length is one year.

VII.        Resources

Texts/Supplemental Reading/References

1. Elementary Statistics, Prentice Hall, 2006.
2. StatCrunch Software

VIII.        Suggested Activities

Appropriate activities are listed in the curriculum map.

IX.        Methodologies

The following methods of instruction are suggested: teacher guided explorations, working in groups/working with a partner, working with manipulatives and discovery activities.

X.        Interdisciplinary Connections

Connections are made to music during the study of harmonic sequences.  Data analysis applications to science and business problems are frequent throughout the course. Connections are also made by means of formulas used in computer programming classes.  Writing assignments and portfolios strengthen the connection between mathematics and language arts literacy and fine arts.  

XI.         Differentiating Instruction for Students with Special Needs: Students with Disabilities, Students at Risk, Students with 504s, English Language Learners, and Gifted & Talented Students

Differentiating instruction is a flexible process that includes the planning and design of instruction, how that instruction is delivered, and how student progress is measured. Teachers recognize that students can learn in multiple ways as they celebrate students’ prior knowledge. By providing appropriately challenging learning, teachers can maximize success for all students.

Differentiating in this course includes but is not limited to:

Differentiation for Support (ELL, Special Education, Students at Risk, Students with 504s)

• Peer mentoring on problems
• Differentiated teacher feedback on assignments
• Modeling out problems on whiteboard
• Visual aids as we project problems on whiteboard
• Study guides
• Tiered assignments
• Scaffolding of materials and assignments
• Re-teaching and review
• Guided note taking
• Exemplars of varied performance levels
• Multi-media approach to accommodating various learning styles

• Supplemental reading material for independent study
• Flexible grouping
• Tiered assignments
• Topic selection by interest
• Enhanced expectations for independent study
• Elevated questioning techniques using Webb's Depth of Knowledge matrix

XII.        Professional Development

The teacher will continue to improve expertise through participation in a variety of professional development opportunities.

XII.        Curriculum Map/Pacing Guide