# RUTHERFORD HIGH SCHOOL Rutherford, New Jersey COURSE OUTLINE STATISTICS AND PROBABILITY

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1 RUTHERFORD HIGH SCHOOL Rutherford, New Jersey COURSE OUTLINE STATISTICS AND PROBABILITY I. INTRODUCTION According to the Common Core Standards (2010), Decisions or predictions are often based on data numbers in context. These decisions or predictions would be easy if the data always sent a clear message, but the message is often obscured by variability. Statistics provides tools for describing variability in data and for making informed decisions that take it into account. 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 A. SKILLS The student will: 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.

2 B. CONTENT The student will: 1. Summarize, represent, and interpret data on a single count or measurement variable by: a. Representing data with plots on the real number line (dot plots, histograms, and box plots). b. Using 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. c. Interpreting differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). d. Using the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. e. Recognizing that there are data sets for which such a procedure is not appropriate. f. Using calculators, spreadsheets, and tables to estimate areas under the normal curve. 2. Summarize, represent, and interpret data on two categorical and quantitative variables by: a. Summarizing categorical data for two categories in two-way frequency tables. b. Interpreting relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). c. Recognizing possible associations and trends in the data. d. Representing data on two quantitative variables on a scatter plot, and describe how the variables are related. e. Fitting a function to the data; using functions fitted to data to solve problems in the context of the data. f. Using given functions or choose a function suggested by the context, emphasizing linear, quadratic, and exponential models. g. Informally assessing the fit of a function by plotting and analyzing residuals. h. Fitting a linear function for a scatter plot that suggests a linear association. 3. Interpret linear models by: a. Interpreting the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. b. Computing (using technology) and interpreting the correlation coefficient of a linear fit. c. Distinguishing between correlation and causation.

3 4. Understand and evaluate random processes underlying statistical experiments by: a. Using statistics as a process for making inferences about population parameters based on a random sample from that population. b. Deciding if a specified model is consistent with results from a given data-generating process, e.g., using simulation. 5. Make inferences and justify conclusions from sample surveys, experiments, and observational studies by: a. Recognizing the purposes of and differences among sample surveys, experiments, and observational studies; explaining how randomization relates to each. b. Using data from a sample survey to estimate a population mean or proportion; developing a margin of error through the use of simulation models for random sampling. c. Using data from a randomized experiment to compare two treatments; using simulations to decide if differences between parameters are significant. d. Evaluating reports based on data. 6. Understand independence and conditional probability and use them to interpret data by: a. Describing 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 ). b. Understanding that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and using this characterization to determine if they are independent. c. Understanding 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. d. Constructing and interpreting 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. e. Recognizing and explaining the concepts of conditional probability and independence in everyday language and everyday situations. 7. Use the rules of probability to compute probabilities of compound events in a uniform probability model by: a. Finding the conditional probability of A given B as the fraction of B s outcomes that also belong to A, and interpreting the answer in terms of the model. b. Applying the Addition Rule, P (A or B) = P (A) + P (B) P (A and B), and interpreting the answer in terms of the model.

4 c. Applying the general Multiplication Rule in a uniform probability model, P (A and B) = P (A) P (B A) = P (B) P (A B), and interpreting the answer in terms of the model. d. Using permutations and combinations to compute probabilities of compound events and solving problems. 8. Calculate expected values and use them to solve problems by: a. Defining a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graphing the corresponding probability distribution using the same graphical displays as for data distributions. b. Calculating the expected value of a random variable; interpreting it as the mean of the probability distribution. c. Developing a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; finding the expected value. d. Developing a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; finding the expected value. 9. Use probability to evaluate outcomes of decisions by: a. Weighing the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. b. Finding the expected payoff for a game of chance. c. Evaluating and comparing strategies on the basis of expected values. d. Using probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). e. Analyzing decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). III. PROFICIENCY LEVELS Statistics and Probability is available to students who have successfully completed Algebra 2 or its equivalent. IV. METHODS OF ASSESSMENT Students will be evaluated by a variety of assessment tools and strategies, which include teacher-made tests and quizzes, homework, notebook (portfolio), computer labs, projects that model probability and statistics in the real world, presentations and a final exam. Students will also be encouraged to assess their own work in order to strive for the highest level of achievement they can attain. Through perseverance, a strong work ethic, and regular participation, students can gain self-confidence in their ability to do mathematics and often improve their overall marking period grades.

5 V. GROUPING Statistics and Probability is a heterogeneously grouped junior/senior level course. Students may take this course in place of or in addition to PreCalculus or College Math as a fourth year mathematics course. VI. ARTICULATION/SCOPE The length of the course is one year. VII. RESOURCES A. Text Elementary Statistics, Prentice Hall, B. Resources Statistics: A First Course, McGraw-Hill, Investigating Probability and Statistics, Jones, C. Software TI-82, TI-83 Programs SAT Software D. Manipulatives Dice Spinners Cards VIII. METHODOLOGIES Students in this course will use technology on a daily basis in the form of the TI-82/TI- 83+ Graphing Calculator. Through discovery exercises and laboratory explorations, they will discover many of the concepts for themselves. They will take an active part in using various algebraic manipulatives and laboratory projects in cooperative learning situations, thus applying teamwork to the learning process. ( A.4, C.1, E.2, B.5, D.2, E.1)

6 IX. SUGGESTED ACTIVITIES Class activities include graphing calculator and software explorations, projects that include data collection and mathematical modeling, portfolio activities, Internet explorations, journal writing, and working with manipulatives to model concepts. Other suggested activities appear in the attached curriculum map. 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. ( A.9, E.5, E.8, F.5, C.3, B.4) XI. DIFFERENTIATING INSTRUCTION FOR STUDENTS WITH SPECIAL NEEDS: STUDENTS WITH DISABILITIES, 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. Examples of Strategies and Practices that Support: Students with Disabilities Use of visual and multi-sensory formats Use of assisted technology Use of prompts Modification of content and student products Testing accommodations Authentic assessments Gifted & Talented Students Adjusting the pace of lessons Curriculum compacting Inquiry-based instruction Independent study Higher-order thinking skills Interest-based content Student-driven

7 Real-world problems and scenarios English Language Learners Pre-teaching of vocabulary and concepts Visual learning, including graphic organizers Use of cognates to increase comprehension Teacher modeling Pairing students with beginning English language skills with students who have more advanced English language skills Scaffolding word walls sentence frames think-pair-share cooperative learning groups teacher think-alouds XII. PROFESSIONAL DEVELOPMENT Teachers shall continue to improve their expertise by participating in a variety of professional development opportunities made available by the Board of Education and other organizations. XIII. CURRICULUM MAP Month Topic Activities September Rules of probability Probability Lab - Create a sample October Expected values Probability Lab Design a survey to collect data for an experiment November Probability to evaluate outcomes of decisions Probability Lab - Roll of the Dice December Independence and conditional probability Frequency Distribution Project using the Internet January Data on a single count or Q2 Cumulative Test measurement variable February Data on two categorical and quantitative variables Statistics Lab collect data about dice rolling March Linear models Statistics Lab - Business April Random processes underlying statistical experiments Use of Calculator - generate random numbers and develop random sample

8 May Sample surveys, Statistics Lab - Z-Score experiments, and observational studies June Review for Final Exam Culminating Projects Final Exam Revised 2015

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