Advanced Placement Statistics Syllabus Course Design This AP Statistics course is taught as a non-calculus-based, activity-driven course in which students actively construct their own understanding of the concepts and methods of statistics. Students explore the realms of descriptive and inferential statistics. The major conceptual themes studied are “Exploring Data,” “Planning and Design of a Study,” “Anticipating Patterns,” and “Statistical Inference”, as outlined by the College Board. Technology, with particular emphasis on the TI-83+, TI-84, TI-89 calculators and Fathom and Minitab software, is incorporated into the regular class activities. The course also employs research on real-world data and newspaper/online journal research to promote student awareness of statistical concepts and judgments in the world around them. Pre-requisites for the course, unit concepts, textbook and resource usage, technology adaptation, teaching strategies, and assessments are designed to suit the College Board’s AP Statistics Course Description. Primary Textbook* Peck, Roxy, Chris Olsen, Jay Devore. Introduction to Statistics and Data Analysis. Belmont, CA:Thomson Brooks/Cole, 2001 (This book is referenced as POD in this document) Supplemental Textbooks Bock, David E., Paul F. Velleman, and Richard D. DeVeaux. Stats: Modeling the World. Boston: Pearson/Addison-Wesley, 2004. Moore, David S. and George P. McCabe. Introduction to Practice of Statistics, 4th edition (or later). New York: W.H. Freeman Co., 2003. Watkins, Ann, Richard Scheaffer, and George W. Cobb. Statistics in Action: Understanding a World of Data. Emeryville, CA: Key Curriculum Press, 2004. Yates, Daniel S., David S. Moore, and Daren S. Starnes. The Practice of Statistics. New York: W.H. Freeman, 2003. Yates, Daniel S., David S. Moore, and Daren S. Starnes. The Practice of Statistics. New York: W.H. Freeman, 1999. Course Outline The following is an outline of the major topics covered by the AP Examination in Statistics. The course provides instruction in each of the following four broad conceptual themes outlined in the AP Statistics Course Description with appropriate emphasis on each: I. Exploring Data: Describing patterns and departures from patterns (20%-30%) Exploratory analysis of data makes use of graphical and numerical techniques to study patterns and departures from patterns. Emphasis should be placed on interpreting information from graphical and numerical displays and summaries. II. Sampling and Experimentation: Planning and conducting a study (10%-15%) Data must be collected according to a well-developed plan if valid information on a conjecture is to be obtained. This plan includes clarifying the question and deciding upon a method of data collection and analysis. III. Anticipating Patterns: Producing models using probability theory and simulation (20%-30%) Probability is the tool used for anticipating what the distribution of data should look like under a given model. IV. Statistical Inference: Estimating population parameters and testing hypotheses (30%-40%) Statistical inference guides the selection of appropriate models. Chapter from POD* and approx. Time Chapter title from POD* text AP Statistics Topic Covered Intro to Statistics Statistics and Data Analysis 1&2 2 weeks The nature and role of Variability II A Overview of methods of data collection 1. Census 2. Sample survey 3. Experiment 4. Observational study II B Planning and conducting surveys 1. Characteristics of a well-designed and well-conducted survey 2. Populations, samples, and random selection 3. Sources of bias in sampling and surveys 4. Sampling methods, including simple random sampling, stratified random sampling, and cluster sampling Resources and Activities 1st 6 weeks project: Vocabulary in action TI-83 Enhanced Statistic Calculator Activities (completed at various points throughout the year) Article taken from http://apcentral.colle geboard.com on AP exam format and acceptable calculator list, etc http://apcentral.colle geboard.com/membe rs/article/1,3046,152171-0-8357,00.html Correlation and Causation Recommend ed Exercises from POD* text Read pp. 1-10 p.6: 1.3 – 7 Types of Data The Data Analysis Process Collecting Data Sensibly: Observation and experimentation II C Planning and conducting experiments 1. Characteristics of a well-designed and well-conducted experiment 2. Treatments, control groups, experimental units, random assignments, and replication 3. Sources of bias and confounding, including placebo effect and blinding 4. Completely randomized design 5. Randomized block design, including matched pairs design 3 2 weeks Read pp. 11-26 p. 14: 2.1-4 p. 20: 2.5, 7, 8 II D Generalizability of results and types of conclusions that can be drawn from observational studies, experiments, and surveys Sampling Simple comparative Experiments Displaying Categorical Data: Frequency Distributions, Bar Charts, and Pie Charts Displaying Numerical Data: Dotplots and Stemand-Leaf Displays Displaying Numerical Data: Frequency Distributions and Histograms Interpreting the Results of Statistical Analyses http://www.cut-theknot.org/do_you_kno w/misuse.shtml Random Rectangles I A. Constructing and interpreting graphical displays of distributions of univariate data (dotplot, stemplot, histogram, cumulative frequency plot) 1. Center and spread 2. Clusters and gaps 3. Outliers and other unusual features 4. Shape I C. Comparing distributions of univariate data (dotplots, back-to-back stemplots, parallel boxplots) 1. Comparing center and spread: within group, between group variation 2. Comparing clusters and gaps 3. Comparing outliers and other unusual features 4. Comparing shapes I E Exploring categorical data: frequency tables 4. Comparing distributions using bar charts “Getting to Know You” Questionnaire The Game of Greed Read pp. 27-35 p. 26: 2.11-21 odd Read pp. p. 35: 2.23-39 odd p. 54: 3.2, 7 p. 62: 3.12-19 p. 79: 3.21, 23, 26, 27, 31, 34 Quantitative Data project (sampling and describing distributions) p. 33: 3.40-42, 45, 49, 50 4 2 weeks 5 2 weeks Describing the Center of a Data Set Describing Variability in a Data Set I B. Summarizing distributions of univariate data 1. Measuring center: median, mean I B Summarizing distributions of univariate data 2. Measuring spread: range, interquartile range, standard deviation Summarizing a Data Set: Box Plots I B Summarizing distributions of univariate data Interpreting the Center and Variability: Chebyshev's Rule, the Empirical Rule, and z scores Interpreting the Results of Statistical Analyses: ScatterPlots p. 110: 4.2, 3, 5, 14 p. 120: 4.20, 22, 25 p. 125: 4.29-34 http://en.wikipedia.or g/wiki/SAT and http://en.wikipedia.or g/wiki/Iq I D Exploring bivariate data 1. Analyzing patterns in scatterplots p. 150: 5.1-8 Correlation and Linearity I D Exploring bivariate data 2. Correlation and linearity p. 163: 5.9, 1214, 16-20 Fitting a Line to Bivariate Data I D Exploring bivariate data 3. Least squares regression line monopoly game activity p. 174: 5.26-29, 32-35 Assessing the Fit of a Line I D Exploring bivariate data 4. Residual plots, outliers, and influential points I D Exploring bivariate data 5. Transformations to achieve linearity: logarithmic and power transformations Reading Computer Outputs p. 188: 5.37-41, 43, 47 2nd 6 weeks project: Transformations (APEX VS) p. 206: 5.52, 54, 55, 58-59 Nonlinear Relationships and Transformations Chance Experiments Definition of Probability, Law of Large Numbers 6 2 weeks 4. Using boxplots 5. The effect of changing units on summary measures I B Summarizing distributions of univariate data 3. Measuring position: quartiles, percentiles, standardized scores (zscores) “How Fast is Your Heart Beating?” (YMS p. 4) Basic Properties of Probability Conditional Probability, Independence p. 233: 6.7-10, 12 III A Probability as relative frequency 1. Interpreting probability, including long-run relative frequency interpretation 2. Law of Large Numbers’ concept III A Probability as relative frequency 3. Addition rule, multiplication rule, conditional probability, and independence III B Combining independent Random Variables 1. Notion of independence versus dependence. Three Coin Pony Three Point, One Point Game p. 248: 6.14, 15, 17, 18, 20-23, 28 p. 259: 6.29-35 p. 267: 6.36-38, 43-45, 47, 48, 51 General Probability Rules - Addition Rule, Multiplication Rule Estimating Probabilities Empirically and Using Simulation Random Variables Probability Distributions for Discrete and Continuous The Mean and Standard Deviation of a Random Variable Linear Combinations 7 2 weeks 8 2 weeks The Binomial and Geometric Distributions The Normal Distributions Properties and the Use of Tables Checking for Normality Constructing Normal Probability Plots Using the Normal Distribution to Approximate a Discrete Distribution Statistics and Sampling Variability Sampling Distribution of a Sample Mean Sampling Distribution of a Sample Proportion p. 280: 6.53-56, 58, 60-63 III A Probability as relative frequency 5. Simulation of random behavior and probability distributions Heads Up! Project (simulation) III A Probability as relative frequency 4. Discrete random variables and their probability distributions, including binomial and geometric Casino Lab - Craps, Roulette, Blackjack, Monte's Dilemma, etc. III A Probability as relative frequency 6. Mean (expected value) and standard deviation of a random variable, and linear transformation of a random variable p. 309: 7.1, 5, 7 p. 314: 7.9, 11, 15, 19 p. 333: 7.28-32, 38-40 III B Combining independent random variables 2. Mean and standard deviation for sums and differences of independent random variables III A Probability as relative frequency 4. Discrete random variables and their probability distributions, including binomial and geometric p. 344: 7.44, 45, 47, 49 p. 345: 7.50, 51, 54-56 p. 363: 7.59-67 odd; 68-72, 75; 62, 68-72, 75 III C The normal distribution 1. Properties of the normal distribution 2. Using tables of the normal distribution 3. The normal distribution as a model for measurements p. 381: 7.88-94 p. 409: 8.1-4, 79 III D Sampling distributions 2. Sampling distribution of a sample mean 3. Central Limit Theorem III D Sampling distributions 1. Sampling distribution of a sample proportion Rice Virtual Lab (sampling distribution of sample means) ESP Card activity (sampling distribution of sample proportion) p. 420: 8.14-17, 20, 23, 25, 26 p. 426: 8.27-29, 31-33 Point Estimation, Confidence Intervals Large Sample Confidence Interval for a Proportion 9 2 weeks Large Sample Confidence Interval for a Mean 10 2 weeks Hypotheses and Test Procedures, Errors in Hypothesis Testing, Power and Probability of Type II Error Hypotheses Testing for a Population Proportion Hypotheses Testing for a Population Mean Difference Between Two Means (Independent Samples) Difference Between Two Means (Paired Samples) 11 2 weeks Difference Between Two Proportions IV A Estimation (point estimators and confidence intervals) 2. Properties of point estimators, including unbiasedness and variability IV A Estimation (point estimators and confidence intervals) 1. Estimating population parameters and margin of errors 3. Logic of confidence intervals, meaning of confidence level and confidence intervals, and properties of confidence intervals. 4. Large sample confidence interval for a proportion III D Sampling distributions 7. t-distributions IV A Estimation (point estimators and confidence intervals) 5. Large sample confidence interval for a mean IV B Tests of significance 1. Logic of significance testing, null and alternative hypotheses; pvalues; one- and two-sided tests; concepts of Type I and Type II errors; concept of power IV B Tests of significance 2. Large sample test for a proportion Hill of beans activity (confidence interval for a proportion) One sample proportion confidence interval project p.467 (9.34, 9.38, 9.45, 9.46, 9.47) Prosecutor's Fallacy Activity p. 481 (10.4, 10.6) DNA Fingerprinting Activity “Is One Side of a Coin Heavier?” IV B Tests of significance 3. Large sample test for a mean III D Sampling distributions 5. Sampling distribution of a difference between two independent sample means IV A Estimation (point estimators and confidence intervals) 7. Large sample confidence interval for a difference between two means (unpaired and paired) IV B Tests of significance 5. Large sample test for a difference between two means (unpaired and paired) III D Sampling distributions 4. Sampling distribution of a difference between two independent sample proportions IV A Estimation (point estimators and confidence intervals) 6. Large sample confidence interval for a difference between two proportions p. 452: 9.10-13, 16, 17, 19 Designing experiments with paper helicopters project p. 500 (10.22, 10.27, 10.28, 10.36) 12 2 weeks 13 2 weeks 4 weeks μσπχ 14 1 week 15 1 week Chi-Squared Tests for Univariate Categorical Data Tests for Homogeneity and Independence In a Two-Way Table The Simple Linear Regression Model Inferences Concerning the Slope of a Population Regression Line Checking Model Adequacy Review for the AP Exam AP Statistics Exam Multiple Regression Analysis Analysis of Variance IV B Tests of significance 4. Large sample test for a difference between two proportions III D Sampling distributions 8. Chi-square distribution IV B Tests of significance 6. Chi-square test for goodness of fit, homogeneity of proportions, and independence (one- and twoway tables) III D Sampling distributions 8. Confidence interval for the slope of a least-squares regression line. “M&M Activity” Chi-square and Linear t test project IV B Tests of significance 7. Test for the slope of a leastsquares regression line 1997 – 2006 AP Statistics Free Response questions and available multiple choice questions. Supplemental AP practice resources, APEX online review and practice assessments, Student Conceptual Outline (mentioned later). Teaching Strategies and Philosophies Non-calculus based discovery with the aid of technology Advanced Placement Statistics is non-calculus based and seeks to provide students with a student-centered, activity-based, learning environment where discovery and exploration of descriptive and inferential statistics concepts drive intellectual experiences. The instructor serves as a facilitator and guide to learning. Students are directed to explore aspects of the statistical process by forming hypotheses, designing procedures to test those hypotheses using appropriate statistical techniques, analyzing problems and data, and drawing conclusions. It is important that all concepts and applications are rationalized by having students frequently reference realworld data and newspaper/online journal articles. Statistics should not only remain within the scope of the educational environment, but be applied directly to students’ lives. Therefore, real data is used where possible during the course, which translates into numbers with a context. An effort is made to revisit some of the same data sets, examples and projects utilized in realizing descriptive statistical concepts, as students grasp the inferential statistical concepts. The re-use of certain data sets and examples throughout the course helps ease the transition from some of the more intuitive descriptive concepts to the more unfamiliar inferential tests. For example, data is used from the random rectangle activity which explores different methods of sampling, to determine good methods of sampling. The same data collected for each method of sampling is used to create graphs (like dot-plots, box-plots, etc...) that students use to describe and compare center, unusual features, shape and spread of the data. The data is also employed to introduce the bell-shape and eventually extrapolate to the normal curve. Emphasis is placed on the importance of the normal distribution (and binomial distribution) to the study of statistics, when data allows it. Students are trained to carry out the process of simulation where appropriate in the course. Students utilize simulation and the normal curve in discovering sampling distributions, and later on in performing inferential tests. They concentrate on strategy and tools of basic data analysis before the instructor imparts inferential ideas to them. This enables the learner to judge facts and figures presented to them in problems of scientific inference by enlisting appropriate methods, conducting statistical analyses and developing inferences. Success in this course and on the AP exam is, to a great extent, dependant on the students’ ability to communicate statistics through writing. As a result, writing is an integral part of the course. Students are taught how to write about data they gather and about concepts like sampling distributions, randomization, random variables, etc…. They also learn how to write about statistical processes. Appropriate use of statistical vocabulary is always stressed. The instructor encourages students to focus on strategy and not just skill. For example, on inference problems they must recognize the difference between samples, sampling distributions and populations. They must be able to distinguish between sample and proportion. Students learn to verify the conditions required for use of a statistical procedure before they carry it out. Emphasis is placed on the strategy of reading for understanding. The learner discovers that all statistics problems come with context. Students utilize the primary text, supplementary texts, the internet, journals, and actual AP exam questions to develop their reading strategies and to discover statistics in context. The abilities to learn, communicate, apply, connect and retain all aspects of the statistical process rely vastly on an individual’s capacity to organize his/her lessons into notes, assignments, and reviews. Cumulative review exercises and homework assignments are built into the course. This allows students to practice selecting the best statistical method on the AP test when faced with problems based on any portion of the course’s content. For review purposes, each student creates a Student AP Conceptual Outline as the course advances. An excerpt of an example is shown towards the end of this document. The exam review process is structured to maximize student performance. AP free response and multiple-choice questions from previous AP exams are incorporated in all unit lessons and tests. They are also utilized, along with other exam preparation resources, during the weeks of review leading up to the AP test. Technology is an invaluable device when it comes to teaching and learning statistics. Since many of the students are familiar with technology and some of its applications in their daily lives, the instructor builds off their awareness when it comes to technology integration. Technology facilitates the process of learning to work with data. Students use the TI graphing calculators and software like Minitab and Fathom so they can focus their efforts on the development of concepts rather than on calculations. On the other hand, emphasis is placed on the importance of accuracy and precision of calculations in drawing conclusions. Software offers students experience using a representative statistical package and provides them with the opportunity to read and analyze generic computer output. Students utilize applets in this course to run simulations and discover underlying notions. The campus has purchased access for each AP student and teacher to http://www.apexvs.com. This web-based resource provides flash animation lessons that include statistical models and simulations. It also provides activities and online assessments that compliment the topics in the AP Statistics Outline of Concepts. Students are provided with access to theses technologies that provide empirical evidence of concepts, as well as provide effective dynamic demonstrations. Through the use of calculators and computers, students enhance their development of statistical understanding by exploring and analyzing data, assessing models, and performing simulations. Assessment Competency -Based Assessment At the beginning of the course, each student is provided with a syllabus that includes an outline and timeline of all AP Statistics course concepts, a summary of the types of course exams and projects, policies for grading and rubric explanations, rationalization of reading and writing requirements, synopsis of technology use, and information on the College Board and the Advanced Placement Test. Students are made aware of the AP Central website early in the course and guided through a tour of its available resources that include the AP Statistics Course Description and syllabus, AP Statistics exam tips for students, past AP Statistics tests, sample responses, and scoring guides. Students are informed of the holistic grading scale that is used throughout the course and on the AP test for scoring their free response answers. Throughout the course, students are assessed in a variety of ways. Some are teacher-designed, activity-based assessments that including simulations and interpretation of data. Assigned independent-study projects require students to design, administer, collect data from, analyze, and write about surveys, experiments and observational studies. Often students investigate data with the aid of software and then write a report that describes the investigation and what was learned. Students are molded to draw on tools they have learned in class to analyze information from statistically based articles. They follow through on their analysis by writing a critical report on their findings. Students are assessed heavily on their ability to verify the conditions required for use of a statistical procedure before they carry it out They are assessed acutely on their ability to read, design, analyze and communicate methods, results and interpretations of the problem being investigated, and use of proper statistical jargon. These skills are modeled and taught throughout the course. Tests are designed to be similar in form and difficulty level to the AP exam, consisting of multiple-choice and free response questions, in order to prepare students for the format of the actual exam. Students are permitted access to an AP Statistics test formula sheet on exams to prepare them to use this resource efficiently when they take the exam. These assessments emphasize: Written and oral communication of statistical concepts Connections between Design, Analysis and Interpretation in the context of the situation or application The use of technology to enhance the development of statistical understanding Group cooperation and collaboration Student AP Conceptual Outline This outline is not intended only as notes, but as a tool to help the students read, summarize and review resources at the end of every topic. It serves as a review sheet, vocabulary definitions sheet, and statistics manual which helps students gain a full command of the language of statistics, a skill essential to mastery and AP test success. The following is an excerpt from a student’s outline: I. Exploring Data: Observing patterns and departures from patterns Exploratory analysis of data makes use of graphical and numerical techniques to study patterns and departures from patterns. Emphasis should be placed on interpreting information from graphical and numerical displays and summaries. A. Interpreting graphical displays of distributions of univariate data (dotplot, stemplot, histogram, cumulative frequency plot) 1. Center and Spread – The center of a distribution is often called the median or the middle of the data; it can also be called the mean which is an average of the data; the spread of the distribution is called variance and it is how wide the data is distributed about the center 2. Clusters and gaps – clusters in the distribution are places where there is some data separated by gaps; one distribution could have many clusters; gaps in the distribution are holes or places where there is a small amount of data in between the distribution 3. Outliers and other unusual features – individual observations that fall well outside the overall pattern of the data 4. Shape – the overall pattern of the distribution for examples of shapes see I, C, 4 B. Summarizing distributions of univariate (one variable) data 1. Measuring center: a. Median - to find median or the middle of the data: Arrange all observations in order of size, from smallest to largest. If the number n observations is odd, the median M is the center observation in the ordered list. The location of the median is found by counting (n+1)/2 observations up from the bottom of the list. If the number n observations is even, the median M is the average of the two center observations in the ordered list. b. Mean – If n observations are denoted by x1, x2, ..., xn their mean is: x 1 x1 x 2 ... x 3 n or in more compact notation: x 1 xi n Notes and Acknowledgements: Even though the course outline above follows the chapters in the primary text (POD*), the text chapters are supplemented with lessons, activities, problem sets, projects and data from other resources such as the supplemental textbooks mentioned above, articles and journals, the College Board website, and other technology resources. It is important to credit the authors of these textbooks, College Board writers and presenters of the Advanced Placement Institutes for Statistics, for many of the resources implemented in this course. They should also be credited for their expert insight into Advanced Placement and the subject of Statistics, which has influenced the course design, teaching strategies and philosophies, and assessments.