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Roanoke County Public Schools AP Statistics Curriculum Guide 2011 AP Statistics Curriculum Guide 2011 Mathematics Curriculum Guide Revised 2011. Available at www.rcs.k12.va.us. Roanoke County Public Schools does not discriminate with regard to race, color, age, national origin, gender, or handicapping condition in an educational and/or employment policy or practice. Questions and/or complaints should be addressed to the Deputy Superintendent/Title IX Coordinator at (540) 562-3900 ext. 10121 or the Director of Pupil Personnel Services/504 Coordinator at (540) 562-3900 ext. 10181. Acknowledgements The following people have made tremendous contributions to the completion of this curriculum guide and all are appreciated. Edward Donahue William Byrd HS Brian W. Harris Hidden Valley HS Susan Sine Cave Spring HS Roanoke County Public Schools Administration Dr. Lorraine Lange Superintendent Cecil Snead Director of Secondary Instruction Rebecca Eastwood Director of Elementary Instruction Linda Bowden Mathematics Coordinator Preface This curriculum guide is written for the teachers to assist them in using the textbooks/resources in a most effective way. This guide will assist the mathematics teacher in preparing students for the challenges of the twenty-first century. As established by the National Council of Teachers of Mathematics Principles and Standards for School Mathematics, educational goals for students are changing. Students should have many and varied experiences in their mathematical training to help them learn to value mathematics, become confident in their ability to do mathematics, become problem solvers, and learn to communicate and reason mathematically. This guide, along with the available textbook resources, other professional literature, alternative assessment methods, and varied instruction in-service activities will assist the mathematics teacher in continuing to integrate these student goals into the curriculum. AP Statistics Curriculum Guide 2011 Table of Contents Introduction/General Comments ............................................................................................................................................. i Textbook/Resources Overview ................................................................................................................................................ i Sequence of Instruction and Pacing Suggestions ................................................................................................................... ii Sequence of Instruction and Pacing Suggestions .................................................................................................................. iii Mapping for Instruction - First Nine Weeks ............................................................................................................................ 1 Mapping for Instruction - Second Nine Weeks ........................................................................................................................ 5 Mapping for Instruction - Third Nine Weeks ......................................................................................................................... 10 Mapping for Instruction - Fourth Nine Weeks ....................................................................................................................... 17 SOL Blueprints ....................................................................................................................................................................... 19 SOL Enhanced Scope and Sequence ...................................................................................................................................... 19 Supplemental Resources ....................................................................................................................................................... 19 SOL 2009 Framework ............................................................................................................................................................ 21 AP Statistics Curriculum Guide 2011 Introduction/General Comments 1. Please follow the suggested sequence and pacing for this course. 2. It is strongly suggested that teachers use the activities and resources that are included in this guide. Many activities have been selected that encourage the use of graphing calculators or computer. 3. Use of mixed reviews and spiraling on assessments is strongly recommended. 4. Students must be allowed the use of calculators throughout this course and should be introduced to the AP Exam formula sheet and statistical tables as early and frequent as possible. 5. Because of time constraints, there is little time to go over homework in class. Students will be encouraged to check answers in the back of the book and to seek extra help outside of class when needed. Textbook/Resources Overview Course Title: AP Statistics Course Text: The Practice of Statistics – 4th Edition Publisher: W.H. Freeman and Company Supplemental Materials: Teacher’s Classroom Resources: Teacher’s Solution Manual Printed Test Bank Digital Test Bank Lecture Powerpoint Presentations Teacher’s Titanium Resource Binder Teacher’s Resource CD Teacher’s Companion Website – www.whfreeman.com/tps4e i AP Statistics Curriculum Guide 2011 Sequence of Instruction and Pacing Suggestions First Nine Weeks SOL Chapter/Sections/Topic *Time Frame PS.8, PS.9, PS.10 Chapter 4: Designing Studies; sections 1-2 7 blocks PS.1, PS.2, PS.3, PS.7 Chapter 1: Exploring Data; sections 1-3 5 blocks PS.17 Chapter 2: Modeling Distributions of Data; sections 1-2 5 blocks PS.12, PS.13 Chapter 5: Probability: What are the Chances?; sections 1-3 6 blocks *Time Frame is based on 95 minutes of instruction per block. Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for 45 days in the middle schools. First Nine Weeks Total 23 blocks Second Nine Weeks SOL Chapter/Sections/Topic *Time Frame PS.14, PS.15, PS.16 Chapter 6: Random Variables; sections 1-3 8 blocks PS.8, PS.20 Chapter 7 : Sampling Distributions; sections 1-3 7 blocks PS.18, PS.21 Chapter 8: Estimating with Confidence; sections 1-3 7 blocks *Time Frame is based on 95 minutes of instruction per block. Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for 45 days in the middle schools. ii Second Nine Weeks Total 22 blocks AP Statistics Curriculum Guide 2011 Sequence of Instruction and Pacing Suggestions Third Nine Weeks SOL Chapter/Sections/Topic *Time Frame PS.19, PS.21 Chapter 9: Testing a Claim; sections 1-3 7 blocks PS.18, PS.19, PS.21 Chapter 10: Comparing Two Populations or Groups; sections 1-2 5 blocks PS.19 Chapter 11: Inference for Distributions of Categorical Data; sections 1-2 5 blocks PS.4, PS.5 Chapter 3: Describing Relationships; sections 1-2 6 blocks *Time Frame is based on 95 minutes of instruction per block. Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for 45 days in the middle schools. Third Nine Weeks Total 23 blocks Fourth Nine Weeks SOL Chapter/Sections/Topic *Time Frame PS.5, PS.6 Chapter 12: More About Regression; sections 1-2 5 blocks Review and AP Testing (It is STRONGLY urged to complete all new curriculum by mid-April (spring break), leaving ample review time (2-3 weeks)for the AP Exam. “Last minute” topics, such as in Chapter 12, may be taught concurrently with review if needed 10 blocks (Additional time is added because AP students frequently miss multiple classes during AP Exam weeks) End of Year Project 5 blocks Review and Final Exam 2 blocks *Time Frame is based on 95 minutes of instruction per block. Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for 45 days in the middle schools. iii Fourth Nine Weeks Total 22 blocks AP Statistics Curriculum Guide 2011 Mapping for Instruction - First Nine Weeks Chapter 4: Designing Studies SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Introduction to Course & Opening Activity Supporting Materials Teacher’s Titanium Resource Binder Activity: See no evil, hear no evil (TPS, pg. 206) Technology Corner: Choosing an SRS (TPS, pg. 214) Comments It is recommended that the teacher introduce the course through an opening-day activity. 1 Block PS.9 The student will plan and conduct a survey. The plan will address sampling techniques (e.g., simple random and stratified) and methods to reduce bias. Investigate and describe sampling techniques, such as simple random sampling, stratified sampling, and cluster sampling. Determine which sampling technique is best, given a particular context. Given a plan for a survey; identify possible sources of bias, and describe ways to reduce bias. PS.8 The student will describe the methods of data collection in a census, sample survey, experiment, and observational study and identify an appropriate method of solution for a given problem setting. Compare and contrast controlled experiments and observational studies and the conclusions one can draw from each. PS.10 The student will plan and conduct an experiment. The plan will address control, randomization, and measurement of experimental error. NO SOL 4-1 Sampling and Surveys 2 Blocks Teachers and students should avoid the use of the term Lurking Variable. 4-2 Experiments 2.5-3 Blocks Teachers will need to supplement more examples for the matched-pairs design. Students should practice both outlining and diagraming experimental designs. 4-3 Using Studies Wisely (Optional) .5 Block Chapter 4 Test 1 Block 1 Data Ethics as a topic is not addressed on the AP exam. AP Statistics Curriculum Guide 2011 Chapter 1: Exploring Data SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics PS.7 The student, using two-way tables, will analyze categorical data to describe patterns and departures from patterns and to find marginal frequency and relative Introduction frequency, including conditional frequencies. Produce a two-way table as a summary of the information 1-1 Analyzing Categorical obtained from two categorical variables. Data Calculuate marginal, relative, and conditional frequencies in a two-way table. Use marginal, relative, and conditional frequencies to 1 Block analyze data in two-way tables within the context of the data. PS.1 The student will analyze graphical displays of data, including dotplots, stemplots, and histograms, to identify and describe patterns and departures from patterns, using central tendancy, spread, clusters, gaps, and outliers. Appropriate technology will be used to create graphical displays. Create and interpret graphical displays of data, including dotplots, stem-and-leaf plots, and histograms. Examine graphs of data for outliers, and explain the 1-2 Displaying Quantitative outlier(s) within the context of the data. Examine graphs of data, and identify the central tendancy of Data with Graphs the data as well as the spread. Explain the central tendancy and the spread of the data within the context of the data. PS.3 The student will compare distributions of two or more 2 Blocks univariate data sets, analyzing center and spread (within group and between group variations), clusters and gaps, shapes, outliers, or other unusual features. Appropriate technology will be used to generate graphical displays. Compare and contrast two or more univariate data sets by analyzing measures of center and spread within a contextual framework. 2 Supporting Materials Activity: Hiring Discrimination – it just won’t fly (TPS, pg. 5) Comments Teachers should not spend too much time in discussing production of bar graphs & pie charts. Encourage the use of Excel to produce bar graphs and pie charts. Teachers may omit Simpson’s Paradox. Technology Corner: Histogram on the Calculator (TPS, pg. 38) Students should be encouraged to write as many comparisons of data sets as possible during this section. Students will utilize the graphing calculator as the primary tool for displaying data. AP Statistics Curriculum Guide 2011 PS.2 The student will analyze numerical characteristics of univariate data sets to describe patterns and departure from patterns, using mean, median, mode, variance, standard 1-3 Describing Quantitative deviation, interquartile range, range, and outliers. Data with Numbers Appropriate technology will be used to calculate statistics. Interpret mean, median, mode, range, interquartile range, variance, and standard deviation of a univariate data set in 2 Blocks terms of the problem’s context. Identify possible outliers, using an algorithm. Explain ways in which standard deviation addresses dispersion by examining the formula for standard deviation. Technology Corner: Making calculator boxplots (TPS, pg. 61) Technology Corner: Computing numerical summaries with technology (TPS, pg. 65) Students should be encouraged to calculate statistica measures by hand at least once and then should be proficient in using their graphing calculators to arrive at numerical summaries. Teachers should include reading computer output of numerical summaries. Chapter 2: Modeling Distributions of Data SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials 2-1 Describing Location in a Distribution 1.5 Blocks Activity: Where do I stand? (TPS, pg. 84) PS.17 The student will identify properties of a normal distribution and apply the normal distribution to determine 2-2 Normal Distributions probabilies, using a table or graphing calculator. 2.5 Blocks Identify the properties of a normal distribution. Describe how the standard deviation and the mean affect the graph of the normal distribution. Determine the probability of a given event, using the normal distribution. Activity: The Normal Curve applet (TPS, pg. 111) NO SOL Chapters 1 & 2 Test 1 Block 3 Comments Ogives are not a large part of the AP Exam. Students must understand that the 68-95-99.7 rule is for Technology Corner: The standard approximation only. Exact calculations should be carried Normal curve (TPS, pg. 118) out whenever sufficient Technology Corner: From z-scores information is provided. to areas, and vice versa (TPS, pg. 123) Technology Corner: Normal probability plots (TPS, pg. 128) Normal probability plots are not included on the AP exam. AP Statistics Curriculum Guide 2011 Chapter 5: Probability: What Are the Chances? SOL with Essential Knowledge and Skill PS.13 The student will find probabilities (relative frequency and theoretical), including conditional probabilities for events that are either dependent or independent, by applying the “law of large numbers” concept, the additional rule, and the multiplication rule. Find conditional probabilities for dependent, independent, and mutually exclusive events. Textbook Chapters/Sections/Topics 5-1 Randomness, Probability, and Simulation 1 Block PS.12 The student will identify and describe two or more events as complementary, dependent, independent, and/or mutually exclusive. Define and give contextual examples of complementary, dependent, independent, and mutually exclusive events. Givent two or more events in a problem setting, determine if 5-2 Probability Rules the events are complementary, dependent, independent, and/or mutually exclusive. PS.13 The student will find probabilities (relative frequency 1.5 Blocks and theoretical), including conditional probabilities for events that are either dependent or independent, by applying the “law of large numbers” concept, the additional rule, and the multiplication rule. Find conditional probabilities for dependent, independent, and mutually exclusive events. PS.12 The student will identify and describe two or more events as complementary, dependent, independent, and/or mutually exclusive. Define and give contextual examples of complementary, dependent, independent, and mutually exclusive events. Givent two or more events in a problem setting, determine if 5-3 Conditional Probability the events are complementary, dependent, independent, and Independence and/or mutually exclusive. PS.13 The student will find probabilities (relative frequency 2.5 Blocks and theoretical), including conditional probabilities for events that are either dependent or independent, by applying the “law of large numbers” concept, the additional rule, and the multiplication rule. Find conditional probabilities for dependent, independent, and mutually exclusive events. Chapter 5 Test 1 Block 4 Supporting Materials Comments Activity: Probability applet (TPS, pg. 284) Simulation is introduced here and should be an integral part of the course from this point forward. Students should be shown how Venn diagrams may be a useful tool for calculating probabilities. Students should be shown where probability formulas can be found on the AP Exam formula sheet. Appropriate terminology and notation should be encouraged throughout this section. Students should be shown how Tree diagrams may be a useful tool for calculating probabilities. Students should be shown where probability formulas can be found on the the AP Exam formula sheet. Appropriate terminology and notation should be encouraged throughout this section. AP Statistics Curriculum Guide 2011 Mapping for Instruction - Second Nine Weeks Chapter 6: Random Variables SOL with Essential Knowledge and Skill PS.16 The student will identify random variables as independent or dependent and find the mean and standard deviation for sums and differences of independent random variables. Compare and contrast independent and dependent random variables. Find the standard deviation for sums and differences of independent random variables. PS.16 The student will identify random variables as independent or dependent and find the mean and standard deviation for sums and differences of independent random variables. Compare and contrast independent and dependent random variables. Find the standard deviation for sums and differences of independent random variables. PS.14 The student will develop, interpret, and apply the binomial probability distribution for discrete random variables, including computing the mean and standard deviation for the binomial variable. Develop the binomial probability distribution within a realworld context. Calclulate the mean and standard deviation for the binomial variable. Use the binomial distribution to calculate probabilities associated with experiments for which there are only two possible outcomes. PS.15 The student will simulate probability distributions, including binomial and geometric. Design and conduct an experiment that simulates a binomial distribution. Textbook Chapters/Sections/Topics 6-1 Discrete and Continuous Random Variables Supporting Materials Activity: Bottled Water versus Tap Water (TPS, pg. 340) Technology Corner: Analyzing random variables on the calculator (TPS, pg. 348) Technology Corner: Simulating with randNorm 2 Blocks 6-2 Transforming and Combining Random Variables Comments The AP Exam only addresses independent random variables. Students are only required to know the rules for combining independent random variables. 2 Blocks 6-3 Binomial and Geometric Random Variables pg. 388) 3 Blocks Chapter 6 Test 1 Block 5 The AP Exam does not include geometric random variables Technology Corner: Binomial coefficients on the calculator (TPS, explicitly. Technology Corner: Binomial probability on the calculator (TPS, pg. 389) Technology Corner: Geometric probability on the calculator (TPS, pg. 400) Students should be tasked to find binomial probabilities in a variety of contexts. The binomial probability formula should be taught but not relied upon heavily in calculations. Appropriate notation and calculator notation should be reinforced throughout this section. AP Statistics Curriculum Guide 2011 Chapter 7: Sampling Distributions SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics PS.8 The student will describe the methods of data collection in a census, sample survey, experiment, and observational study and identify an appropriate method of solution for a given problem setting. Compare and contrast population and sample, and parameter and statistic. 7-1 What is a Sampling PS.20 The student will identify the meaning of sampling Distribution? distribution with reference to random variable, sampling statistic, and parameter and explain the Central Limit Theorem. This will include sampling distribution of a sample proportion, a sample mean, a difference between 1.5 Blocks two sample proportions, and a difference between two sample means. Describe the effect of sample size on the sampling distribution and on related probabilities. Identify and describe the characteristics of a sampling distribution of a sample proportion, mean, difference between two sample proportions, or difference between two sample means. PS.20 The student will identify the meaning of sampling distribution with reference to random variable, sampling statistic, and parameter and explain the Central Limit Theorem. This will include sampling distribution of a 7-2 Sample Proportions sample proportion, a sample mean, a difference between two sample proportions, and a difference between two sample means. 2 Blocks Describe the effect of sample size on the sampling distribution and on related probabilities. Use the normal approximation to calculate probabilities of sample statistics falling within a given interval. Identify and describe the characteristics of a sampling distribution of a sample proportion, mean, difference between two sample proportions, or difference between two sample means. 6 Supporting Materials Comments Activity: The German tank problem Students should be tasked frequently to identify the shape, (TPS, pg. 414) center, and spread of a given sampling distribution. Students should be tasked regularly to compare and contrast population and sample distributions. Simulation of sampling distributions is encouraged to help students make deep connections with the material. Teachers should stress the conditions that must be met in order to use the normal approximation to the sampling distribution of the sample proportion. AP Statistics Curriculum Guide 2011 PS.20 The student will identify the meaning of sampling distribution with reference to random variable, sampling statistic, and parameter and explain the Central Limit Theorem. This will include sampling distribution of a sample proportion, a sample mean, a difference between two sample proportions, and a difference between two 7-3 Sample Means sample means. Describe the use of the Central Limit Theorem for drawing inferences about a population parameter based on a 2.5 Blocks sample statistic. Describe the effect of sample size on the sampling distribution and on related probabilities. Use the normal approximation to calculate probabilities of sample statistics falling within a given interval. Identify and describe the characteristics of a sampling distribution of a sample proportion, mean, difference between two sample proportions, or difference between two sample means. Chapter 7 Test 1 Block 7 Teachers should stress the conditions that must be met in order to use the normal approximation to the sampling distribution of the sample mean. Teachers should stress that the Central Limit theorem only applies to the sampling distribution of the mean and not that of the sample proportion. AP Statistics Curriculum Guide 2011 Chapter 8: Estimating with Confidence SOL with Essential Knowledge and Skill PS.18 The student, given data from a large sample, will find and interpret point estimates and confidence intervals for parameters. The parameters will include proportion and mean, difference between two proportions, and difference between to means (independent and paired). Construct confidence intervals to estimate a population parameter, such as a proportion or the difference between two proportions; or a mean or the difference between two means. Interpret confidence intervals in the context of the data. Explain the importance of random sampling for confidence intervals. Calculate point estimates for parameters, and discuss the limitations of point estimates. PS.18 The student, given data from a large sample, will find and interpret point estimates and confidence intervals for parameters. The parameters will include proportion and mean, difference between two proportions, and difference between to means (independent and paired). Construct confidence intervals to estimate a population parameter, such as a proportion or the difference between two proportions; or a mean or the difference between two means. Interpret confidence intervals in the context of the data. Textbook Chapters/Sections/Topics Supporting Materials 8-1 Confidence Intervals: The Basics Activity: The mystery mean (TPS, pg. 468) Activity: The Confidence Interval applet (TPS, pg. 473) Comments Students should simulate multiple confidence intervals to develop an understanding of confidence level. 1 Block 8-2 Estimating a Population Proportion 2 Blocks 8 If the teacher has not yet Technology Corner: Confidence interval for a population proportion introduced the student to the four-step inference process, it is (TPS, pg. 492) highly recommended that it begin at this point and continue through AP testing. Students should be encouraged to know both the formulas and calculator functions for all inference procedures. AP Statistics Curriculum Guide 2011 PS.18 The student, given data from a large sample, will find and interpret point estimates and confidence intervals for parameters. The parameters will include proportion and mean, difference between two proportions, and difference between to means (independent and paired). Construct confidence intervals to estimate a population 8-3 Estimating the parameter, such as a proportion or the difference between Population Mean two proportions; or a mean or the difference between two means. 3 Blocks Interpret confidence intervals in the context of the data. PS.21 The student will identify properties of a t-distribution and apply t-distributions to single-sample and two-sample (independent and matched pairs) t-procedures, using tables or graphing calculators. Identify the properties of a t-distribution Compare and contrast a t-distribution and a normal distribution Use a t-test for single-sample and two-sample data. Chapter 8 Test 1 Block 9 Technology Corner: Inverse t on the calculator (TPS, pg. 506) Technology Corner: One-sample t intervals for the mean on the calculator (TPS, pg. 514) Students should be encouraged to know both the formulas and calculator functions for all inference procedures. AP Statistics Curriculum Guide 2011 Mapping for Instruction - Third Nine Weeks Chapter 9: Testing a Claim SOL with Essential Knowledge and Skill PS.19 The student will apply and interpret the logic of a hypothesis-testing procedure. Tests will include large sample test for proportion, mean, difference between two proportions, and difference between two means (independent and paired) and Chi-squared test for goodness of fit, homogeneity of proportions, and independence. Use hypothesis-testing procedures to determine whether or not to reject the null hypothesis. The null hypothesis may address proportion, mean, difference between two proportions or two means, goodness of fit, homogeneity of proportions, and independence. Compare and contrast Type I and Type II errors. Explain how and why the hypothesis-testing procedure allows one to reach a statistical decision. PS.19 The student will apply and interpret the logic of a hypothesis-testing procedure. Tests will include large sample test for proportion, mean, difference between two proportions, and difference between two means (independent and paired) and Chi-squared test for goodness of fit, homogeneity of proportions, and independence. Use hypothesis-testing procedures to determine whether or not to reject the null hypothesis. The null hypothesis may address proportion, mean, difference between two proportions or two means, goodness of fit, homogeneity of proportions, and independence. Compare and contrast Type I and Type II errors. Explain how and why the hypothesis-testing procedure allows one to reach a statistical decision. Textbook Chapters/Sections/Topics 9-1 Significance Tests: The Basics Supporting Materials Comments Activity: I’m a great free-throw shooter! (TPS, pg. 528) The calculation of the probability of a Type II error is an optional topic, however students must understand the relationship between power and the probability of a Type II error. Activity: Investigating power (TPS, pg. 543) 1 Block 9-2 Tests about a Population Proportion 2.5 Blocks 10 Technology Corner: Oneproportion z test on calculator (TPS, pg. 555) Students should be encouraged to know both the formulas and calculator functions for all inference procedures. AP Statistics Curriculum Guide 2011 PS.19 The student will apply and interpret the logic of a hypothesis-testing procedure. Tests will include large sample test for proportion, mean, difference between two proportions, and difference between two means (independent and paired) and Chi-squared test for goodness of fit, homogeneity of proportions, and independence. Use hypothesis-testing procedures to determine whether or 9-3 Tests about a Population Mean not to reject the null hypothesis. The null hypothesis may address proportion, mean, difference between two proportions or two means, goodness of fit, homogeneity of proportions, and independence. 2.5 Blocks Compare and contrast Type I and Type II errors. Explain how and why the hypothesis-testing procedure allows one to reach a statistical decision. PS.21 The student will identify properties of a t-distribution and apply t-distributions to single-sample and two-sample (independent and matched pairs) t-procedures, using tables or graphing calculators. Identify the properties of a t-distribution Compare and contrast a t-distribution and a normal distribution Use a t-test for single-sample and two-sample data. Chapter 9 Test 1 Block 11 Technology Corner: Computing P-values from t distributions on the calculator (TPS, pg. 569) Technology Corner: One-sample t test on the calculator (TPS, pg. 573) Students should be encouraged to know both the formulas and calculator functions for all inference procedures. AP Statistics Curriculum Guide 2011 Chapter 10: Comparing Two Populations or Groups SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics PS.18 The student, given data from a large sample, will find and interpret point estimates and confidence intervals for parameters. The parameters will include proportion and mean, difference between two proportions, and difference between to means (independent and paired). Construct confidence intervals to estimate a population parameter, such as a proportion or the difference between 10-1 Comparing Two two proportions; or a mean or the difference between two Proportions means. Interpret confidence intervals in the context of the data. PS.19 The student will apply and interpret the logic of a hypothesis-testing procedure. Tests will include large 2 Blocks sample test for proportion, mean, difference between two proportions, and difference between two means (independent and paired) and Chi-squared test for goodness of fit, homogeneity of proportions, and independence. Use hypothesis-testing procedures to determine whether or not to reject the null hypothesis. The null hypothesis may address proportion, mean, difference between two proportions or two means, goodness of fit, homogeneity of proportions, and independence. Compare and contrast Type I and Type II errors. Explain how and why the hypothesis-testing procedure allows one to reach a statistical decision. 12 Supporting Materials Activity: Is yawning contagious? (TPS, pg. 602) Technology Corner: Confidence interval for a difference in proportions (TPS, pg. 611) Technology Corner: Significance test for a difference in proportions (TPS, pg. 618) Comments Students should be encouraged to know both the formulas and calculator functions for all inference procedures. AP Statistics Curriculum Guide 2011 PS.18 The student, given data from a large sample, will find and interpret point estimates and confidence intervals for parameters. The parameters will include proportion and mean, difference between two proportions, and difference between to means (independent and paired). Construct confidence intervals to estimate a population parameter, such as a proportion or the difference between two proportions; or a mean or the difference between two means. Interpret confidence intervals in the context of the data. PS.19 The student will apply and interpret the logic of a hypothesis-testing procedure. Tests will include large 10-2 Comparing Two Means sample test for proportion, mean, difference between two proportions, and difference between two means (independent and paired) and Chi-squared test for 2 Blocks goodness of fit, homogeneity of proportions, and independence. Use hypothesis-testing procedures to determine whether or not to reject the null hypothesis. The null hypothesis may address proportion, mean, difference between two proportions or two means, goodness of fit, homogeneity of proportions, and independence. Compare and contrast Type I and Type II errors. Explain how and why the hypothesis-testing procedure allows one to reach a statistical decision. PS.21 The student will identify properties of a t-distribution and apply t-distributions to single-sample and two-sample (independent and matched pairs) t-procedures, using tables or graphing calculators. Identify the properties of a t-distribution Compare and contrast a t-distribution and a normal distribution Use a t-test for single-sample and two-sample data. Chapter 10 Test 1 Block 13 Technology Corner: Two-sample Students should not be using the t intervals on the calculator (TPS, Pooled, two-sample tprocedures. pg. 636) Technology Corner: Two-sample t tests with computer software and calculators (TPS, pg. 642) Students should be encouraged to know both the formulas and calculator functions for all inference procedures. AP Statistics Curriculum Guide 2011 Chapter 11: Inference for Distributions of Categorical Data SOL with Essential Knowledge and Skill PS.19 The student will apply and interpret the logic of a hypothesis-testing procedure. Tests will include large sample test for proportion, mean, difference between two proportions, and difference between two means (independent and paired) and Chi-squared test for goodness of fit, homogeneity of proportions, and independence. Use the Chi-squared test for goodness of fit to decide if the population being analyzed fits a particular distribution pattern. Use hypothesis-testing procedures to determine whether or not to reject the null hypothesis. The null hypothesis may address proportion, mean, difference between two proportions or two means, goodness of fit, homogeneity of proportions, and independence. PS.19 The student will apply and interpret the logic of a hypothesis-testing procedure. Tests will include large sample test for proportion, mean, difference between two proportions, and difference between two means (independent and paired) and Chi-squared test for goodness of fit, homogeneity of proportions, and independence. Use the Chi-squared test for goodness of fit to decide if the population being analyzed fits a particular distribution pattern. Use hypothesis-testing procedures to determine whether or not to reject the null hypothesis. The null hypothesis may address proportion, mean, difference between two proportions or two means, goodness of fit, homogeneity of proportions, and independence. Textbook Chapters/Sections/Topics Supporting Materials 11-1 Chi-Square Goodnessof-Fit Tests Activity: The candy man can (TPS, pg. 676) Technology Corner: Finding Pvalues for chi-square tests on the calculator (TPS, pg. 683) Comments Students should be encouraged to know both the formulas and calculator functions for all inference procedures. 1.5 Blocks 11-2 Inference for Relationships 2.5 Blocks Chapter 11 Test 1 Block 14 Technology Corner: Chi-square goodness-of-fit test on the calculator (TPS, pg. 687) Technology Corner: Chi-square tests for two-way tables on the calculator (TPS, pg. 705) Students should be encouraged to know both the formulas and calculator functions for all inference procedures. AP Statistics Curriculum Guide 2011 Chapter 3: Describing Relationships SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics PS.4 The student will analyze scatterplots to identify and describe the relationship between two variables, using shape; strength of relationship; clusters; positive, negative, or no association; outliers; and influential points. Appropriate technology will be used to generate scatterplots and identify outliers and influential points. Examine scatterplots of data, and describe skewness, 3-1 Scatterplots and kurtosis, and correlation within the context of the data. Describe and explain any unusual features of the data, such Correlation as clusters, gaps, or outliers, within the context of the data. Identify influential data points (observations that have great effect on a line of best fit because of extreme x-values) and 2 Blocks describe the effect of the influential points. PS.5 The student will find and interpret linear correlation, use the method of least squares regression to model the linear relationship between two variables, and use the residual plots to assess linearity. Appropriate technology will be used to compute correlation coefficients and residual plots. Calculate a correlation coefficient Explain how the correlation coefficient, r, measures association by looking at its formula. 15 Supporting Materials Activity: CSI Stats: The case of the missing cookies (TPS, pg. 142) Activity: Correlation and regression applet (TPS, pg. 152) Technology Corner: Scatterplots on the calculator (TPS, pg. 149) Comments To calculate correlation on the calculator, access the calculator’s catalog and select Diagnostics On. Then run regression analysis. Students should be shown where the correlation formula can be found on the the AP Exam formula sheet. Students should understand that correlation does not imply causation. AP Statistics Curriculum Guide PS.5 The student will find and interpret linear correlation, use the method of least squares regression to model the linear relationship between two variables, and use the residual plots to assess linearity. Appropriate technology will be used to compute correlation coefficients and residual plots. Use regression lines to make predictions, and identify the limitations of the predictions Use residual plots to determine if a linear model is satisfactory for describing the relationship between two variables. Describe the errors inherent in extrapolation beyond the range of the data. Use least squares regression to find the equation of the line of best fit for a set of data. Explain how least squares regression generates the equation of the line of best fit by examining the formulas used in computation. 2011 3-2 Least-Squares Regression Activity: Investigating properties of the least-squares regression line (TPS, pg. 170) Students should be shown where the regression formulas can be found on the the AP Exam formula sheet. Technology Corner: Leastsquares regression lines on the calculator (TPS, pg. 170) Teachers will need to supplement the use of computer output so that students may gain exposure prior to the AP Exam. Technology Corner: Residual plots and s on the calculator (TPS, pg. 178) 3 Blocks Chapter 3 Test 1 Block 16 AP Statistics Curriculum Guide 2011 Mapping for Instruction - Fourth Nine Weeks Chapter 12: More about Regression SOL with Essential Knowledge and Skill PS.5 The student will find and interpret linear correlation, use the method of least squares regression to model the linear relationship between two variables, and use the residual plots to assess linearity. Appropriate technology will be used to compute correlation coefficients and residual plots. Use regression lines to make predictions, and identify the limitations of the predictions Use residual plots to determine if a linear model is satisfactory for describing the relationship between two variables. Describe the errors inherent in extrapolation beyond the range of the data. Use least squares regression to find the equation of the line of best fit for a set of data. Explain how least squares regression generates the equation of the line of best fit by examining the formulas used in computation. Textbook Chapters/Sections/Topics Supporting Materials Activity: The helicopter experiment (TPS, pg. 738) Technology Corner: Regression inference on the calculator (TPS, pg. 756) 12-1 Inference for Linear Regression 1.5 Blocks PS.6 The student will make logarithmic and power transformations to achieve linearity. Appropriate 12-2 Transforming to technology will be used. Achieve Linearity Apply a logarithmic transformation to the data. Explain how a logarithmic transformation works to achieve a linear relationship between variables. 2.5 Blocks Apply a power transformation to data. Explain how a power transformation works to achieve a linear relationship between variables. Chapter 12 Test 1 Block 17 Comments Teachers will need to supplement the use of computer output so that students may gain exposure prior to the AP Exam. It is not critical that students memorize verifications for confidence intervals and tests. Typically these verifications are given on the AP Exam. Technology Corner: Transforming This section may be covered concurrently with AP Exam to achieve linearity on the reviews. calculator (TPS, pg. 782) AP Statistics Curriculum Guide SOL with Essential Knowledge and Skill 2011 Textbook Chapters/Sections/Topics AP Review and Testing Supporting Materials 10 Blocks Final Project 5 Blocks Review and Final Exam 2 Blocks 18 Cumulative AP Practice Test (TPS, pg. 799) Inference Summary (TPS, back cover) AP Exam Tips (TPS,Appendix A) Comments It is strongly recommended that teachers give students a mockAP Exam in preparation for the actual exam. Consider using the latest set of released Free Response questions coupled with the 2007 released multiple choice questions. AP Statistics Curriculum Guide 2011 SOL Blueprints SOL Enhanced Scope and Sequence Supplemental Resources Links to other teacher’s websites http://statpages.org http://wiki.stat.ucla.edu/socr/index.php/EBook http://wiki.stat.ucla.edu/socr/index.php/SOCR_Data http://socr.ucla.edu/ www.herkimershideaway.org (Sanderson Smith) http://mrwaddell.net/apstats/teachers.html (provides links to other teacher’s sites) http://web.mac.com/statsmonkey/StatsMonkey/Statsmonkey.html (Jason Molesky) http://www.mastermathmentor.com/ (Stu Schwartz) answer keys provided via CD courtesy of Cave Spring High Math dept funds 19 AP Statistics Curriculum Guide 2011 http://home.htva.net/~bock/ (Dave Bock) http://www.bbn-school.org/us/math/ap_stats/ (links to applets by topic) http://bcs.whfreeman.com/yates2e/default.asp?s=&n=&i=&v=&o=&ns=0&t=&uid=0&rau=0 (the books site but only 2nd edition is available) http://it.stlawu.edu/~rlock/10sites.html (links to “top ten” stats websites) http://www.stat.duke.edu/sites/java.html (links to applets by topic) http://www.dartmouth.edu/~chance/ChanceLecture/AudioVideo.html (all about chance) Statistical data websites: http://www.census.gov/ http://cdc.gov/DataStatistics/ http://nces.ed.gov/ Professional organizations http://www.collegeboard.org http://www.amstat.org/ http://www.nationalmathandscience.org/programs Crossword Puzzles made into ACTIVE Flipcharts provided via CD. 20 AP Statistics Curriculum Guide 2011 SOL 2009 Framework Probability and Statistics 21 AP Statistics Curriculum Guide 2011 Copyright © 2009 by the Virginia Department of Education P.O. Box 2120 Richmond, Virginia 23218-2120 http://www.doe.virginia.gov All rights reserved. Reproduction of these materials for instructional purposes in public school classrooms in Virginia is permitted. Superintendent of Public Instruction Patricia I. Wright, Ed.D. Assistant Superintendent for Instruction Linda M. Wallinger, Ph.D. Office of Elementary Instruction Mark R. Allan, Ph.D., Director Deborah P. Wickham, Ph.D., Mathematics Specialist Office of Middle and High School Instruction Michael F. Bolling, Mathematics Coordinator Acknowledgements The Virginia Department of Education wishes to express sincere thanks to Deborah Kiger Bliss, Lois A. Williams, Ed.D., and Felicia Dyke, Ph.D. who assisted in the development of the 2009 Mathematics Standards of Learning Curriculum Framework. NOTICE The Virginia Department of Education does not unlawfully discriminate on the basis of race, color, sex, national origin, age, or disability in employment or in its educational programs or services. The 2009 Mathematics Curriculum Framework can be found in PDF and Microsoft Word file formats on the Virginia Department of Education’s Web site at http://www.doe.virginia.gov. 22 Virginia Mathematics Standards of Learning Curriculum Framework 2009 Introduction The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies the Mathematics Standards of Learning by defining the content knowledge, skills, and understandings that are measured by the Standards of Learning assessments. The Curriculum Framework provides additional guidance to school divisions and their teachers as they develop an instructional program appropriate for their students. It assists teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and describing the intellectual skills students need to use. This supplemental framework delineates in greater specificity the content that all teachers should teach and all students should learn. Each topic in the Mathematics Standards of Learning Curriculum Framework is developed around the Standards of Learning. The format of the Curriculum Framework facilitates teacher planning by identifying the key concepts, knowledge and skills that should be the focus of instruction for each standard. The Curriculum Framework is divided into two columns: Essential Understandings and Essential Knowledge and Skills. The purpose of each column is explained below. Essential Understandings This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the Standards of Learning. Essential Knowledge and Skills Each standard is expanded in the Essential Knowledge and Skills column. What each student should know and be able to do in each standard is outlined. This is not meant to be an exhaustive list nor a list that limits what is taught in the classroom. It is meant to be the key knowledge and skills that define the standard. The Curriculum Framework serves as a guide for Standards of Learning assessment development. Assessment items may not and should not be a verbatim reflection of the information presented in the Curriculum Framework. Students are expected to continue to apply knowledge and skills from Standards of Learning presented in previous grades as they build mathematical expertise. 23 TOPIC: DESCRIPTIVE STATISTICS PROBABILITY AND STATISTICS STANDARD PS.1 The student will analyze graphical displays of univariate data, including dotplots, stemplots, and histograms, to identify and describe patterns and departures from patterns, using central tendency, spread, clusters, gaps, and outliers. Appropriate technology will be used to create graphical displays. ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS Data are collected for a purpose and have meaning in a context. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Measures of central tendency describe how the data cluster or group. Create and interpret graphical displays of data, including dotplots, stem-and-leaf plots, and histograms. Measures of dispersion describe how the data spread (disperse) around the center of the data. Examine graphs of data for clusters and gaps, and relate those phenomena to the data in context. Graphical displays of data may be analyzed informally. Data analysis must take place within the context of the problem. Examine graphs of data for outliers, and explain the outlier(s) within the context of the data. Examine graphs of data and identify the central tendency of the data as well as the spread. Explain the central tendency and the spread of the data within the context of the data. 24 TOPIC: DESCRIPTIVE STATISTICS PROBABILITY AND STATISTICS STANDARD PS.2 The student will analyze numerical characteristics of univariate data sets to describe patterns and departure from patterns, using mean, median, mode, variance, standard deviation, interquartile range, range, and outliers. ESSENTIAL UNDERSTANDINGS Data are collected for a purpose and have meaning within a context. Analysis of the descriptive statistical information generated by a univariate data set should include the interplay between central tendency and dispersion as well as among specific measures. Data points identified algorithmically as outliers should not be excluded from the data unless sufficient evidence exists to show them to be in error. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to 25 Interpret mean, median, mode, range, interquartile range, variance, and standard deviation of a univariate data set in terms of the problem’s context. Identify possible outliers, using an algorithm. Explain the influence of outliers on a univariate data set. Explain ways in which standard deviation addresses dispersion by examining the formula for standard deviation. TOPIC: DESCRIPTIVE STATISTICS PROBABILITY AND STATISTICS STANDARD PS.3 The student will compare distributions of two or more univariate data sets, analyzing center and spread (within group and between group variations), clusters and gaps, shapes, outliers, or other unusual features. ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS Data are collected for a purpose and have meaning in a context. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Statistical tendency refers to typical cases but not necessarily to individual cases. 26 Compare and contrast two or more univariate data sets by analyzing measures of center and spread within a contextual framework. Describe any unusual features of the data, such as clusters, gaps, or outliers, within the context of the data. Analyze in context kurtosis and skewness in conjunction with other descriptive measures. TOPIC: DESCRIPTIVE STATISTICS PROBABILITY AND STATISTICS STANDARD PS.4 The student will analyze scatterplots to identify and describe the relationship between two variables, using shape; strength of relationship; clusters; positive, negative, or no association; outliers; and influential points. ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to A scatterplot serves two purposes: – to determine if there is a useful relationship between two variables, and – to determine the family of equations that describes the relationship. Data are collected for a purpose and have meaning in a context. Association between two variables considers both the direction and strength of the association. The strength of an association between two variables reflects how accurately the value of one variable can be predicted based on the value of the other variable. Outliers are observations with large residuals and do not follow the pattern apparent in the other data points. 27 Examine scatterplots of data, and describe skewness, kurtosis, and correlation within the context of the data. Describe and explain any unusual features of the data, such as clusters, gaps, or outliers, within the context of the data. Identify influential data points (observations that have great effect on a line of best fit because of extreme x-values) and describe the effect of the influential points. TOPIC: DESCRIPTIVE STATISTICS PROBABILITY AND STATISTICS STANDARD PS.5 The student will find and interpret linear correlation, use the method of least squares regression to model the linear relationship between two variables, and use the residual plots to assess linearity. ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS Data are collected for a purpose and have meaning in a context. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Least squares regression generates the equation of the line that minimizes the sum of the squared distances from the data points to the line. Calculate a correlation coefficient. Each data point may be considered to be comprised of two parts: fit (the part explained by the model) and residual (the result of chance variation or of variables not measured). Explain how the correlation coefficient, r, measures association by looking at its formula. Use regression lines to make predictions, and identify the limitations of the predictions. Residual = Actual – Fitted A correlation coefficient measures the degree of association between two variables that are related linearly. Use residual plots to determine if a linear model is satisfactory for describing the relationship between two variables. Describe the errors inherent in extrapolation beyond the range of the data. Use least squares regression to find the equation of the line of best fit for a set of data. Explain how least squares regression generates the equation of the line of best fit by examining the formulas used in computation. Two variables may be strongly associated without a cause-andeffect relationship existing between them. 28 TOPIC: DESCRIPTIVE STATISTICS PROBABILITY AND STATISTICS STANDARD PS.6 The student will make logarithmic and power transformations to achieve linearity. ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to A logarithmic transformation reduces positive skewness because it compresses the upper tail of the distribution while stretching the lower tail. Nonlinear transformations do not preserve relative spacing between data points. 29 Apply a logarithmic transformation to data. Explain how a logarithmic transformation works to achieve a linear relationship between variables. Apply a power transformation to data. Explain how a power transformation works to achieve a linear relationship between variables. TOPIC: DESCRIPTIVE STATISTICS PROBABILITY AND STATISTICS STANDARD PS.7 The student, using two-way tables, will analyze categorical data to describe patterns and departure from patterns and to find marginal frequency and relative frequencies, including conditional frequencies. ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS Data are collected for a purpose and have meaning in a context. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Simpson’s paradox refers to the fact that aggregate proportions can reverse the direction of the relationship seen in the individual parts. Two categorical variables are independent if the conditional frequencies of one variable are the same for every category of the other variable. 30 Produce a two-way table as a summary of the information obtained from two categorical variables. Calculate marginal, relative, and conditional frequencies in a two-way table. Use marginal, relative, and conditional frequencies to analyze data in two-way tables within the context of the data. TOPIC: DATA COLLECTION PROBABILITY AND STATISTICS STANDARD PS.8 The student will describe the methods of data collection in a census, sample survey, experiment, and observational study and identify an appropriate method of solution for a given problem setting. ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to The value of a sample statistic varies from sample to sample if the simple random samples are taken repeatedly from the population of interest. Poor data collection can lead to misleading and meaningless conclusions. 31 Compare and contrast controlled experiments and observational studies and the conclusions one can draw from each. Compare and contrast population and sample and parameter and statistic. Identify biased sampling methods. Describe simple random sampling. Select a data collection method appropriate for a given context. TOPIC: DATA COLLECTION PROBABILITY AND STATISTICS STANDARD PS.9 The student will plan and conduct a survey. The plan will address sampling techniques (e.g., simple random and stratified) and methods to reduce bias. ESSENTIAL UNDERSTANDINGS The purpose of sampling is to provide sufficient information so that population characteristics may be inferred. Inherent bias diminishes as sample size increases. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to 32 Investigate and describe sampling techniques, such as simple random sampling, stratified sampling, and cluster sampling. Determine which sampling technique is best, given a particular context. Plan a survey to answer a question or address an issue. Given a plan for a survey, identify possible sources of bias, and describe ways to reduce bias. Design a survey instrument. Conduct a survey. TOPIC: DATA COLLECTION PROBABILITY AND STATISTICS STANDARD PS.10 The student will plan and conduct an experiment. The plan will address control, randomization, and measurement of experimental error. ESSENTIAL UNDERSTANDINGS Experiments must be carefully designed in order to detect a cause-and-effect relationship between variables. Principles of experimental design include comparison with a control group, randomization, and blindness. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to 33 Plan and conduct an experiment. The experimental design should address control, randomization, and minimization of experimental error. TOPIC: PROBABILITY PROBABILITY AND STATISTICS STANDARD PS.11 The student will identify and describe two or more events as complementary, dependent, independent, and/or mutually exclusive. ESSENTIAL UNDERSTANDINGS The complement of event A consists of all outcomes in which event A does not occur. Two events, A and B, are independent if the occurrence of one does not affect the probability of the occurrence of the other. If A and B are not independent, then they are said to be dependent. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Events A and B are mutually exclusive if they cannot occur simultaneously. 34 Define and give contextual examples of complementary, dependent, independent, and mutually exclusive events. Given two or more events in a problem setting, determine if the events are complementary, dependent, independent, and/or mutually exclusive. TOPIC: PROBABILITY PROBABILITY AND STATISTICS STANDARD PS.12 The student will find probabilities (relative frequency and theoretical), including conditional probabilities for events that are either dependent or independent, by applying the Law of Large Numbers concept, the addition rule, and the multiplication rule. ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS Data are collected for a purpose and have meaning in a context. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Venn diagrams may be used to find conditional probabilities. Calculate relative frequency and expected frequency. The Law of Large Numbers states that as a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability. Find conditional probabilities for dependent, independent, and mutually exclusive events. 35 TOPIC: PROBABILITY PROBABILITY AND STATISTICS STANDARD PS.13 The student will develop, interpret, and apply the binomial probability distribution for discrete random variables, including computing the mean and standard deviation for the binomial variable. ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to A probability distribution is a complete listing of all possible outcomes of an experiment together with their probabilities. The procedure has a fixed number of independent trials. A random variable assumes different values depending on the event outcome. A probability distribution combines descriptive statistical techniques and probabilities to form a theoretical model of behavior. 36 Develop the binomial probability distribution within a realworld context. Calculate the mean and standard deviation for the binomial variable. Use the binomial distribution to calculate probabilities associated with experiments for which there are only two possible outcomes. TOPIC: PROBABILITY PROBABILITY AND STATISTICS STANDARD PS.14 The student will simulate probability distributions, including binomial and geometric. ESSENTIAL UNDERSTANDINGS A probability distribution combines descriptive methods and probabilities to form a theoretical model of behavior. A probability distribution gives the probability for each value of the random variable. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to 37 Design and conduct an experiment that simulates a binomial distribution. Design and conduct an experiment that simulates a geometric distribution. TOPIC: PROBABILITY PROBABILITY AND STATISTICS STANDARD PS.15 The student will identify random variables as independent or dependent and find the mean and standard deviations for sums and differences of independent random variables. ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to A random variable is a variable that has a single numerical value, determined by chance, for each outcome of a procedure. 38 Compare and contrast independent and dependent random variables. Find the standard deviation for sums and differences of independent random variables. TOPIC: PROBABILITY PROBABILITY AND STATISTICS STANDARD PS.16 The student will identify properties of a normal distribution and apply the normal distribution to determine probabilities, using a table or graphing calculator. ESSENTIAL UNDERSTANDINGS The normal distribution curve is a family of symmetrical curves defined by the mean and the standard deviation. Areas under the curve represent probabilities associated with continuous distributions. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to The normal curve is a probability distribution and the total area under the curve is 1. 39 Identify the properties of a normal probability distribution. Describe how the standard deviation and the mean affect the graph of the normal distribution. Determine the probability of a given event, using the normal distribution. TOPIC: INFERENTIAL STATISTICS PROBABILITY AND STATISTICS STANDARD PS.17 The student, given data from a large sample, will find and interpret point estimates and confidence intervals for parameters. The parameters will include proportion and mean, difference between two proportions, and difference between two means (independent and paired). ESSENTIAL UNDERSTANDINGS A primary goal of sampling is to estimate the value of a parameter based on a statistic. Confidence intervals use the sample statistic to construct an interval of values that one can be reasonably certain contains the true (unknown) parameter. Confidence intervals and tests of significance are complementary procedures. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Paired comparisons experimental design allows control for possible effects of extraneous variables. 40 Construct confidence intervals to estimate a population parameter, such as a proportion or the difference between two proportions; or a mean or the difference between two means. Select a value for alpha (Type I error) for a confidence interval. Interpret confidence intervals in the context of the data. Explain the importance of random sampling for confidence intervals. Calculate point estimates for parameters and discuss the limitations of point estimates. TOPIC: INFERENTIAL STATISTICS PROBABILITY AND STATISTICS STANDARD PS.18 The student will apply and interpret the logic of a hypothesis-testing procedure. Tests will include large sample test for proportion, mean, difference between two proportions, and difference between two means (independent and paired) and Chi-squared tests for goodness of fit, homogeneity of proportions, and independence. ESSENTIAL UNDERSTANDINGS Confidence intervals and tests of significance are complementary procedures. Paired comparisons experimental design allows control for possible effects of extraneous variables. Tests of significance assess the extent to which sample data support a hypothesis about a population parameter. The purpose of a goodness of fit test is to decide if the sample results are consistent with results that would have been obtained if a random sample had been selected from a population with a known distribution. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Practical significance and statistical significance are not necessarily congruent. 41 Use the Chi-squared test for goodness of fit to decide if the population being analyzed fits a particular distribution pattern. Use hypothesis-testing procedures to determine whether or not to reject the null hypothesis. The null hypothesis may address proportion, mean, difference between two proportions or two means, goodness of fit, homogeneity of proportions, and independence. Compare and contrast Type I and Type II errors. Explain how and why the hypothesis-testing procedure allows one to reach a statistical decision. TOPIC: INFERENTIAL STATISTICS PROBABILITY AND STATISTICS STANDARD PS.19 The student will identify the meaning of sampling distribution with reference to random variable, sampling statistic, and parameter and explain the Central Limit Theorem. This will include sampling distribution of a sample proportion, a sample mean, a difference between two sample proportions, and a difference between two sample means. ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to The Central Limit Theorem states: – The mean of the sampling distribution of means is equal to the population mean. – If the sample size is sufficiently large, the sampling distribution approximates the normal probability distribution. – If the population is normally distributed, the sampling distribution is normal regardless of sample size. Sampling distributions have less variability with larger sample sizes. 42 Describe the use of the Central Limit Theorem for drawing inferences about a population parameter based on a sample statistic. Describe the effect of sample size on the sampling distribution and on related probabilities. Use the normal approximation to calculate probabilities of sample statistics falling within a given interval. Identify and describe the characteristics of a sampling distribution of a sample proportion, mean, difference between two sample proportions, or difference between two sample means. TOPIC: INFERENTIAL STATISTICS PROBABILITY AND STATISTICS STANDARD PS.20 The student will identify properties of a t-distribution and apply t-distributions to single-sample and two-sample (independent and matched pairs) t-procedures, using tables or graphing calculators. ESSENTIAL UNDERSTANDINGS Paired comparisons experimental design allows control for possible effects of extraneous variables. The sampling distribution of means with a small sample size follows a t-distribution. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to 43 Identify the properties of a t-distribution. Compare and contrast a t-distribution and a normal distribution. Use a t-test for single-sample and two-sample data.
