AP Statistics Syllabus Course Information
Text: Coyner, Scott, et al. Statistics. CPM Education Program, 2017
Course Summary
Statistics is the study of data – how to gather it, analyze it, interpret, and draw conclusions from it. In the first semester of AP Statistics, students will learn statistical techniques to analyze and interpret data, use probability and two-way tables to explore the ideas of independence and likelihood, learn to design studies and experiments that minimize bias and variability, and build the concept of a probability distribution to explore more complex ideas in probability that arise in data analysis. In the second semester, students will use probability distributions to build the idea of a sampling distribution as a measure of bias and variability in an estimate, then use sampling distributions to motivate an understanding of statistical inference – confidence intervals and hypothesis tests – including inference for proportions, means, categorical distributions using chi-squared distributions, and slopes of linear regression equations.
Assignments and Projects
Students will work together in teams nearly every day to work through complex problems that motivate understanding, a technique called problem-based learning. By working through complex problems together, students can support each other in tricky conceptual ideas and check each other’s work, with the teacher serving as a support and guide as necessary. Communication and cooperation will be vital for success in the classroom! Homework and assessments are spiraled, meaning they contain material from earlier lessons and chapters as well as the most recent one, to help maintain comfort with earlier chapters and prepare for the AP exam. In addition to homework and problems from the text, at least one AP Free Response problems from a previous year will be completed and scored in class and at home each chapter. Finally, one to two projects will be completed each semester: see the course outline for more details.
Technology TI-84+ or equivalent graphing calculators will be used throughout the course in class and at home. In addition, computers, tablets, or mobile devices can be used to access online applets at http://stats.cpm.org and http://www.desmos.com to support instruction. Students will also use tools such as Google Apps for Education or Microsoft Office to help analyze and present data. An online problem generator will also be used to generate additional random practice problems for students to work and aid in studying.
Course Outline
Chapters 1-6 should be completed in the first semester and Chapters 7-12 in the second (before the AP exam). If time remains in the year after the AP exam, additional projects or units will be explored. Lessons are completed at a rate of approximately 1 per standard length class day.
Chapter 1 – Representing Data
In this chapter students learn how to display, analyze, and compare categorical and quantitative data using graphical displays and summary statistics. Calculators and online tools are used when appropriate. Lesson AP Topics 1.1.1
I.A, I.E.1, I.E.4
1.1.2
I.A.1 – I.A.4, I.E.1
1.1.3
I.E.1 – I.E.3, III.B.1
1.2.1
I.A.1 – I.A.4, I.B.1
1.2.2
I.A.1, I.B.1, I.B.2, I.B.4
1.2.3
I.A.1, I.B.1, I.B.2, I.B.4, III.D.2
1.2.4
I.B.1, I.B.2, I.B.4, I.C.1 – I.C.4
1.3.1
I.B.1 – I.B.4
1.3.2
I.B.1 – I.B.4
1.3.3
I.B.3, I.B.5
Lesson Summary
Visualizing Information – Students collect data and look at the uses, strengths, and weaknesses of bar graphs, dot plots, two-way tables, Venn diagrams, and scatterplots. Quantitative and Categorical displays – Students explore certain chart types more thoroughly, including histograms, stem-and-leaf plots, bar charts, and Venn diagrams. Types of Data and Variables – Students explore data displays and practice choosing a display type to use. They are introduced to the idea of associated variables. Choosing Mean or Median – Students use an online tool to compare datasets visually with histograms and boxplots and are introduced to the summary statistics of mean, median, range and IQR. Mean vs. median advantages are discussed. Variance and Standard Deviation – Students are introduced to the mean absolute deviation and the standard deviation as measures of spread. Graphing calculators are used to calculate those values and compare with the IQR and range. Sample Variance and Sample Standard Deviation – Students explore the sample variance and sample standard deviation as better estimates of population spread when data is a sample. This is done through exhaustive calculation. Investigating Data Representations – Students model a golf tournament. Data is gathered by tossing pennies, then analyzed thoroughly. Percentiles – Students explore percentiles as measures of position in a distribution. They interpret ogives (cumulative relative frequency graphs) and compare to histograms. Z-Scores – students learn to standardize data to create z-scores and compare values in different data sets. Linear Transformations – Students apply linear transformations to quantitative data sets and observe the effects on summary statistics and graphs.
Chapter 2 – Two-variable Quantitative Data
Students represent paired quantitative data using scatterplots and linear regression models. Graphing calculators, Desmos activities, and the Bivariate Explorer eTool at http://stats.cpm.org are used throughout. Lesson AP Topics
Lesson Summary Scatterplots and Association – Students review scatterplots and 2.1.1 I.D.1 discuss strength, form, and direction of an association. Line of Best Fit – Students create lines of best fit by hand and 2.1.2 I.D.1 interpret the slope and y-intercept of the lines in context. Residuals – Students discuss the idea of a residual and interpret them 2.1.3 I.D.1, I.D.4 in context. The Least Squares Regression Line – Students uncover the idea of the least squares regression line by starting with the idea of minimizing 2.1.4 I.D.1, I.D.3 the standard deviation of the residuals. They use a Desmos eTool to explore the concept visually. Using Technology to Find the LSRL – Students use graphing calculators and eTool to make scatterplots and find regression lines. 2.1.5 I.D.1, I.D.3 They learn to read the slope and y-intercept from Minitab-style computer regression output. The Correlation Coefficient – Students explore the rise of the correlation coefficient as the slope of the standardized data sets as well 2.2.1 I.D.1 – I.D.3 as by its definition. They then use an eTool to explore how it connects to strength of an association. Behavior of Correlation and the LSRL – Students explore the relationship between the correlation coefficient and the least squares 2.2.2 I.D.2, I.D.3 regression line, and use that to motivate thinking about how the correlation is affected by various data changes. Residual Plots – Students explore the idea of a residual plot as a way 2.2.3 I.D.1 – I.D.4 of deciding if an association is linear in form. Calculators and an eTool are used. Association is Not Causation – Students use scatterplots to realize 2.2.4 II.D that association does not imply causation. Interpreting Correlation in Context – Students learn the 2.2.5 I.D.1 – I.D.4 interpretation of correlation in context through the derivation of the coefficient of determination, R2 . Chapter 1 and 2 Data Collection Project Students find one or more data sets based on a topic of interest to them and write a 2 to 4 page report that includes three charts – one categorical (e.g. bar chart), one univariate quantitative (e.g. histogram), and one scatterplot – along with descriptions of and and connections between the displays. Reports are assessed on the statistical quality of the charts, appropriateness of the data, correct vocabulary, and depth of connection and analysis.
Chapter 3 – Multivariable Categorical Data
Students use two-way tables, Venn diagrams, and tree diagrams to explore relationships when they have multiple categorical variables. From there, students explore probability, including conditional probability and the idea of independence or association of events. Finally students use simulation to model complex events as well as begin to explore the idea of variation in sampling and how it can be controlled through sample size. Lesson AP Topics
3.1.1
I.E.1, I.E.2, III.A.1 – III.A.3
3.1.2
I.E.1 – I.E.3, III.A.1 – III.A.3
3.1.3
I.E.1 – I.E.3, III.A.1 – III.A.3
3.1.4
I.E.1–I.E.3
3.1.5
I.E.1 – I.E.3, III.A.1 – III.A.3
3.2.1
III.A.1 – III.A.3, III.A.5
3.2.2
III.A.1 – III.A.3, III.A.5
3.2.3
III.A.1 – III.A.5
Lesson Summary Probabilities and Two-Way Frequency Tables – Students build and compute probabilities with Venn diagrams and frequency tables. Probability terms such as sample space, outcome, and event are introduced. Association and Conditional Relative Frequency Tables – Student are introduced to conditional relative frequency tables and use them to decide if two events are independent or associated. Probability Notation – Students are introduced to probability notation and develop the general addition formula and Bayes’ theorem for conditional probabilities. Relative Frequency Tables and Conditional Probabilities – Students use tree diagrams to go back and forth between conditional probabilities, frequency tables, relative frequency tables, and conditional relative frequency tables. Analyzing False Positives – Students explore the base rate fallacy; counterintuitive situations where “false positives” occur frequently compared to true positives. Probability Trees – Students explore using probability trees to explore situations with more independent variables than can be shown on a two-way table. Problem Solving with Categorical Data – Students put together everything they have learned about probability to answer a variety of questions. Simulations of Probability – Students use coins and the random number feature of graphing calculators to perform simulations of difficult-to-calculate probabilities.
Chapter 4 – Studies and Experiments
Students explore survey design, potential sources of bias, and experimental design. At several points in the chapter students come up with their own survey questions to ask and are tasked with collecting and analyzing their own data. Lesson AP Topics 4.1.1
4.1.2
4.1.3
4.1.4
4.1.5 (opt.)
4.2.1
4.2.2
4.2.3
4.2.4 (opt.)
Lesson Summary Survey Design I – Students explore survey questions and the ways in which they might introduce response bias. As teams, they come II.B.3 up with a first draft of a research question and survey questions they would like answered. Samples and the Role of Randomness – Students explore the importance of randomness in limiting bias and discuss how II.B.2 – non-random sampling systems can result in bias. Students decide if II.B.4 a simple random sample is a reasonable choice for the study they designed in the last lesson. Sampling When an SRS is Not Possible – Students explore II.B.3, II.B.4 alternatives to a simple random sample, including cluster samples, stratified samples, and systematic samples. Observational Studies and Experiments – Students are II.A.1 – introduced to the term “experiment” as they compare and contrast II.A.4, II.C.3, various study designs and analyze what conclusions are reasonable II.D from each. Survey Design II – Students present the results of the survey they designed in lesson 4.1.1 and 4.1.2. Other students will follow a II.B.1 – rubric on good survey design to critique their classmates. II.B.4 (Depending on timing, this lesson may be moved later in the chapter or semester) Cause and Effect with Experiments – Students learn the II.C.1, II.C.2, important features of an experiment – control, randomization, and II.C.4 replication – and see how that provides evidence of a causal relationship between variables. Experimental Design I – Students investigate sources of confounding and learn how to control confounding variables, I.E.4, II.C.3, including through the use of blocking. A simulation eTool is used II.C.5, II.D to quickly generate various possible outcomes of an experiment with and without blocking for the sake of comparison. Experimental Design II – Students use what they’ve learned II.C.3, II.D about experimental design to critique and improve an experiment testing reaction time. Experimental Design III – Students conduct a blind experiment in II.C.1–II.C.4, class to determine whether they can taste the difference between II.D bottle and tap water. The design of the experiment is critiqued and discussed at each stage.
Chapter 4 Survey Project
Partially explained in lessons 4.1.1, 4.1.2, and 4.1.5 above. In this project, small groups of students decide on a research question with two or three associated survey questions, perform the survey using good survey design, and analyze the results, including through the use of appropriate charts. The results are then presented to the class. Presentations are assessed on the quality of the questions and survey design with regards to bias, the appropriateness of charts, the quality of analysis, and the clarity of the presentation.
Chapter 5 – Probability Density Functions, including Normal
Students are introduced to the idea of a random variable and a density curve as a representation of a continuous random variable. The normal probability curve is explored in great depth. Lesson AP Topics Lesson Summary Relative Frequency Histograms and Random Variables – Students review relative frequency histograms, 5.1.1 I.A, III including creating them on calculators and with online tools, and use them to answer questions. The concept of a random variable is introduced. Introduction to Density Functions – By looking at density histograms – relative frequency histograms with a bin width of 1 5.1.2 I.B.1 – students make the connection between area and probability. This is used to introduce the concept of a density function, beginning with the uniform density function. The Normal Probability Density Function – Students are introduced to the normal probability density function and 5.1.3 III.C.1, III.C.3 cumulative density function, the empirical rule, and how to find cumulative densities using calculators. The Inverse Normal Function – Students use their calculators 5.2.1 III.C.1, III.C.3 to work back and forth between positions on a normal curve and normal probabilities. The Standard Normal Distribution and z-Scores – Students review the concept of z-scores and use it to build the standard 5.2.2 III.C.1 – III.C.3 normal curve, which is then used to compare normal curves from different scenarios. 5.2.3 III.C.1 – III.C.3 Additional Practice Problems
Chapter 6 – Discrete Probability Distributions
Students are introduced to discrete random variables and probability distributions, spend time exploring linear transformations and combinations of random variables, and then explore the important examples of binomial and geometric distributions. Graphing calculators are used extensively for calculations of binomial and geometric probabilities. Lesson AP Topics 6.1.1
6.1.2
6.1.3
6.2.1 6.2.2 6.2.3
6.2.4
6.2.5
6.3.1 6.3.2
Lesson Summary Mean and Variance of a Discrete Random Variable – Students III.A.4, III.A.6 are introduced to discrete random variables and learn to calculate the expected value and variance of a discrete random variable. Linear Combinations of Independent Random Variables – Students explore what happens to the mean, variance, and III.B.1, III.B.2 standard deviation of random variables under linear transformations and combinations. Exploring the Variability of X – X – In this lesson, students address head-on the common confusion about why X – X is not III.A.5, III.B.2 simply zero in the context of random variables. Solidifies understanding of random variables vs. standard algebraic variables. Introducing the Binomial Setting – Students are introduced to III.A.4 the binomial setting in the context of guessing on a multiple-choice test. Binomial Probability Density Function – Students derive the III.A.4 probability density function for binomial situations. Exploring Binomial pdf and cdf – Students are introduced to III.A.4 calculator functions for the binomial probability and cumulative density functions and use them to solve problems. Shape, Center, and Spread of the Binomial Distribution – Students re-examine the binomial distribution as a III.A.4, III.A.6 complete distribution and explore its center (expected value) and spread. Normal Approximation to the Binomial Distribution – III.A.4, III.C Students explore under what conditions the binomial distribution can be reasonably approximated by the normal distribution. Introduction to the Geometric Distribution – Students are III.A.4, III.A.6 introduced to the geometric probability distribution and compare and contrast it with the binomial distribution. III.A.4 Binomial and Geometric Practice
Chapter 7 – Categorical Sampling
Students are introduced to the concept of a sampling distribution as a way of measuring variability of a statistic between samples. After an initial exploration, the chapter focuses on sampling distributions of proportions and uses it to motivate the concept of a confidence interval for a population proportion. Simulation technology is used extensively in this chapter. Lesson AP Topics Lesson Summary III.A.5, III.D.1, Introduction to Sampling Distributions – Students create 100 7.1.1 III.D.6, different samples of chocolate candies to explore the idea of IV.A.1 – IV.A.3 sampling variability and a sampling distribution. III.A.5, III.D.1, Simulating Sampling Distributions of Sample Proportions – 7.1.2 III.D.6, Students use an online tool to simulate sample proportions and IV.A.1 – IV.A.3 get a sense of how proportions vary from sample to sample Formulas for the Sampling Distributions of Sample III.A.4 – III.A.6, Proportions – Students connect the idea of a sampling 7.1.3 III.D.1, III.D.6, distribution for proportions to a binomial distribution and derive IV.A.1, IV.A.2, formulas that govern the center and spread of the distribution. Confidence Interval for a Population Proportion – Students III.D.1, III.D.6, are introduced to the idea of a confidence interval and use their 7.2.1 IV.A.1 – IV.A.3 formulas to find 95% confidence intervals for a population proportion. Confidence Levels for Confidence Intervals – Students learn III.D.1, III.D.6, to use the normal curve to create confidence intervals with 7.2.2 IV.A.1 – IV.A.4 different confidence levels. They explore the full meaning of a confidence interval. Changing the Margin of Error in Confidence Intervals – Students learn to interpret confidence intervals and 7.2.3 IV.A.1 – IV.A.4 confidence levels using an eTool, and explore how changing confidence level and sample size can affect the margin of error. Evaluating Claims with Confidence Intervals – Students IV.A.1 – IV.A.4, 7.2.4 explore how confidence intervals can help evaluate claims about IV.B.1, IV.B.2 proportions. Simulation Project – Chapters 7 and 8
In this project students are tasked with exploring and documenting what happens if the “Large Population Condition”, also known as the 10% condition, is NOT met in situations involving proportion inference. To do this, students are allowed to use two eTools – one that simulates sampling distributions for proportions and one that simulates confidence intervals – from populations of any size. Students turn in a paper or presentation documenting their findings, with appropriate graphical displays and vocabulary, and try to come up with a way to correct the problem.
Chapter 8 – Hypothesis testing with proportions
Students are introduced to the big ideas of hypothesis tests and explore the concept through the lens of sample proportions and the difference between two independent proportions. Lesson 8.1.1
AP Topics III.A.5, III.D.1, III.D.6, IV.B.1, IV.B.2
8.1.2
IV.B.1, IV.B.2
8.1.3
IV.B.1, IV.B.2
8.2.1
IV.B.1, IV.B.2
8.2.2
IV.B.1, IV.B.2
8.3.1
III.D.4, IV.A.4, IV.A.5, IV.B.1
8.3.2
IV.B.1, IV.B.2, IV.B.3
8.3.3
IV.A.4, IV.A.5, IV.B.1
Lesson Summary
Introduction to Hypothesis Testing – Students are introduced to the big ideas of a hypothesis test through a simulation exercise. Hypothesis Tests for Proportions – Students work through a one-tailed proportion for a hypothesis test. Alternative Hypotheses and Two-Tailed Tests – Students learn to decide which type of hypothesis to write in one-tailed situations and explore two-tailed situations. Types of Errors in Hypothesis Testing – Students are exposed to the vocabulary for errors in hypothesis tests. Power of a Test – Students are guided through the calculation of power for a hypothesis test in proportions then use an eTool to explore ways to affect power. The Difference Between Two Proportions – Students derive formulas for the sampling distribution of the difference between two proportions, and create confidence intervals for the difference. Two-Sample Proportion Hypothesis Tests – Students build on the formulas for the difference between two proportions to do hypothesis tests for the difference between two proportions. More Proportion Inference – Practice
Chapter 9 – Chi-squared procedures
Students are introduced to the concepts of a chi-squared distribution by first calculating and combining several independent proportion tests into one test. The three types of chi-squared tests are then explored and practiced. Lesson
AP Topics
9.1.1
III.D.8, IV.B.3, IV.B.6
9.1.2
IV.B.6
9.1.3
IV.B.6
9.2.1
IV.B.6
Lesson Summary Introduction to the Chi-Squared Distribution – Students are motivated to explore the chi-squared distribution in order to compare three different proportions and decide if any one is different from the others. Chi-Squared Goodness of Fit – An example about Benford’s Law is used to motivate the chi-squared goodness of fit procedures. More Applications of Chi-Squared Goodness of Fit Chi-Squared Test for Independence – The chi-squared test for independence is introduced.
9.2.2
IV.B.6
9.2.3
IV.B.6
Chi-Squared Test for Homogeneity of Proportions – Students compare and contrast the chi-squared tests for homogeneity and independence. Practicing and Recognizing Chi-Squared Inference Procedures
Chapter 10 – Quantitative sampling / 1-sample mean inference
This chapter begins with heavy use of simulations to explore the sampling distributions of several quantitative variables, including both measures of center and spread. Focus then moves to inference on the mean and the derivation of procedures to find confidence intervals and perform hypothesis tests for a single mean. Lesson
AP Topics
10.1.1
III.A.5, III.D.2, III.D.6, IV.A.1 – IV.A.3
10.1.2
III.A.5, III.D.2, III.D.6, IV.A.1 – IV.A.3
10.2.1
III.A.5, III.D.2, III.D.3, III.D.6, IV.A.1 – IV.A.3
10.2.2
III.A.5, III.D.2, III.D.6, IV.A.1, IV.A.6
10.3.1
III.A.5, III.D.2, III.D.6, III.D.7, IV.A.1 – IV.A.3
10.3.2
III.D.2, III.D.7, IV.A.6
10.3.3
III.D.2, III.D.7, IV.B.4
Lesson Summary Quantitative Sampling Distributions – Students use an eTool to explore the sampling distributions of means and medians for a small sample in an interesting distribution. Bias and variability of estimators are defined. More Sampling Distributions –More simulation is used to explore sampling distributions of other statistics, including range, variance, and standard deviation, confirming that the sample variance is an unbiased estimator for the population variance. The Central Limit Theorem – Students use simulation to explore the conditions under which the sampling distribution for a mean is approximately normal. A formula is derived for the standard deviation of the sampling distribution of a sample mean. Using the Normal Distribution with Means – Students calculate probabilities and confidence intervals for sample means with a known population standard deviation. Introducing the t-Distribution – With more simulations, students explore the sampling distribution of the t-statistic and recognize how using it can help create confidence intervals that work more precisely. Calculating Confidence Intervals for μ – Students use the t-distribution and their calculators to create confidence intervals for a population mean. z-Tests and t-Tests for the Population Means – Students perform hypothesis tests for a single population mean.
Chapter 11 – Comparing means
In this chapter, students compare and contrast paired and independent data situations from surveys and experiments, and learn to perform inference procedures with means for both paired and independent data. Lesson AP Topics Lesson Summary Paired and Independent Data from Surveys and Experiments II.C.1, II.C.5, – Repeating the reaction time experiment from chapter 4, 11.1.1 IV.B.4, IV.B.5 students explore the value of matched pairs procedures vs. independence procedures, and analyze when they are possible. II.C.1, II.C.5, Paired Inference Procedures – Building on knowledge from 11.1.2 IV.A.6, IV.A.7, chapter 10, students build confidence intervals and perform tests IV.B.5 on the means of paired differences. Tests for the Difference of Two Means – Students derive a III.D.5, IV.B.1, 11.1.3 formula the standard error of the difference in two means and IV.B.5 perform a complete test on such a difference. Two-Sample Mean Inference with Experiments and II.C.1, II.C.5, Two-Sample Confidence Intervals – Students continue 11.1.4 IV.A.7, IV.B.1, practicing, and explore the differences in conditions between IV.B.5 observational studies and experiments. IV.B.1, IV.B.4, Inference in Different Situations – Additional practice 11.2.1 IV.B.5 problems on mean inference. IV.B.1, IV.B.2, Identifying and Implementing an Appropriate Test – 11.2.2 IV.B.4 – IV.B.6 Additional practice problems on all forms of inference
Chapter 12 – Regression inference / Transforming for Linearity
This final chapter before the AP exam solidifies topics in scatterplots and linear regression. Lesson AP Topics Lesson Summary Sampling Distribution of the Slope of the Regression Line – 12.1.1 IIID.6, IV.A.8 Students explore the sampling distribution for slope of a regression line and the conditions under which it is predictable Inference for the Slope of the Regression Line – Students use 12.1.2 IV.A.8, IV.B.7 computer output to create confidence itnervals and perform tests on slope of a regression line Transforming Data to Achieve Linearity – Students explore 12.2.1 I.D.1 – I.D.5 several mathematical methods to transform non-linear scatterplot data into linear data for the sake of regressions Using Logarithms to Achieve Linearity – Students do more 12.2.2 I.D.1 – I.D.5 work on linearizing non-linear data.
