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20886932260600AP Statistics00AP Statistics6714490113347500251460858520Updated 2014 Anne Arundel County Public Schools9500095000Updated 2014 Anne Arundel County Public Schools AP StatisticsCourse NavigationRationaleImplementation InstructionsScope and SequenceAssessment ExpectationsNational/State Standards:Acorn Guide AP StatisticsUnit NavigationUnit IUnit IIUnit IIIUnit IVUnit VSuggested Number of Days Per Unit: Unit I: Exploring Data: Describing patterns and departures from patterns. (20%-30%; 19 days)Unit II: Sampling and Experimentation: Planning and conducting a study. (10%-15%; 7 days)Unit III: Anticipating Patterns: Exploring random phenomena using probability and simulation. (20%-30%; 18 days)Unit IV: Statistical Inference: Estimating population parameters and testing hypotheses. (30%-40%; 27 days)Unit V: AP Exam ReviewThe percentages in parentheses for each unit indicate the coverage for that content area in the AP exam. The number of days allotted to each unit includes time for review and assessment.Unit I OverviewUnit Title: Exploring Data: Describing patterns and departures from patterns. Exploratory analysis of data makes use of graphical and numerical techniques to study patterns and departures from patterns. Emphasis should be placed on interpreting information from graphical and numerical displays and summaries. This Unit covers Chapter 1, 2 and 3 from The Practice of Statistics 5th Edition by Starnes, Tabor, Yates and Moore. Return to Unit NavigationStandards for Unit: (Click to view standards brace map) Maryland Content Standards: N/ACommon Core State Standards:Reading Standards: Writing Standards:Speaking and Listening Standards (opportunities to address speaking and listening standards are dependent upon which instructional strategies you choose and are noted in the center column):Indicators for Unit from 2010 College Board AP Statistics Course Description:Constructing and interpreting graphical displays of distributions of univariate data (dotplot, stemplot, histogram, cumulative frequency plot)Center and SpreadClusters and gapsOutliers and unusual featuresShapeSummarizing distributions of univariate data Measuring center: median, meanMeasuring spread: range, interquartile range, standard deviationMeasuring position: quartiles, percentiles, standardized scores (z-scores)Using boxplotsThe effect of changing units on summary measuresComparing distributions of univariate data (dotplots, back-to-back stemplots, parallel boxplots)Comparing center and spreadComparing clusters and gapsComparing outliers and unusual featuresComparing shape Exploring bivariate dataAnalyzing patterns in scatterplotsCorrelation and linearityLeast-squares regression lineResidual plots, outliers, and influential pointsTransformations to achieve linearity; logarithmic and power transformationsExploring categorical data Frequency tables and bar chartsMarginal and joint frequencies for two-way tablesConditional relative frequencies and associationComparing distributions using bar charts The normal distributionProperties of the normal distributionUsing tables of the normal distributionThe normal distribution as a model for measurementVertical Alignment: Click Here to View Vertical Alignment Multi-Flow MapEssential Questions: What is the distinction between quantitative and categorical variables? How do their data displays differ?How can the collection, organization, and display of data help to evaluate claims and make predictions about real-life situations?How can the understanding and use of measures of central tendency be useful for interpreting and drawing conclusions about data?How can data be analyzed so that misleading representations and interpretations be recognized and inferences will be reasonable?How do you use techniques of exploratory data analysis to study patterns and departures from patterns of data distributions?The Big Idea:Students will learn to display, describe and interpret data in meaningful ways.Globalization / Relevance:As our society becomes more data-driven, understanding of data display and analysis techniques are essential in all disciplines.Desired Student Learning Outcomes Instructional Delivery and Resources Chapter 1Exploring DataSection 1.1Analyzing Categorical Data(2 days)Learning Outcomes:Display and analyze categorical data.Calculate and display the marginal and conditional distribution of a categorical variable from a two-way table.Describe the association between two categorical variables by comparing appropriate conditional distributions.Vocabulary: individuals, variable,categorical variable,quantitative variable,distribution, frequency,relative frequency, piecharts, bar graphs, twoway table, marginaldistribution, conditionaldistribution, side-by-sidebar graph (segmented bargraph), association Critical Content and Skills: Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): Lesson Seeds (Essential Lesson Components) Identify categorical data and determine which data display (pie chart or bar graph) is more appropriate.Identify what makes some graphs of categorical data deceptive.Calculate and display the marginal distribution of a categorical variable from a two-way table.Calculate and display the conditional distribution of a categorical variable from a two-way table.Calculate conditional distributions from two-way tables to look for a possible association between two-categorical variables (the idea of independence).UDL/DI-Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Section 1.1 PowerpointOption #2: Against All Odds: What is Statistics? video, lesson resources, and interactive tools: Resources:Section 1.1Literary Connections: Other General Resources:Assessment EOL 1We collect these data from 50 male students. Which variable is categorical?eye color head circumferencehours of homework last weeknumber of cigarettes smoked dailynumber of TV sets at homeAssessment EOL 2 Which of the following distributions are more likely to be skewed to the right than skewed to the left?Household incomesHome PricesAges of teenage driversII onlyI and II I and IIIII and IIII, II and IIIAssessment EOL 3Section 1.1 EOL 3Desired Student Learning Outcomes Instructional Delivery and Resources Chapter 1Exploring DataSection 1.2Displaying Quantitative Data with Graphs(2 days)Learning Outcomes:Display the distribution of a quantitative variable with an appropriate graph.Describe the overall pattern (shape, center, and spread) of a distribution and identify any major departures from the pattern (outliers). Compare distributions of quantitative data using data displays.Vocabulary: dotplot, stemplot, histogram, shape, center, spread, outlier, symmetric, skewed, modesCritical Content and Skills:Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): Lesson Seeds (Essential Lesson Components) Make and interpret dotplots, stemplots, and histograms of quantitative data.Create and analyze histograms with technology.Calculator Directions: Histograms on the Calculator pg. 36 Describe quantitative distribution (Shape, Outliers, Center and Spread- SOCS) Identify the shape of a distribution as roughly symmetric, skewed, approximately normal, bimodal, pare distributions of quantitative data from dotplots, stemplots, back-to-back stemplots, and histograms.Be sure to discuss what makes some distributions displays deceptive. Misleading Histogram #1Misleading Histogram #2UDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Section 1.2 PowerpointOption #2: This activity can be used to collect data from students in class. Students can then use this data to create data displays and calculate numeric summaries to make comparisons.HYPERLINK ""Directions: Texting CompetitionTexting Pangrams for CompetitionOption #3: These applets can be used to demonstrate how changing the bin width can greatly affect the appearance of a histogram: #4: Against All Odds: Stemplots video, lesson resources, and interactive tools: #5: Against All Odds: Histograms video, lesson resources, and interactive tools: Resources:Section 1.2 Literary Connections: Other General Resources:Assessment EOL 1Which is true of the data shown in the histogram:I. The distribution is skewed right.II. The mean is probably smaller than the median.III. We should use median and IQR to summarize these data. A) I onlyB) II onlyC) III only D) II and III onlyE) I, II and IIIAssessment EOL 2AP FRQ 2005B #1- AP FRQ Scoring Guidelines 2005BAssessment EOL 3AP FRQ 2006 #1- AP FRQ Scoring Guidelines 2006Desired Student Learning Outcomes Instructional Delivery and Resources Chapter 1Exploring DataSection 1.3Describing Quantitative Data with Numbers(3 days, including review and assessment)Learning Outcomes:Describe the center and spread of quantitative data numerically. Choose the most appropriate measure of center and spread in a given setting.Use appropriate graphs and numerical summaries to compare distributions of quantitative data.Vocabulary:variability, mean (x), median, five number summary, quartiles, interquartile range, range, boxplot, variance, standard deviation, resistantCritical Content and Skills:Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): Lesson Seeds (Essential Lesson Components) Calculate the mean and median of a data distribution. Understand the effects of outliers on measures of central tendency. Activity: Mean as a “balance point” Calculate the measures of spread: range, IQR, standard deviation.Identify outliers using the 1.5×IQR rule. Make and interpret boxplots of quantitative data. Use data displays and numeric evidence to determine the most appropriate measures of center and spread to describe a quantitative distribution. Use appropriate graphs and numerical summaries to compare distributions of quantitative data.Analyze how adding and/or multiplying by a constant affects the center and/or spread of a variable’s distribution.Use technology to compute numerical summary and create appropriate data displays.Calculator Directions: Making Box plots pg. 59 and Computing Numerical Summaries pg. 63 Include an assessment of all Chapter 1 topics.UDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Section 1.3 PowerpointOption #2: Linear Transformation Video ()Option #3: Against All Odds: Measures of Center video, lesson resources, and interactive tools: #4: Against All Odds: Boxplots video, lesson resources, and interactive tools: #5: Against All Odds: Standard Deviation video, lesson resources, and interactive tools: Resources:Section 1.3 Literary Connections: Other General Resources:Assessment EOL 1The five-number summary of credit hoursfor 24 students in a statistics class is: MinQ1MedianQ3Max13.015.016.518.022.0Which statement is true?A) There are no outliers in the data.B) There is at least on low outlier in the data.C) There is at least one high outlier in the data.D) There are both low and high outliers in the data.E) None of the above.Assessment EOL 2Suppose 10% of a data set lie between 40 and 60. If 5 is first added to the each value in the set and then each result is doubled, which of the following is true?10% of the resulting data will lie between 85 and 125.10% of the resulting data will lie between 90 and 130. 15% of the resulting data will lie between 80 and 120. 20% of the resulting data will lie between 45 and 65.30% of the resulting data will lie between 85 and 125.Assessment EOL 3AP FRQ 2007B #1- Scoring GuidelinesAP FRQ 2001 #1- Scoring GuidelinesAP FRQ 2004 #1- Scoring Guidelines Desired Student Learning Outcomes Instructional Delivery and Resources Chapter 2Modeling Distributions of DataSection 2.1Describing Location in a Distribution(2 days)Learning Outcomes:Find, estimate and interpret percentiles and individual values within a distribution of data.Find and interpret the standardized score of an individual value within a pare values from two different distributions using a z-score.Examine and understand the effect of transforming data distributions.Vocabulary:percentiles, z-scores, cumulative relative frequency graph, transform dataCritical Content and Skills: Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): Lesson Seeds (Essential Lesson Components) Find and interpret the percentile of an individual value within a distribution of data.Activity: Where do I stand?Estimate percentiles and individual values using a cumulative relative frequency graph.Find and interpret the standardized score (z-score) of an individual value within a distribution of data. Explain how z-scores can be used to compare values from different distributions. Example: Jenny Takes Another Test (p. 91) Describe the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center and spread of a distribution. Data Exploration: The speed of Light (p. 98) UDL/DI-Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Section 2.1 NotesOption #2: Shifting and Scaling Textbook Resources:Section 2.1 Literary Connections: Other General Resources:Assessment EOL 1Scores on the ACT college entrance exam follow a bell-shaped distribution with the mean 18 and standard deviation 6. Wayne’s standardized score was -1. What was Wayne’s actual ACT score?Assessment EOL 2The 70 highest dams in the world have an average height of 206 meters with a standard deviation of 35 meters. The Hoover and Grand Coulee dams have heights of 221 and 68 meters, respectively. The Russian dams, the Nurek and Charvak, have heights with z-scores of +2.69 and -1.13, respectively. List the dams in order of ascending size.Charvak, Grand Coulee, Hoover, NurekCharvak, Grand Coulee, Nurek, HooverGrand Coulee, Charvak, Hoover, NurekGrand Coulee, Charvak, Nurek, HooverGrand Coulee, Hoover, Charvak, NurekAssessment EOL 3Costs for standard veterinary services at a local animal hospital follow a Normal model with a mean of $80 and a standard deviation of $20.Draw a clearly label the Normal model.Is it unusual to have a veterinary bill of $130? Use statistical analysis to explain your answer. Desired Student Learning Outcomes Instructional Delivery and Resources Chapter 2Modeling Distributions of DataSection 2.2Density Curves and Normal Distributions(4 days, including review and assessment)Learning Outcomes:Explore and interpret the Empirical Rule (68-95-99.7) in a Normal Distribution.Recognize when the Normal model is appropriate and find the percentage of observations falling within a specified interval in a Normal model using appropriate technologyUse technology to determine the specific value that corresponds to a given percentile (invNorm).Vocabulary:density curve, standard deviation, normal distribution, normal curves, empirical rule (68-95-99.7 rule), normal probability plotCritical Content and Skills:Can Know Content and Skills:All Normal model probabilities can be readily accessed using technology, however some students may benefit from seeing how Table A is used. Background Information for Teachers (should NOT be used with students): Lesson Seeds (Essential Lesson Components) Estimate the relative locations of the median and mean on a density curve.Use the 68-95-99.7 rule to estimate areas (proportions of values) in a Normal Distribution.Use Table A or technology to find (i) the proportion of z-values in a specified interval, or (ii) a z-score from a percentile in the standard Normal distributionCalculator Directions: From Z-Scores to Areas and Vice Versa (pg. 116) Use Table A or technology to find (i) the proportion of values in a specified interval, or (ii) the value that corresponds to a given percentile in any Normal distribution. Determine whether a distribution of data is approximately Normal from graphical and numerical evidence Calculator Directions: Normal Probability Plot ( pg. 125) Include an assessment of all Chapter 2 topics.UDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Section 2.2 NotesOption #2: Working with Z-Scores Leveled Normal Model PracticeOption #3: Directions: Z-Puzzles Z-PuzzleOption #4: Probability Plot WorksheetTextbook Resources:Section 2.2 Literary Connections: Other General Resources:These applets can be used to either calculate the area under the Normal model, or to determine the z-score for a given percentile: Normal Probability Plot Creator uses Microsoft Excel to create a Normal probability plot which can then be copied into other documents.Assessment EOL 1The distribution of heights of adult American males is approximately Normal with mean 69 inches and standard deviation of 2.5 inches. Draw an accurate sketch of the distribution of men’s heights. Be sure to label the mean, as well as the points 1, 2, and 3 standard deviations away from the mean on the horizontal axis. Assessment EOL 2The mean income per household in a certain state is $9500 with a standard deviation of $1750. The middle 95% of incomes are between what two values?Assessment EOL 3AP FRQ 2003 #3 (Parts a and b only)- Scoring GuidelinesDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 3Describing RelationshipsSection 3.1Scatterplots and Correlation (2 days) Learning Outcomes:Display the relationship between two quantitative variables using scatterplotsDescribe the relationship between two quantitative variables, using the correlation coefficient as part of the description.Vocabulary:scatterplot, explanatory variable, response variable, direction, form, strength, positive association, negative association, correlationCritical Content and Skills: Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): Lesson Seeds (Essential Lesson Components) Identify explanatory and response variables in situations where one variable helps to explain or influences the other. Make a scatterplot to display the relationship between two quantitative variables.Calculator Directions: Scatterplots on the calculator pg. 150Describe the direction, form, and strength of a relationship displayed in a scatterplot and identify outliers in a scatterplot.Interpret the correlation.Understand the basic properties of correlation, including how the correlation is influenced by outliers.Explain why association does not imply causation. Data Exploration: The SAT essay: Is longer better? (pg. 157)UDL/DI-Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Chapter 3.1 NotesOption #2: CSI STAT: The Case of the Missing CookiesOption #3: These applets allow students to guess correlation values by looking at data sets: Textbook Resources:Section 3.1 Literary Connections: Other General Resources:HYPERLINK "" \t "_blank"Bivariate Data ClassworkHYPERLINK "" \t "_blank"Bivariate Data HomeworkHYPERLINK "" \t "_blank"Correlation PracticeBivariate Data WarmupAssessment EOL 1 In a scatterplot of the average price of an automobile and the miles driven, you expect to see:very little association.a weak negative association.a strong negative association.a weak positive association.a strong positive association.Assessment EOL 2If the point in the upper right corner is removed from the data set, then what will happen to the slope of the line of best fit (b) and to the correlation of (r)? both will increaseboth will decreaseb will increase, and r will decreaseb will decrease, and r will increaseboth will remain the same. Assessment EOL 3Data show that there is a positive association between the population of 17 European countries and the number of stork pairs in those countries.Briefly explain what “positive association” means in this context.Wildlife advocates want the stork population to grow, so they approach the governments of these countries to encourage their citizens to have children. As a statistician, what do you think of this plan? Explain briefly. Desired Student Learning Outcomes Instructional Delivery and Resources Chapter 3Describing RelationshipsSection 3.2Least-Squares Regression (2 days) Learning Outcomes:Construct and interpret a linear model for the relationship between two quantitative variables using summary statistics and correlation, and technology.Construct and interpret residual plots to assess whether a linear model is appropriate.Interpret the standard deviation of the residuals and r2 to assess the least-squares regression line models the relationship. Vocabulary:regression line, slope, y-intercept, extrapolation, least-squares regression line, residual plot, standard deviation of the residuals, coefficient of determinations, outliers, influential observationsCritical Content and Skills:Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): Lesson Seeds (Essential Lesson Components) Interpret the slope and y-intercept of a least-squares regression line.Use the least-squares regression line to predict y for a given x. Explain the dangers of extrapolation.Calculate and interpret residuals.Explain the concept of least squares.Determine the equation of a least-squares regression line using technology or computer output.Construct and interpret residual plots to assess whether a linear model is appropriate.Interpret the standard deviation of the residuals and R2 and use these values to assess how well the least-squares regression line models the relationship between two variables. Describe how the slope, y intercept, standard deviation of the residuals, and R2are influenced by outliers. Find the slope and y intercept of the least-squares regression line from the means and standard deviations of x and y and their correlation. UDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Chapter 3.2 Notes (Student), Chapter 3.2 Notes (Teacher)Option #2: Data Exploration Activity: "r" Is Not Enough Option #3: Least Squares Regression Demonstration – This online application can be used to demonstrate the least squares process, and allows students to try to manually adjust a line in an attempt to find the least squares.Textbook Resources:Section 3.2 Literary Connections: Other General Resources: HYPERLINK "" \t "_blank" Regression Classwork HYPERLINK "" \t "_blank" Regression Homework HYPERLINK "" \t "_blank" Regression Practice HYPERLINK "" \t "_blank" Regression Warmup HYPERLINK "" \t "_blank" Regression Analysis Classwork HYPERLINK "" \t "_blank" Computer Outputs WarmupVerifying Goodness of Fit Classwork HYPERLINK "" \t "_blank" Verifying Goodness of Fit Practice HYPERLINK "" \t "_blank" Verifying Goodness of Fit Homework HYPERLINK "" \t "_blank" Regression Review Warmup HYPERLINK "" \t "_blank" Regression ReviewDrunk Driving Fatalities – Data that can be used for instruction or assessmentAssessment EOL 1 All but one of the statements below contain a mistake. Which one could be true?The correlation between height and weight is 0.569 inches per pound.The correlation between weight and length of foot is 0.488.The correlation between the bread of a dog and its weight is 0.435.The correlation between gender and age is -0.171.If the correlation between blood alcohol level and reaction time is 0.73, the correlation between reaction time and blood alcohol level is -0.73. Assessment EOL 2According the The World Almanac and Book of Facts 2004, the debt per capita for the years 1990-2001 gives you the following scatterplot:Regression output gives us the equation of the regression line as Debt=-2231226+1128(year) with a R2=98.8%.Explain in context what the slope of the line means.Explain in context what R2means.You decide to look at the residual plot to examine the relationship further. Based on the residual plot below, does a linear model seem to be appropriate? Explain.Assessment EOL 3AP FRQ 2002 #4- Scoring GuidelinesDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 12More About RegressionSection 12.2Transforming to Achieve Linearity(2 days, including review and assessment)Learning Outcomes:Use transformations involving logarithms to find a power model or an exponential model that describes the relationship between two variables, and use the model to make predictions.Determine which of several transformations does a better job of producing a linear relationship.Vocabulary:transforming, power model, logarithm, exponential model, Critical Content and Skills: Recognize when data is non-linear.Select and perform the correct transformation technique.Interpret the transformed regression in context.Can Know Content and Skills:In the interest of time, skip the first part of this section in the textbook which discusses using powers and roots to transform data.Background Information for Teachers (should NOT be used with students):Section 12.2 should be taught as a part of Chapter 3, as it deals directly with these concepts.Your focus for this section should be on selecting the best transformation technique and the use and interpretation of the resulting regression equation. Do not worry as much about the mathematical reasons that different techniques work.Discourage students from using scatterplots of non-linear data to identify the specific non-linear form of the data, as non-linear scatterplots can be misleading. Also discourage students from using “guess and check” to identify the correct transformation technique, as it can be cumbersome and time-consuming. Instead, encourage students to look for clues provided in the problem: the problem stem, equations of non-linear models, scatterplots of transformed data, or residual plots of transformed data.Lesson SeedsRemind students that all of our work with bivariate data has been reserved for linear relationships. Give students the opportunity to recognize data that is non-linear. When data is non-linear, we use transformation techniques to linearize the data. Once data is linear, we can make predictions and draw interpretations using the techniques that we learned throughout Chapter 3.Tell students that they will primarily see three different non-linear forms: exponential, logarithmic, and power. Discourage the use of scatterplots to recognize which non-linear form is present, as these can be misleading. Instead, encourage students to read each problem carefully, as they will often be told which non-linear form is the best model.Review the parent functions of non-linear forms. If students are not directly told which non-linear form exists, they will usually be provided with the non-linear equation. Give students the opportunity to recognize the three different forms from their equations.Exponential: y=abxLogarithmic: y=a+blnxPower: y=axpIn particular, remind students that the difference between an exponential and power equation is the location of the independent variable, x.Tell students that non-linear data is transformed into a linear relationship by taking the logarithm of either all x-values, all y-values, or both x and y values in the data set. (Both common and natural logarithms will work, as long as students are consistent in which one they use.) The non-linear form of the data dictates which transformation to use:Exponential: Leave x unchanged, take logarithm of y. (x, log y)Logarithmic: Take logarithm of x, leave y unchanged. (log x, y)Power: Take logarithm of x and y. (log x, log y)Once data is transformed, students can use the linear regression function on their calculators to write a regression equation. Remind students that their equation must include the transformation used.Exponential: logy=b0+b1xLogarithmic: y=b0+b1logxPower: logy=b0+b1logxTechnology Corner 30: “Transforming to Achieve Linearity on the Calculator” gives instructions on how to use the calculator to transform data. The instructions describe a “guess and check” to determine the best transformation method. Remind students that the best transformation method can often be found using clues in the problem.Give students an opportunity to practice using transformed models to make predictions when given a value of x. Refer to Example: “Go Fish!” on Pages 772-774. Students may require a brief review of exponentiation to solve a logarithmic rm students that they may still interpret slope, r or R2 in a transformed regression equation, as long as they include the logarithm with the correct variable. This is not addressed in our textbook. Refer to the examples below:Slope of an exponential relationship: “Each year, we predict the natural log of the number of transistors to increase by 0.366.”R2 of a power relationship: “99.9% of the variation in log(weight) can be explained by log(length).”Include a review of all Chapter 3 topics (including Section 12.2).UDL/DI-Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Option #2: Option #3:Textbook Resources:Section 12.2Literary Connections: Other General Resources:AMSCO’s AP Statistics (student resource), Pages 60-64Barron’s AP Statistics (teacher resource), Pages 120-123 HYPERLINK "" Transformation Notes and Practice HYPERLINK "" \t "_blank" Linearization Homework HYPERLINK "" \t "_blank" Linearization ClassworkLinearization WarmupAssessment EOL 1What is the appropriate transformation method for linearizing data that has each of the following relationships?Exponential?Logarithmic?Power?Assessment EOL 2Data was collected to determine if the length of an alligator can be used to predict the alligator’s weight. In order to linearize the data, a statistician took the logarithm of the weights and achieved the following residual plot.Did the statistician use an appropriate re-expression technique?What was the form of the original data (before transformation)?Assessment EOL 3Two measures x and y were taken on 18 subjects. The first of two regressions, Regression I, yielded and had the following residual plot.The second regression, Regression II, yielded and had the following residual plot.Which of the following conclusions is best supported by the evidence above?There is a linear relationship between x and y, and Regression I yields a better fit.There is a linear relationship between x and y, and Regression II yields a better fit.There is a negative correlation between x and y.There is a nonlinear relationship between x Click to Return to Course NavigationStudent Reflection: What methods or strategies helped you learn and understand the content the best? What would help you learn the math concepts better?Teacher Reflection:How do you know that you were effective? What questions, connected to the lesson standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?What strategies were highly effective for students’ success?What resources might be helpful to assist in instruction?Teacher Feedback and SharingTeachers: Please contribute feedback, questions, or comments to the Curriculum Resource Office.Unit II OverviewUnit Title: Sampling and Experimentation: Planning and Conducting a Study Data must be collected according to a well-developed plan if valid information on a conjecture is to be obtained. This plan includes clarifying the question and deciding upon a method of data collection and analysis. The unit corresponds to Chapter 4 in The Practice of Statistics 5th Edition by Starnes, Tabor, Yates, and Moore.Return to Unit NavigationStandards for Unit: (Click to view standards brace map) Maryland Content Standards: N/ACommon Core State Standards:Reading Standards: Writing Standards:Speaking and Listening Standards (opportunities to address speaking and listening standards are dependent upon which instructional strategies you choose and are noted in the center column):Indicators for Unit from 2010 College Board AP Statistics Course Description: Overview of methods of data collectionCensusSample SurveyExperiment Observational StudyPlanning and Conducting SurveysCharacteristics of a well-designed and well-conducted surveyPopulations, samples, and random selectionSources of bias in sampling and surveysSampling methods, including simple random sampling, stratified random sampling and cluster samplingPlanning and conducting experimentsCharacteristics of a well-designed and well-conducted experimentTreatments, control groups, experimental units, random assignments and replicationSources of bias and confounding, including placebo effect and blinding Completely randomized designRandomized block design, including matched pairs designGeneralizability of results and types of conclusions that can be drawn from observational studies, experiments, and surveys Vertical Alignment: Click Here to View Vertical Alignment Multi-Flow MapEssential Questions: How do we obtain data?Why is it important to be mindful of bias in data collection and how can bias be reduced?How does the sampling method impact conclusions that can be drawn?What are the principles of experimental design and why are they important?The Big Idea:Students will learn proper methods of data collection.Globalization / Relevance:In all disciplines, the conclusions that we may draw from data are dependent on the methods used to collect the data.Desired Student Learning Outcomes Instructional Delivery and Resources Chapter 4Designing StudiesSection 4.1Sampling and SurveysSection 4.2Experiments (1 day) Learning Outcomes:Identify population and sample in a statistical studyDistinguish between an observational study and an experiment Describe the scope of inference that is appropriate in a statistical study Vocabulary:population, sample, census, observational study, experiment, scope of inference, causation, associationCritical Content and Skills:Can Know Content and Skills:Background Information for Teachers (should NOT be used with students):Lesson Seeds (Essential Lesson Components) 1.Provide an introduction to the idea of drawing conclusions based on data“Activity: See no evil, hear no evil?” on page 209 is an experiment that students conduct with a partner on memory recall. Data for the class is compiled graphically, and then students draw conclusions based on the results.2.Define population, census, and sampleExplain the difference between census and sample surveyCensus vs Sample animation “Example: Sampling Hardwood and Humans” on page 210 requires students to identify the population and the sample from a given context3.Provide an overview of the types of studies used to collect data and note that each type will be discussed in detail throughout the unitObservational study, includes census and sample surveys (Some textbooks identify census and sample surveys as a third type of study.)ExperimentDiscuss the critical difference between the two types of studies (page 235)“Example: Does Taking Hormones Reduce Heart Attack Risk after Menopause?” on pages 235-236 is an activity that illustrates the differences between the two typesBegin discussing the scope of inference with respect to each type of study. Correlation and causation will be discussed in depth throughout the unit.Sample surveys are often used to make generalizations about a particular population, so great care must be taken in the sample selection.Observational studies can establish the existence of an association between two variables. An association does not necessarily indicate cause and effect, so use caution when stating conclusions of an observational study. An experiment is not intended to make generalizations about a population. It tells whether a certain treatment caused a specific response in a group of subjects, with replication being critical.UDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1:Against all Odds: Census and Sampling video, lesson resources, and interactive tools: #2: Observational and Experimental Studies worksheet Textbook Resources:Section 4.1 pages 208-211Section 4.2 pages 234-236 Literary Connections: Other General Resources:AMSCO’s AP Statistics (student resource), pages 74-76; 92-93Barron’s AP Statistics (teacher resource), pages 161-168Assessment EOL 11.Selected terms from Unit 2 Quizlet2.Identify the population and sample in each situation.A radio station asks listeners to call in with their opinion regarding a political issue.A hardware store employee inspects a box of 50 hammers from a truckload of hammers before accepting the shipment.An environmentalist collects a jar of sand from 23 beaches to examine the level of pollutants along the shoreline of the Chesapeake Bay.Assessment EOL 2What is the difference between an observational study and an experiment?Explain the difference between association and causation.Assessment EOL 3AP FRQ 1999 #3a - scoring guideAP FRQ 2007 #5a - sample responseDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 4Designing StudiesSection 4.1 Sampling and Surveys(2 days) Learning Outcomes:Identify voluntary and convenience samplesDistinguish simple random from stratified or cluster samplesUnderstand the differences in undercoverage, non-response, question wording and how they lead to bias Vocabulary:selection bias, response bias, variability, convenience sample, voluntary response, random, simple random, stratified, systematic, cluster, multistage sample, undercoverage, nonresponseCritical Content and Skills:Can Know Content and Skills:Background Information for Teachers (should NOT be used with students):Students are often confused about the difference between cluster vs. stratified sampling. They also struggle with multi-stage. Be sure to allow time for clarity. Lesson Seeds (Essential Lesson Components) 1.Provide an overview of the various sampling methods. The text breaks these methods into two categories: bad and good.Bad samples: convenience, voluntary responseGood samples all involve random sampling: simple random sample, stratified, cluster, and multistage. Systematic is another design that is not mentioned in the text.Types of Sample Designs worksheet and examples HYPERLINK "" \t "_blank" Thinking Map on Types of Sampling MethodsDefine selection bias (defined as bias on page 212)Be sure to discuss the dangers of bias when selecting a sample and the types of sampling methods that lead to bias, which is a systematic favoring of a particular outcome that leads to an underestimate or overestimate of the value of interest.It is important for students to recognize selection bias and be able to describe its effect on the value of interest (see “AP Exam Tip” on page 212).Point out that even a representative sample may produce data that is biased, if the responses of those selected are influenced in any way. This type of bias is called response bias and will be discuss later in the section.2.Discuss inference for sampling in more depthIf a sample does not fairly represent the target population, then making inferences from the sample data would be misleading“Example: Going to Class” on pages 223-224 provides an example of sampling variability. This activity also leads students to conclude that sample size is important, and larger samples provide less variable results.The distinction between bias and variability is important.Bias deals with the systematic favoring of a particular outcome, so it will lead to an overestimate or underestimate of the value of interest. Increasing the sample size does not reduce bias, if there is an error in the sampling method, the wording of the question, undercoverage, etc.Variability, also called sampling error, is always present and deals with the fact that different samples will produce different sample results. Sampling variability can never be eliminated, but it can be reduced by increasing the sample size.3.Discuss potential problems that arise with samples and surveysDefine undercoverage and nonresponseDefine response bias and provide examples of surveys that produce biased resultsInclude examples of poorly worded survey questions, such as leading questions, or the order of the questionsDiscuss the implications of a survey conducted face-to-face versus an anonymous surveyUDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Against all Odds: Samples and Surveys video, lesson resources, and interactive tools: #2:JellyBlubber ColonyActivity to help students see the importance of random samplingOption #3: Generating Random DigitsActivity to reinforce concept of randomness Option #4: Seek and Solve on Sampling Methods - Scavenger hunt of vocabulary termsSampling Vocabulary Relay Race – definitions and terms (alternative to scavenger hunt)Option #5: Sampling JigsawStudents work in groups to learn about various sampling techniques and apply the various techniques to a real world problemOption #6: Sample Survey Project This project requires students to work in pairs to investigate the impact of bias in a sample survey.Textbook Resources:Section 4.1 Pages 211 - 228Literary Connections: Other General Resources:AMSCO’s AP Statistics (student resource), Pages 76-84Barron’s AP Statistics (teacher resource), Pages 169-185Assessment EOL 1Selected terms from Unit 2 QuizletAssessment EOL 21.What is the difference between “simple random sample” and “random”?2.Does randomization ensure the data is reliable? Explain your position.Assessment EOL 3AP FRQ 2010B #2 - sample responseAP FRQ 2008 #2 - sample responsesDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 4Designing StudiesSection 4.2 Experiments Section 4.3Using Studies Wisely(4 days, including review and assessment) Learning Outcomes:Explain the concept of confounding and how it limits the ability to make cause and effect conclusionsIdentify experimental units, explanatory and response variables, and treatmentsExplain the importance and purpose of comparison, random assignment, control, and replicationDesign and create a completely randomized design, including blocking and matched pairs, for an experimentDescribe the placebo effect and the purpose of blinding.Vocabulary:explanatory variable, response variable, confounding, treatment, factors, levels, experimental units, subjects, random assignment, comparison, replication, completely randomized design, placebo, placebo effect, double-blind, statistically significant, block design, matched pairs design, randomized block designCritical Content and Skills:Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): Lesson Seeds (Essential Lesson Components) 1.Explore the nature of experiments as a way to understand cause and effect relationshipsDefine explanatory and response variablesDefine confounding and explain that a well-designed experiment will minimize the effects of confounding variables using a principle called control. Observational studies do not control confounding variables, so the effects of these variables are mixed together with the effects of other variables.2.Discuss the terminology of experiments Define important terms: treatment, experimental units, subjects, factors, levels“Example: When Will I Ever Use this Stuff?” on pages 237-238 is an activity that will enable students to apply the terminology of experimental design to a real-life example.“Example: TV Advertising” om page 239 is an activity where students design an experiment with multiple explanatory variables3.Discuss elements of good experimental design and factors that lead to poor designsAsk students to identify problems that could lead to poor experimental design“Example: Are Online SAT Prep Courses Effective?” on page 240 provides an example of a poorly designed experimentProvide other examples of poorly designed experiments and ask students to identify the flaws in the designAsk students to brainstorm characteristics that need to be present in a well-designed experiment. Define principles of experimental design, listed on page 242. (Note: some texts consider comparison to be part of randomization.) The following provide examples of different experimental designs.“Example: The Physicians Health Study” on pages 243-244“Example: Conserving Energy” on pages 245-246Discuss other factors to consider in experimental design, utilize examples in text to illustrateDefine placebo and placebo effectDiscuss the idea of (single and double) blinding in an experimentDefine blocking and matching in experimental design. Matching is a type of blocked design. Be sure to provide examples of each type of design, as many students confuse the two.Review terms related to experimental design: Experimental Design Vocabulary4.Explain inference for well-designed experimentsDefine statistically significantUse the Activity “Distracted Driving” on page 249 to build much of the discussion around experimentsReiterate the distinction between causation and correlationProvide several examples that challenge establishing causation: “Example: Do Center Brake Lights Reduce Rear-End Crashes” on page 286 and “Example: Does Smoking Cause Lung Cancer?” on page 269Causation versus Correlation – Mathgraphic explaining the difference between correlation and causationUnusual Relationships – Provides examples of associations between unexpected variables to help students distinguish between causation and correlationEvidence from poorly designed experiments should be viewed with cautionDiscuss ethical considerations in experimental design5.Review and assess topics in Chapter 4UDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Against all Odds: Designing Experiments video, lesson resources, and interactive tools: #2:Blocking Dogs – A simulation activity utilizing a blocked designOption #3: Bears in Space – Simulation Experiment with gummy bears Option #4: Pepsi Challenge: As a class activity, have students design a matched-pairs, double blind experiment to replicate the Pepsi Challenge of the 1980s to test which cola students actually prefer. Take a poll of the class prior to the experiment to determine which cola students claim to prefer then perform the experiment. See background information and 1981 Commercial clip for more information. Option #4:Paper Color Experiment – Students design an experiment to test the question, “Does paper color distract reader from a task?” (alternative to the Pepsi Challenge)Textbook Resources:Section 4.2 Section 4.3 Literary Connections: Research and reflect on the ethical implications from the Tuskegee Syphilis Experiment – Timeline and Research Implications. The article posted on the Center for Disease Control‘s website provides a timeline of the experiment, which lasted 40 years, as well as its impact on the development of legislation to protect human subjects of biomedical and behavioral research.Have students read and reflect on the Milgram experiments reading either:Milgram, S. (1973). The perils of obedience. Harper’s Magazine, 62-77.Milgram, S. (1974). Obedience to authority: An experimental view. HarpercollinsOther General Resources:AMSCO’s AP Statistics (student resource), Pages 84-91Barron’s AP Statistics (teacher resource), Pages 187-207Assessment EOL 1Selected terms from Unit 2 QuizletAssessment EOL 2It has been reported that cats kept as pets tend to be overweight. The Veterinarian Association has claimed that diet and exercise will help these cats get in shape. The Association of Veterinarians propose three different diets and two different exercise programs. In Diet 1 the pet owner controls the portion of cat food; Diet 2 the pet owner only feeds the cat fish; and Diet 3 the veterinarian provides a mixture of fresh vegetables that will be given with the cat food. With the exercise plans, Plan A has the owner having three 30-minute play sessions a week; Plan B has the owner having 20-minute play sessions daily. Eighty cat owners volunteer to take part in an experiment to help their overweight cats lose weight.Identify the following:a) the subjectsb) the factor(s) and the number of level(s) for eachc) the number of treatmentsAssessment EOL 3AP FRQ 2006 #5 - sample responsesAP FRQ 2005 #1, #5a - sample responsesAP FRQ 2010 #1a - sample response Click to Return to Course NavigationStudent Reflection: 1. What methods or strategies helped you learn and understand the content the best?2. What would help you learn the math concepts better?Teacher Reflection:1.How do you know that you were effective? What questions, connected to the lesson standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?2.What strategies were highly effective for students’ success?3.What resources might be helpful to assist in instruction?Teacher Feedback and SharingTeachers: Please contribute feedback, questions, or comments to the Curriculum Resource Office.Unit III OverviewUnit Title: Anticipating Patterns: Exploring random phenomena using probability and simulationProbability is the tool used for anticipating what the distribution of data should look like under a given model. This Unit covers Chapters 5, 6 and 7 from The Practice of Statistics 5th Edition by Starnes, Tabor, Yates and Moore.Return to Unit NavigationStandards for Unit: (Click to view standards brace map) Maryland Content Standards: N/ACommon Core State Standards:Reading Standards: Writing Standards:Speaking and Listening Standards (opportunities to address speaking and listening standards are dependent upon which instructional strategies you choose and are noted in the center column):Indicators for Unit from 2010 College Board AP Statistics Course Description:ProbabilityInterpreting probability, including long-run relative frequency interpretation“Law of Large Numbers” conceptAddition rule, multiplication rule, conditional probability and independenceDiscrete random variables and their probability distributions, including binomial and geometricSimulation of random behavior and probability distributionsMean (expected value) and standard deviation of a random variable, and linear transformations of a random variableCombining independent random variablesNotion of independence versus dependenceMean and standard deviation for sums and differences of independent random variablesThe normal distribution (refer to Unit 1)Properties of the normal distributionUsing tables of the normal distributionThe normal distribution as a model for measurementsSampling distributionsSampling distribution of a sample proportionSampling distribution of a sample meanCentral Limit TheoremSampling distribution of a difference between two independent sample proportions (refer to Unit 4)Sampling distribution of a difference between two independent sample means (refer to Unit 4)Simulation of sampling distributionst-distribution (refer to Unit 4)Chi-square distribution (refer to Unit 4)Vertical Alignment: Click Here to View Vertical Alignment Multi-Flow MapEssential Questions: What does probability tell us?How is probability calculated?What is a probability distribution?How do mean (expected value) and standard deviation describe a random variable?What are sampling distributions and how are they used?The Big Idea:Students will learn to calculate the probability of various events.Globalization / Relevance:This unit studies and predicts patterns from repeated events. By anticipating these patterns, we can notice and draw conclusions from any deviations from predicted results.Desired Student Learning Outcomes Instructional Delivery and Resources Chapter 5Probability: What Are the Chances?Section 5.1Randomness, Probability and Simulation(2 days)Learning Outcomes:Interpret probability as a long-run relative frequencyUse simulation to model chance behaviorVocabulary:probability, law of large numbers, simulationCritical Content and Skills: Differentiate between short-run and long-run behaviorCreate and interpret the results of simulationsCan Know Content and Skills:The paragraph immediately following the example on Page 298 can be used to preview the rationale behind the 10% Condition (Page 401).The “Think About It” on Page 298 can be used to preview the rationale behind significance tests (Chapter 9).Background Information for Teachers (should NOT be used with students):If time is a concern, place the most emphasis on recognizing a correctly designed simulation. Next priority should be designing a simulation (without necessarily conducting it). Make actually conducting the simulation the lowest priority.Lesson SeedsDiscuss the difference between short-run and long-run behavior. Define the law of large numbers. Differentiate between the law of large numbers and the invalid “law of averages”. Define probability as the long-run relative frequency of an event.Differentiate between the layman’s use of the word “random” and its use in statistics. Demonstrate the inability of people to be truly random.Activity: “Investigating Randomness”, Page 293Example: “Runs in Coin Tossing”, Page 293Create a list of common random devices (slips of paper, spinner, dice, coin flip, table of random digits, calculator random number generator) and note the advantages and limitations of each.Describe how random devices are used in simulations.Some simulations require a fixed number of random events.Activity: “Hiring Discrimination – It Just Won’t Fly!”, Page 5Activity: “Distracted Driving”, Page 249Activity: “The ‘1 in 6 Wins’ Game”, Page 288Some simulations require an unspecified number of random events, continuing until a desired outcome is obtained.Example: “Golden Ticket Parking Lottery”, Page 296Example: “NASCAR Cards and Cereal Boxes”, Page 297Emphasize that each repetition of a simulation has a result that is recorded, such as the number of “successes” obtained or the number of attempts needed to achieve a goal. Simulations are repeated many times and the distribution of the results are analyzed.UDL/DI-Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: The Monty Hall Problem – A motivator activity that leads to discussions of probability and simulations. Links to websites and videos are included.Option #2: Simulations - Pass the problemOption #3: TI Simulation and Law of Large Numbers Activity – Students simulate coin tosses to explore whether “lucky streaks” really exist.Textbook Resources:Section 5.1Technology Corner: “Choosing an SRS”, Page 215, describes how to use the random number generator function on the graphing calculatorLiterary Connections: Other General Resources:AMSCO’s AP Statistics (student resource), Pages 144-145Barron’s AP Statistics (teacher resource), Pages 221-223 HYPERLINK "" Modeling Outcomes PracticeHYPERLINK ""Simulations PracticeFamily Size Simulation – This applet simulates how big a family must be in order to have two children of each gender.Assessment EOL 1Which of the following could be used to simulate winning a game which has a winning rate of 3/8?On a table of random digits, assign 1, 2, and 3 as wins. The remaining digits are losses.On a table of random digits, assign 1, 2, and 3 as wins. Assign 4, 5, 6, 7, and 8 as losses. Skip 9 and 0.Roll a die 8 times. Count the number of times a 1, 2 or 3 is rolled.Assessment EOL 2The Mars candy company starts a marketing campaign that puts a plastic game piece in each bag of M&Ms. 30% of the pieces show the letter ‘M’, 20% show the symbol ‘&’, and the rest say “try again”. When you collect a set of 3 symbols M, &, and M, you can turn them in for a free bag of candy.Design a simulation that could be used to determine the average number of bags needed in order to be a winner.Design a simulation that could be used to determine the percent of time that a person who purchases 6 bags is a winner.Assessment EOL 3AP FRQ 2014 #2c - Sample responses not available at this time.AP FRQ 2001 #3 - Sample ResponsesAP FRQ 1998 #6abc - Scoring GuidelinesDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 5Probability: What Are the Chances?Section 5.2Probability Rules(1 day)Learning Outcomes:Describe a probability model for a chance process.Use basic probability rules, including the complement rule and addition rule for mutually exclusive events.Use a two-way table or Venn diagram to model a chance process and calculate probabilities involving two events.Use the general addition rule to calculate probabilities.Vocabulary:sample space, probability model, event, complement, mutually exclusive (disjoint), addition rule, general addition rule, Venn diagramCritical Content and Skills:Identify valid probability distributionsRead and use probability notationCalculate the probability of one or multiple eventsCan Know Content and Skills:Background Information for Teachers (should NOT be used with students):Our textbook uses the term mutually exclusive to describe two events that cannot both occur. However, other sources use the term disjoint. Be sure your students are comfortable with both terms.Lesson SeedsDefine a probability model as a list of all possible outcomes (sample space) and the probability of each.Describe and calculate probabilities using the Basic Probability Rules from pages 307 and 308. (Consider making this a student-led activity, as many students are familiar with these rules.)Emphasize that in mathematics, “or” is inclusive. In other words, “senior or female” includes students who are seniors, female, or both.Describe how to identify mutually exclusive (disjoint) events. Show that A and B are mutually exclusive if PA and B=0.Demonstrate the use of two-way tables and Venn diagrams to calculate probabilities. Use these to develop the general addition rule: PA or B=PA+PB-P(A and B).Introduce commonly used notation: Ac, ∩, ∪In statistics, referring to these symbols as “not”, “and” and “or” may be more valuable than the formal “complement”, “intersection” and “union”.Some resources use the symbols ~A or ?A for the complement.Section 5.3 will provide students with a better opportunity to practice and use this notation.UDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Against All Odds: Introduction to Probability video, lesson resources, and interactive tools: Option #2: Against All Odds: Probability Models video, lesson resources, and interactive tools: #3:Textbook Resources:Section 5.2Literary Connections: Other General Resources:AMSCO’s AP Statistics (student resource), Pages 98-100, 107Barron’s AP Statistics (teacher resource), Pages 209-211Assessment EOL 1Which of the following is a valid probability distribution?A.ABCD1000B.ABCD0.40.40.40.4C.ABCD1.20.30-0.5D.ABCD0.10.20.30.3Assessment EOL 2Movies are rated R for many reasons, but the primary reasons are sexual content and violence. Suppose 40% of movies are rated R for sexual content, 55% for violence and 35% for both.a)What is the probability that a movie is rated R for violence but not sexual content?A.15%B.20%C.35%D.40%E.60%b)What is the probability that a movie is rated R for reasons other than violence or sexual content?Assessment EOL 3A study by the University of Texas examined a sample of 626 people to see if there was an increased risk of contracting hepatitis C associated with having a tattoo. If the subject had a tattoo, researchers asked whether it had been done in a commercial tattoo parlor or elsewhere. The table below summarizes their findings:Tattoo done in parlorTattoo done elsewhereNo tattooHas hepatitis C17818No hepatitis C3553495a)What is the probability a surveyed person has a tattoo?b)What is the probability a surveyed person has no tattoo and no hepatitis C?c)What is the probability a surveyed person has a tattoo done elsewhere or has hepatitis C?Desired Student Learning Outcomes Instructional Delivery and Resources Chapter 5Probability: What Are the Chances?Section 5.3Conditional Probability and Independence(4 days, including review and assessment)Learning Outcomes:Calculate and interpret conditional probabilities.Use the general multiplication rule to calculate probabilities.Use tree diagrams to model a chance process and calculate probabilities involving two or more events.Determine if two or more events are independent.When appropriate, use the multiplication rule for independent events to compute probabilities.Vocabulary:conditional probability, general multiplication rule, tree diagram, independent events, multiplication rule for independent eventsCritical Content and Skills:Calculate conditional probabilityUse tree diagrams to calculate probabilityUse probability to determine whether two events are independentUse the multiplication ruleCan Know Content and Skills:Background Information for Teachers (should NOT be used with students):Students tend to struggle most in determining which probability strategy to use. As such, place as much emphasis as possible on how to identify the strategy to use for a particular problem:Addition Rule and General Addition RuleVenn DiagramConditional Probability (from a table)Tree DiagramMultiplication Rule and General Multiplication RuleFormulasDetermining IndependenceEncourage students to reserve formulas for “naked math” problems or to determine independence. Students can mistakenly combine formulas with one of the other strategies. An example of a commonly made mistake can be found on BlackBoard.Students often confuse the concepts of mutually exclusive and independent. Be sure to reinforce the distinction between the two. The Think About It: “Is there a connection between mutually exclusive and independent?” may be helpful to students’ understanding.Lesson SeedsDemonstrate how conditional probability can be calculated from a table. Often it is helpful to determine the denominator first. Use the Example: “Who Has Pierced Ears?”, Page 318, to demonstrate that conditional probability is not commutative. (P(A|B)≠P(B|A))Use the worksheet from BlackBoard to demonstrate probabilities in which items are selected without replacement. Note that, unless otherwise indicated, items are assumed to be selected without replacement.Demonstrate the use of tree diagrams and the general multiplication rule. Use Example: “Who Visits YouTube?”, Page 324, and Example: “Mammograms”, Page 325, to demonstrate the process of reversing the conditioning.Demonstrate how probabilities can be used to determine whether two events are independent.Demonstrate how the multiplication rule can be used when two events are independent.Use Example: “Rapid HIV Testing”, Page 329, to demonstrate that “at least one” and “none” are complements: Pat least one outcome is A=1-P(none of the outcomes are A).Use the worksheet from BlackBoard to provide students an opportunity to practice “naked math” probability problems using the two probability formulas provided on the AP Test. PA∪B=PA+PB-PA∩BPAB=PA∩BPBInclude an assessment of all Chapter 5 topics.DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: At Least One Exploration – This worksheet helps students to discover how to use the complement rule to calculate “at least one” probabilities. (Answer Key) Option #2: Practice with Skittles– Practice with the addition and multiplication rules.Option #3: Reverse the Conditional DI – This worksheet has the same problem on both sides. The front side shows the problem “as the AP exam would write it”. The back side breaks the problem down into more manageable steps.Option #4: TI Calculator Independence Activity – An exploration of independence, the multiplication rule, and conditional probability.Textbook Resources:Section 5.3Literary Connections: Other General Resources:AMSCO’s AP Statistics (student resource), Pages 104-106Barron’s AP Statistics (teacher resource), Pages 212-214, 253-254Assessment EOL 1Suppose 14% of students at a large school are left-handed.What is the probability of randomly selecting four students and finding that at least one was a leftie?If you are surveying students at random, what is the probability that the fourth students is the first surveyed student who is left handed?There are eighteen boys and fourteen girls in Mr. Algebra’s class. Mr. Algebra is standing at the door greeting students as they arrive.a)What is the probability that the first three students who arrive are boys?b)What is the probability that out of the first four students who arrive, at least one is a girl?c)What is the probability that the first girl to arrive is the third student overall?Assessment EOL 21.Suppose P(X) = 0.5 and P(Y) = 0.35.If P(X|Y) = 0.60, what is P(Y|X)? A.0.58B.0.7C.0.18D.0.21E.0.422.Police reports about last year’s traffic accidents indicated that 40% involved speeding, 25% involved alcohol and 10% involved both risk factors. Do these two risk factors appear to be independent?3.An automobile service station specializes in oil changes and tire replacements. Eighty percent of its customers request an oil change. Of those who request an oil change, 20 percent also request a tire replacement. Of those who do not request an oil change, 35% request a tire replacement. Suppose Jason has requested a tire replacement. What is the probability that he also requested an oil change?4.Forty people were asked if they’ve recently eaten at McDonalds or Subway. 24 had eaten at Subway. Fourteen had eaten at McDonald’s. Nine had eaten at both. What is the probability that a randomly selected Subway customer had also eaten at McDonalds?Assessment EOL 3AP FRQ 2014 #2ab - Sample responses not available at this time.AP FRQ 2011 #2 - Sample ResponsesAP FRQ 2011B #3 - Sample ResponsesAP FRQ 2010B #5abc - Sample ResponsesAP FRQ 2009B #2 - Sample ResponsesAP FRQ 2006B #3 - Sample ResponsesAP FRQ 2003B #2 - Scoring GuidelinesAP FRQ 2004 #3bc - Scoring GuidelinesDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 6Random VariablesSection 6.1Discrete and Continuous Random Variables(2 days)Learning Outcomes:Compute probabilities using the probability distribution of a discrete random variable.Calculate and interpret the mean (expected value) of a discrete random variable.Calculate and interpret the standard deviation of a discrete random pute probabilities using the probability distribution of certain continuous random variables.Vocabulary:probability distribution, random variable, discrete random variables, mean of a discrete random variable, expected value, variance of a random variable, standard deviation of a random variable, continuous random variableCritical Content and Skills:Calculate the mean, standard deviation and variance of a discrete random variable.Use a density curve or Normal model to calculate probabilities.Can Know Content and Skills:Background Information for Teachers (should NOT be used with students):Be sure to take time to spiral back on Normal model calculations as part of your discussion of continuous random variables.Lesson SeedsDemonstrate how to use a probability distribution to calculate probabilities. Explain that when the possible outcomes are numeric, we use a random variable to represent the distribution.Emphasize the properties of discrete random variables listed at the bottom of page 348.Use Example: “Winning (and Losing) at Roulette”, Pages 350 and 351 to demonstrate how the expected value of a discrete random variable is calculated and its meaning in context.Demonstrate the calculation of variance and standard deviation of a random variable.Use Technology Corner: “Analyzing Random Variables on the Calculator”, Page 354, to demonstrate how to calculate the mean, standard deviation and variance of a discrete random variable. Remind students the importance of writing the calculations on free response questions, but encourage them to use calculator commands to actually evaluate the formulas.Demonstrate how to use a Normal model to calculate probabilities based on a continuous random variable.DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Against All Odds: Random Variables video, lesson resources, and interactive tools: Option #2: TI Expected Value Activity – Students use tree diagrams to create a probability distribution and calculate the expected value for earnings of a charity fundraiser.Option #3:Textbook Resources:Section 6.1Literary Connections: Other General Resources:AMSCO’s AP Statistics (student resource), Pages 101, 121, 124-127Barron’s AP Statistics (teacher resource), Pages 223-226Assessment EOL 1The random variable C represents the number of students who forget their homework for a particular class.C0123P(C)0.20.20.50.1What is the probability that at least two students forget their homework?Assessment EOL 21.A biology professor responds to some student questions by email. The probability model below describes the number of emails (E) that the professor may receive from students during a day. E012345P(E)0.050.100.200.250.30xDetermine the value of x.How many emails should the professor expect to receive each day?What is the standard deviation for the number of emails received?2.A potato chip company advertises that bags of chips weigh 9.5oz. The manufacturer knows that the average weight of bags it sells is 10oz., with a standard deviation of 0.27oz. If a bag is randomly selected, what is the probability that it is underweight?Assessment EOL 3AP FRQ 2012 #2abc - Sample ResponsesAP FRQ 2008 #3 - Sample ResponsesAP FRQ 2005 #2 - Scoring GuidelinesAP FRQ 2002B #2 - Scoring GuidelinesDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 6Random VariablesSection 6.2Transforming and Combining Random Variables(2 days)Learning Outcomes:Describe the effects of transforming a random variable by adding or subtracting a constant and multiplying or dividing by a constant.Find the mean and standard deviation of the sum or difference of independent random variables.Find probabilities involving the sum or difference of independent Normal random variables.Vocabulary:linear transformation, independent random variablesCritical Content and Skills:Calculate the mean and standard deviations of random variables that have been shifted, scaled, combined, subtracted and repeated.Calculate probabilities based on combined and subtracted Normal random variables.Can Know Content and Skills:Background Information for Teachers (should NOT be used with students):Lesson SeedsDraw a parallel to Section 2.1, in which students learned that shifting data only affects measures of center/location, while scaling data affects measures of center/location and spread. In a similar way, shifting a random variable affects the mean, and scaling a random variable affects the mean and standard deviation.Explain the formulas for determining the mean, variance and standard deviation of independent random variables that are combined, subtracted and repeated.Provide students with opportunities to use context to differentiate between scaling a random variable by multiplying by a constant and combining repetitions of a random variable. The Think About It: “X1+X2 is Not the Same As 2X”, Page 375, is one example.Demonstrate how to calculate probabilities based on Normal distributions that are combined, subtracted or repeated. Use Example: “Put a Lid on It!”, Page 381, to demonstrate that subtraction can be used to determine the probability that one individual is greater than another. (This concept will be important when students study two-sample hypothesis tests.)DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Random Variable Formula Sheet – A graphic organizer to help students keep track of the various formulas they learn in this section. (Filled In Example) Option #2: Random Variable Rules Quizlet – Online “flashcards” that can be used to help remember the rules and formulasOption #3:Textbook Resources:Section 6.2Literary Connections: Other General Resources:AMSCO’s AP Statistics (student resource), Pages 134-137Barron’s AP Statistics (teacher resource), Pages 255-258Assessment EOL 1If μx=7 and σx=2, what are the mean and standard deviation of 3X-4?Assessment EOL 21.Two stores sell watermelons. At the first store the melons weigh an average of 22 pounds with a standard deviation of 2.5 pounds. At the second store the melons are smaller, with a mean of 18 pounds and a standard deviation of 2 pounds. Assume the weights of watermelons at both stores are normally distributed. You select a melon at random at each store.What is the expected value and standard deviation of the difference in weights?What is the probability that the melon from the first store is heavier than the melon from the second store?2.A small business has a service contract with WeCanFix Inc. for their laser printer. The business has found they need to use the service on average 0.2 times a year with a standard deviation of 0.548. If the service contract was made for a 3-year term, what is the mean and standard deviation of the number of times the printer needs to be serviced?Assessment EOL 3AP FRQ 2006 #3 - Sample ResponsesAP FRQ 2005B #2 - Scoring GuidelinesAP FRQ 2004 #4 - Scoring GuidelinesAP FRQ 2002 #3bc - Scoring GuidelinesAP FRQ 2001 #2 - Scoring GuidelinesAP FRQ 2000 #6b - Scoring GuidelinesAP FRQ 1999 #5 - Scoring GuidelinesDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 6Random VariablesSection 6.3Binomial and Geometric Random Variables(3 days, including review and assessment)Learning Outcomes:Determine whether the conditions for using a binomial random variable are pute and interpret probabilities involving binomial distributions.Calculate the mean and standard deviation of a binomial random variable. Interpret these values in context.Find probabilities involving geometric random variables.When appropriate, use the Normal approximation to the binomial distribution to calculate probabilities.Vocabulary:binomial setting, binomial random variable, binomial distribution, binomial coefficient, factorial, 10% condition, Large Counts condition, geometric setting, geometric probabilityCritical Content and Skills:Recognize binomial settings.Calculate mean, standard deviation and probabilities using binomial distributions.Can Know Content and Skills:Recognize geometric settings and calculate their probabilities. (These probabilities can be calculated using methods learned in Section 5.3.)Determine when the Normal model is appropriate to approximate a binomial probability. (This concept will be developed more extensively when students are learning about sampling distributions.)Can Know Content and Skills:Background Information for Teachers (should NOT be used with students):Lesson SeedsDiscuss the requirements of a binomial setting. Give students opportunities to practice recognizing binomial settings from context. The mnemonic in the margin of Page 388 may be helpful to students.Give students the opportunity to recognize and use the binomial probability formula. The formula is provided on the AP test. Students will likely not need to use the binomial coefficient formula. Instead, encourage students to use the notation nk and Technology Corner: “Binomial Coefficients on the Calculator”, Page 392, to evaluate the coefficient.Use Technology Corner: “Binomial Probability on the Calculator”, Pages 394 and 395, to calculate binomial probabilities on the calculator. Refer to the AP Exam Tip on page 395 with regards to showing work. The text on Page 395 provides a good demonstration on when to use the complement rule. Even though students can use technology to calculate a binomial probability, they will still likely have to recognize proper use of the formula.Demonstrate when and how to use the mean and standard deviation of a binomial random variable.Demonstrate when it is appropriate and how to use a Normal model to approximate a binomial probability. This concept is covered in greater deal in Chapter 7 and can be passed over if time is a concern.Students can use methods from Chapter 5 to calculate geometric probabilities. Do not teach the geometric formulas on pages 404-409 if time is a concern.Include an assessment of all Chapter 6 topics.DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Against All Odds: Binomial Distributions video, lesson resources, and interactive tools: Option #2: TI Binomial Activity – Students explore the survival rate of penguin eggsOption #3: Textbook Resources:Section 6.3Literary Connections: Other General Resources:AMSCO’s AP Statistics (student resource), Pages 113-117, 120, 129-133Barron’s AP Statistics (teacher resource), Pages 216-220, 227Normal Approximation to the Binomial – This applet simultaneous graphs a binomial distribution and the corresponding Normal approximation.Assessment EOL 1Which of these has a binomial model?The number of black cards in a 10-card hand.The colors of the cars in Giant’s parking lot.The number of hits a baseball player gets in 6 times at bat.Whether or not a person likes chocolate.The number of people we survey until we find someone who owns an iPod.Assessment EOL 2The National Association of Retailers reports that 62% of all purchases are now made by credit cards. A local florist assumes this is true for their sales as well. In general, they make 20 sales in one day.What are the mean and standard deviation for the number of credit cards used daily? What is the probability that at least 15 of 20 sales are paid by credit card?Assessment EOL 32004 AP FRQ #3a - Scoring GuidelinesDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 7Sampling DistributionsSection 7.1What Is a Sampling Distribution?(1 day)Learning Outcomes:Distinguish between a parameter and a statistic.Use the sampling distribution of a statistic to evaluate a claim about a parameter.Distinguish among the distribution of a population, the distribution of a sample, and the sampling distribution of a statistic.Determine whether or not a statistic is an unbiased estimator of a population parameter.Describe the relationship between sample size and the variability of a statistic.Vocabulary:parameter, statistic, sampling variability, sampling distribution, population distribution, distribution of sample data, unbiased estimator, biased estimator, variability of a statisticCritical Content and Skills:Understand the concept of a sampling distributionExplain how sample size affects the variability of a sampling distribution.Can Know Content and Skills:Background Information for Teachers (should NOT be used with students):Consider having students create histograms of sampling distributions of different sample sizes from the same population. Have these posted in your classroom so that you can refer back to them throughout the inference unit.Lesson SeedsExplain the difference between a statistic and a parameter. Note the change in notation: x vs. μ, s vs. σ, p vs. p.Use Activity: “Reaching for Chips”, Page 426, to demonstrate the idea of a sampling distribution.Figure 7.3, Page 428, provides an excellent comparison of the distribution of a population, the distribution of a sample, and a sampling distribution.Use Activity: “Sampling Heights”, Page 430, to show how some statistics are unbiased estimators and others are biased estimators.Discuss how larger sample sizes will reduce the variability of the statistic. Figure 7.8, Page 434, can be used to demonstrate the different ways that bias and variability can affect statistics.Use the worksheet from BlackBoard or AP FRQ 2009 #6bc to demonstrate how to use results from a simulated sampling distribution.DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Sampling Distribution Applet – This applet allows you to select a data set and create a sample distribution based on sample sizes of your choosing. Option #2: Option #3:Textbook Resources:Section 7.1Literary Connections: Other General Resources:AMSCO’s AP Statistics (student resource), Pages 150-154Barron’s AP Statistics (teacher resource), Page 289Assessment EOL 1A sample of college students was taken and it was found that the average number of alcoholic drinks consumed in a week is 3.4.Is 3.4 a statistic or a parameter?What notation (symbol) should be used to identify 3.4?Assessment EOL 2Which of the following are true statements?Sample parameters are used to make inferences about population statistics.Statistics from smaller samples have more variability.Parameters are fixed, while statistics vary depending on which sample is chosen.I and III and IIIII and IIII, II, and IIINone are correct.Assessment EOL 3AP FRQ 2009B #5 - Sample ResponsesAP FRQ 2005B #6bcd - Scoring GuidelinesAP FRQ 1998 #6 - Scoring GuidelinesDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 7Sampling DistributionsSection 7.2Sample Proportions(1 day)Learning Outcomes:Find the mean and standard deviation of the sampling distribution of a sample proportion p. Check the 10% condition before calculating σp.Determine if the sampling distribution of p is approximately Normal.If appropriate, use a Normal distribution to calculate probabilities involving p.Vocabulary:sampling distribution of pCritical Content and Skills:Calculate mean and standard deviation of a sampling distribution of sample proportions.Check assumptions and conditions for using the Normal model.Calculate probabilities using sampling distributions of sample proportions.Can Know Content and Skills:Background Information for Teachers (should NOT be used with students):Lesson SeedsUse Activity: “The Candy Machine”, Pages 440 and 441, Example: “Sampling Candies”, Page 442, or the Homer vs. Stewie activity on BlackBoard to demonstrate a sampling distribution of p.Refer to the summary on Page 444 for important aspects of a sampling distribution of sample proportions. Emphasize the need to check the 10% and Large Counts Conditions.Use Example: “Going to College”, Pages 445 and 446, to demonstrate how to determine the probability of obtaining a sample with a p above, below, or between two values.DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Homer vs. Stewie activity – Students create sampling distributions of sample proportions from various sample sizes. Option #2: Option #3:Textbook Resources:Section 7.2Literary Connections: Other General Resources:AMSCO’s AP Statistics (student resource), Pages 156-159Barron’s AP Statistics (teacher resource), Pages 289-292 Assessment EOL 1What are the conditions for using a Normal model to calculate probabilities for events involving p?Assessment EOL 2It is generally believed that electrical problems affect about 14% of new cars. An automobile mechanic conducts diagnostic tests on 128 new cars on the lot. What is the probability that in this group over 18% of the new cars will be found to have electrical problems?Assessment EOL 3A fair coin is tossed 100 times, and comes up tails 60% of the time. A different fair coin is tossed 1000 times, and also comes up tails 60% of the time. Which set of results is more unusual? Explain your reasoning.Desired Student Learning Outcomes Instructional Delivery and Resources Chapter 7Sampling DistributionsSection 7.3Sample Means(2 days, including review and assessment)Learning Outcomes:Find the mean and standard deviation of the sampling distribution of a sample mean x. Check the 10% condition before calculating σx.Explain how the shape of the sampling distribution of x is affected by the shape of the population distribution and the sample size.If appropriate, use a Normal distribution to calculate probabilities involving x.Vocabulary:sampling distribution of x, central limit theorem (CLT)Critical Content and Skills:Calculate mean and standard deviation of a sampling distribution of sample means.Check assumptions and conditions for using the Normal model.Calculate probabilities using sampling distributions of sample means.Can Know Content and Skills:Background Information for Teachers (should NOT be used with students):Lesson SeedsUse Activity: “Penny for Your Thoughts”, Page 450, Example: “Making Money”, Page 451, or the Baseball Salaries activity to demonstrate a sampling distribution of x.Refer to the summary on Page 452 for important aspects of a sampling distribution of sample means.Explain that when the population is Normally distributed, the sampling distribution of sample means is also Normally distributed.Explain that when the population is not Normally distributed, the central limit theorem states that the sampling distribution of sample means is Normally distributed if n is large. Refer to the Large Sample Condition on Page 458.Use Example: “Servicing Air Conditioners”, Page 459, to demonstrate how to determine the probability of obtaining a sample with an x above, below, or between two values.Include an assessment of all Chapter 7 topics.DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Against All Odds: Sampling Distributions video, lesson resources, and interactive tools: #2: Baseball Salaries Activity – Students use actual American League salaries to create sampling distributions of sample size 5, 10 and 20.Option #3: TI 84 Central Limit Theorem Exploration – Students use simulated number cubes to create sampling distributions of sample size 2, 4 and 7.Textbook Resources:Section 7.3Literary Connections: Other General Resources:AMSCO’s AP Statistics (student resource), Pages 165-166Barron’s AP Statistics (teacher resource), Pages 292-296Assessment EOL 1What are the conditions for using a Normal model to calculate probabilities for events involving x?Assessment EOL 2Herpetologists (snake specialists) found that a certain species of reticulated python have an average length of 20.5 feet with a standard deviation of 2.3 feet. The scientists collect a random sample of 35 adult pythons and measure their lengths. What is the probability that the sample mean length was less than 19.5 feet?Assessment EOL 3AP FRQ 2007 #3 - Sample ResponsesAP FRQ 2007B #2 - Sample ResponsesAP FRQ 2004B #3 - Scoring GuidelinesAP FRQ 1998 #1 - Scoring Guidelines Click to Return to Course NavigationStudent Reflection: What methods or strategies helped you learn and understand the content the best? What would help you learn the math concepts better?Teacher Reflection:How do you know that you were effective? What questions, connected to the lesson standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?What strategies were highly effective for students’ success?What resources might be helpful to assist in instruction?Teacher Feedback and SharingTeachers: Please contribute feedback, questions, or comments to the Curriculum Resource Office.Unit IV OverviewUnit Title: Statistical Inference: Estimating population parameters and testing hypothesesThis unit covers point estimators and confidence intervals as well as tests of significance for proportions, means, and the slope of a least-squares regression line. The unit corresponds to Chapters 8 through 12 in The Practice of Statistics 5th Edition by Starnes, Tabor, Yates, and Moore.Return to Unit NavigationStandards for Unit: (Click to view standards brace map) Maryland Content Standards: N/ACommon Core State Standards:Reading Standards: Writing Standards:Speaking and Listening Standards (opportunities to address speaking and listening standards are dependent upon which instructional strategies you choose and are noted in the center column):Indicators for Unit from College Board AP Statistics Course Description: A.Estimation (point estimators and confidence intervals)1.Estimating population parameters and margins of error2.Properties of point estimators, including unbiasedness and variability3.Logic of confidence intervals, meaning of confidence level and confidence intervals, and properties of confidence intervals4.Large sample confidence interval for a proportion5.Large-sample confidence interval for difference between two proportions6.Confidence interval for a mean7.Confidence interval for a difference between two means (paired and unpaired)8.Confidence interval for the slope of a least-squares regression lineB. Tests of Significance1.Logic of significance testing, null and alternative hypotheses; P-values; one- and two-sided tests; concepts of Type I and Type II errors; concept of power2.Large-sample test for proportion3.Large-sample test for difference between two proportions4.Test for a mean5.Test for a difference between two means (paired and unpaired)6.Chi-square test for goodness of fit, homogeneity of proportions, and independence (one- and two-way tables)7.Test for the slope of a least-squares regression lineVertical Alignment: Click Here to View Vertical Alignment Multi-Flow MapEssential Questions: 1.What assumptions do we need to make or what conditions must be met in order to use inference procedures?2.What conclusions can be drawn on the relationship between the margin of error, sample size, and confidence level?3.How can you make predictions about a population based on results from a sample?4.How would you evaluate evidence against a claim about a population based on results from a sample?The Big Idea:Students will learn to use samples to draw conclusions about populations.Globalization / Relevance:This unit studies the ability of statisticians to generalize results from a sample to a larger population. This skill is essential to any discipline in which samples are used to draw conclusions about populations. Desired Student Learning Outcomes Instructional Delivery and Resources Chapter 8Estimating with ConfidenceSection 8.1Confidence Intervals: The Basics(1 day)Learning Outcomes:Determine point estimate and margin of error from confidence intervalInterpret confidence interval in contextInterpret confidence level in contextDescribe how sample size & confidence level affect length of intervalExplain how practical issues like nonresponse, undercoverage, and response bias can affect interpretation of confidence intervalVocabulary: point estimator, point estimate, confidence interval, margin of error, confidence level, bias, variability, critical valueCritical Content and Skills: Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): The Acorn manual suggests teaching confidence intervals for proportions, means, and slope before introducing tests of significance for these parameters. However, the textbook orders these topics differently. For convenience, this unit guide will primarily follow the order of the textbook, so use of the publisher’s supplementary resources may be used more effectively.Lesson Seeds (Essential Lesson Components) 1.Explore the concept of confidence by estimating population parameters and margins of error.“Activity: Mystery Mean” on page 476 provides an introduction to this concept and continues through the sectionReiterate the importance of good sampling procedures and the need to proceed with formal inference only when satisfied that the data deserve such an analysis2.Define point estimator and point estimate“Example: From Batteries to Smoking” on page 478 gives examples of several point estimatorsDiscuss properties of point estimators, including unbiasedness and variabilityRefer back to Figure 7.8 on page 434 for graphic showing relationship between bias and variability3.Logic of confidence intervals, meaning of confidence level and confidence intervals, and properties of confidence intervalsDiscuss the appropriate interpretation of confidence level and interval. It may help to provide good examples as well as examples of improper interpretations to help clarify the meaning. Students often misinterpret a confidence level as the probability of capturing the true parameter.“Example: Who Will Win the Election?” on page 481 gives students an opportunity to interpret a confidence interval, identify the point estimate and margin of error, and draw conclusions from the interval. Students must be able to properly interpret a confidence interval. Refer to the “Think about it” and “Check for Understanding” examples on pages 484 and 485.Determine confidence intervals at different levels for various estimates of a population parameter given the margin of error.Describe how confidence intervals behave with respect to the margin of error and sample size."Activity: The Confidence Intervals Applet” on page 485 provides students the opportunity to explore the relationship between confidence level, sample size, and the length of the confidence interval.Freeman has another confidence interval applet that reinforces the concept of confidence, which can be found at: & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Confidence Interval Foldable Idea – to be completed throughout the unit or at the end as a study guideOption #2: Against All Odds: Confidence Intervals video, lesson resources, and interactive tools: Resources:Section 8.1Literary Connections: Other General Resources:Rossman/Chance also has a Confidence Interval applet:’s AP Statistics (student resource)Section 7.1 pp. 184-188 Barron’s AP Statistics practice assessment problems – teacher resource Overview of Confidence Intervals on p. 319Assessment items pp. 338-339 #1-5Summary of Confidence Interval Procedures – A completed template is attached; however this document can be filled-in by students as procedures are developed throughout the unit. This document includes intervals for one- and two-sample proportions and means.Assessment EOL 1 “Check Your Understanding” page 485Assessment EOL 2A New York Times poll on women’s issues interviewed 1025 women randomly selected from the United States, excluding Alaska and Hawaii. The poll found that 47% of the women said they do not get enough time for themselves.a)The poll announced a margin of error of ±3 percentage points for 95% confidence in its conclusions. What is the 95% confidence interval for the percent of all adult women who think they do not get enough time for themselves?b)Explain to someone who knows no statistics why we can’t just say that 47% of all adult women do not get enough time for themselves.c)Explain clearly what “95% confidence” means.Assessment EOL 3AP FRQ 2010 #3 – sample responseAP FRQ 2009B #6 – sample responseAP FRQ 2009 #4 – sample responseAP FRQ 2008B #2 – sample responseDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 8Estimating with ConfidenceSection 8.2 Estimating a Population Proportion (1 day)Learning Outcomes:State and check conditions and assumptions for inference for proportionsDescribe sampling distribution for sample proportion Determine critical values for calculating a C% confidence interval for population proportion using table or technologyConstruct and interpret confidence interval for a population proportion Determine sample size required to obtain a C% confidence interval for population proportion with a specified margin of errorVocabulary: random, large counts, standard error, critical value, one-sample z interval for proportionsCritical Content and Skills:Students cannot rely solely on using the commands of the graphing calculator to determine critical z values and to construct confidence intervals. They must also be able to utilize the formulas to determine margin of error, standard error, and sample size.Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): Lesson Seeds (Essential Lesson Components) 1.Large sample confidence interval for a proportionDescribe the population and explain in words what the parameter p represents Discuss conditions that must be met to estimate p. Discuss what to do if conditions are not met; this is explained on pages 495-6 of the text. Students must state the conditions and show evidence that they are met for the AP Exam. If conditions are not met, say so, and proceed with caution.Calculate and interpret one-sample z-intervals for a population proportionCalculate the numerical value of the statistic p that estimates p Review the sampling distribution for pCalculate the standard error for the population proportion from the sample statistic p. Be sure to distinguish between standard deviation and standard error, referencing sampling distribution of p. The standard deviation formula cannot be used when the population proportion is unknown. Determine critical z* values for various confidence levels using technology and the standard normal table (optional). Even though students can use technology to calculate confidence intervals, they will still likely have to employ proper use of the formulas for standard error and margin of error.Interpret a confidence interval in the context of the problem. Encourage students to write complete sentences. Students should also be able to use a confidence interval estimate to determine whether a claim made about a population is valid. “Example: The Beads” on pages 498-499 provides an opportunity to use a confidence interval to comment on a claim.2.Choose an appropriate sample size for estimating the population proportion given the confidence level. The margin of error formula requires knowledge of p or p , which are both unknown prior to sampling. To help students understand the use p=0.5 in the calculations, test various values of p in the numerator of the standard error formula p1-pn, beginning with 0.1, 0.2, 0.4, 0.5, 0.6, etc. Students should begin to see a trend in the size of the numerator. Ask them to guess which value will maximize the numerator.Using p=0.5 will ensure the sample is large enough to meet other requirementsUDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Goldfish Activity – Confidence Intervals for One Sample Proportion Option #2:TI Activity: Confidence Intervals for Proportions orTI Activity: Estimating a Population ProportionTextbook Resources:Section 8.2 Literary Connections: Other General Resources:AMSCO’s AP Statistics (student resource)Sections 7.2 & 7.3 pp. 188-199 (includes one- and two-sample intervals for proportions and means) Barron’s AP Statistics practice assessment problems – teacher resource Overview of Confidence Intervals for a Proportion on pp. 320-324Assessment items pp. 339-341 #6-7, 9-13 and p.349-352 #1, 2, 11Assessment EOL 11.A newspaper reported results of a poll, which asked voters which candidate they would choose on election day, stating with 90% confidence that between 53% and 57% of voters prefer Candidate A.a)What is the sample proportion of voters who prefer Candidate A?b)What is the margin of error?c)What is the critical z* value for the confidence interval?d)What was the sample size of the poll?2.To plan the budget for next year a college needs to estimate what impact the current economic downturn might have on student requests for financial aid. Historically this college has provided aid to 35% of its students. Officials look at a random sample of this year’s applications to see what proportion indicate a need for financial aid. Based on these data they create a 90% confidence interval of (32%, 40%). a)Could this confidence interval be used to test the hypothesis H0: p = 0.35 versus HA: p ≠ 0.35 at the α = 0.10 level of significance? Explain.b)For question #8 above what is the margin of error?3.“Check Your Understanding” page 4964.“Check Your Understanding” page 4995.“Check Your Understanding” page 503Assessment EOL 21.In a July 2008 study of 1050 randomly selected smokers, 74% said they would like to give up smoking. At the 95% confidence level, what is the margin of error?2.In a survey of 100 randomly selected residents of a large city during the month of February, 13% reported being unemployed.a)Construct a 95% confident interval estimate for the true proportion of unemployed residents in the city during the month of February.b)By May of the same year, the mayor of the city reported an unemployment rate of only 10%, claiming that his efforts to strengthen the workforce are clearly working. Use the confidence interval to comment on the validity of his claim.Assessment EOL 3AP FRQ 2010B #4ab – sample responseAP FRQ 2003B #6 – sample responseAP FRQ 2003 #6 – sample responseAP FRQ 2005 #4b – sample responseDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 8Estimating with ConfidenceSection 8.3Estimating a Population Mean(3 days, including review and assessment)Learning Outcomes:State and check conditions and assumptions for inference for meansExplain how the t-distributions are different from the standard normal distribution and describe when it is appropriate to applyDescribe sampling distribution for sample mean Determine critical values for calculating a C% confidence interval for population mean using table or technologyConstruct and interpret confidence interval for a population mean Determine sample size required to obtain a C% confidence interval for population mean with a specified margin of errorVocabulary: t-distribution, degrees of freedom, one-sample t intervalCritical Content and Skills:Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): Lesson Seeds (Essential Lesson Components) 1.Confidence interval for a meanDescribe the population and explain in words what the parameter μ represents. Students often have difficulty identifying whether to use procedures for proportions or means, so provide several examples of scenarios that require students to identify the appropriate measure. Distinguish between situations when σ is known and when it is unknown and the distribution that must be used in each case – standard normal or t distribution.Define the t distribution and degrees of freedom and identify characteristics of the distribution. Introduce students to the t distribution table. Some of the older calculators do not have a command that enables students to identify critical t* values from a given probability, so students must be able to utilize the table effectively. Instructions for use of the table and technology are provided on pages 513-514.Discuss conditions that must be met to estimate μ. Students must state the conditions and show evidence that they are met for the AP Exam. If conditions are not met, say so, and proceed with caution.Calculate and interpret one-sample t-intervals for a population meanCalculate the numerical value of the statistic x that estimates μDetermine the standard deviation σx and z test statistic when σ is known, and determine the standard error SEx and t test statistic when σ is unknownDetermine critical t* values (or z* values in rare cases when σ is known) for various confidence levels using the t distribution table or technology.Construct a confidence interval from summary statistics (“Example: Air Pollution” on pages 519-520) and from data (“Example: Video Screen Tension” on pages 520-521) using formulas and technology. It’s also a good idea to provide examples of various computer outputs, so students understand how to interpret results from other technology sources (“Case Closed: Need Help? Give Us a Call!” on page 525).Interpret a confidence interval in the context of the problem. Encourage students to write complete sentences. Students should also be able to use a confidence interval estimate to determine whether a claim made about a population is valid.2.Choose an appropriate sample size for estimating the population mean given the confidence level and margin of error. There is no easy way to do this when the standard deviation of the population is unknown, since the t distribution requires knowledge of the sample size to determine degrees of freedom.Students are only expected to determine sample size when a reasonable estimate for the population standard deviation is given. In this case the standard normal distribution may be used to obtain the critical z* value associated with the given confidence level. (Note: Switching between the standard normal distribution and t distribution can be confusing to students.)3.Review and assess Chapter 8 topicsUDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1:TI Activity: Estimating and Finding Confidence Intervals or TI Activity: Confidence Intervals for Means orTI Activity: Population Mean, sigma UnknownTextbook Resources:Section 8.3Literary Connections: Other General Resources:AMSCO’s AP Statistics (student resource)Sections 7.2 & 7.3 pp. 188-199 (includes one- and two-sample intervals for proportions and means), Section 8.3 p.255 Barron’s AP Statistics (teacher resource) Overview of t distribution pp. 301-302Sample assessment items p. 308 #14-15, p. 346 #31Overview of Confidence Intervals for a Mean on pp. 326-331Assessment items pp. 342-347 #14-19, 28, 32-34 and p. 350 #4-7Assessment EOL 11.When must we use t models instead of z models for inference questions involving means?2.What is the critical t* value for a 90% confidence interval from a sample of 1000?3.“Check Your Understanding” page 5144.“Check Your Understanding” page 5225.“Check Your Understanding” page 524Assessment EOL 2A researcher testing the effectiveness of a weight loss supplement reported that out of 50 participants randomly assigned to take the supplement the average weight lost per week was 2.5 pounds with a standard deviation of 0.75 pounds.a)Determine a 95% confidence interval.b)Interpret the interval in the context of the problem.Assessment EOL 3AP FRQ 2008B #3ab – sample responseAP FRQ 2004 #6ac – sample responseAP FRQ 2004B #5ab – sample responseDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 9Testing a ClaimSection 9.1Significance Tests: The BasicsSection 9.2 Tests about a Population Proportion(2 days)Learning Outcomes:State the null and alternative hypotheses for a significance test about a population parameterInterpret the P-value in contextDetermine whether the results of a study are statistically significant and make an appropriate conclusion using the significance levelInterpret a Type I and Type II error in context and give a consequence of eachInterpret the power of a test and describe what factors affect the power of a testDescribe the relationship among the probability of a Type 1 error (significance level), the probability of a Type II error, and the power of a testVocabulary: significance test, null hypothesis, alternative hypothesis, one-sided alternative, two-sided alternative, P-value, reject versus fail to reject, statistical significance, significance level, Type I error, Type II error, powerCritical Content and Skills:Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): This chapter covers significance tests for a single population proportion and mean. The concept of paired versus unpaired data with respect to means is introduced when dealing with the mean difference between pairs of observations. Type I and Type II errors are defined in Section 9.1 and the concept of power is introduced in Section 9.2 and spiraled throughout the chapter.Lesson Seeds (Essential Lesson Components) 1.Logic of significance testing, null and alternative hypotheses; P-values; one- and two-sided tests; concepts of Type I and Type II errors; concept of powerExplore the concept of significance tests“Case Study: Do You Have a Fever?” on page 537 is an activity that will engage students and introduce them to the idea of evaluating evidence against a claim about a population. There is also an applet referred to in the text for a simulation activity: “I’m a Great Free-Throw Shooter” on pages 538-539. State the null hypothesis and alternative hypothesisDiscuss the difference between one-sided and two-sided alternative hypothesesSpecify that the alternative hypothesis should be stated before collecting or observing the data. Even though students may have access to the results ahead of time, they should determine the alternative hypothesis based on what they suspect is true. This is explained at the bottom of page 540.Discuss the basic outline and reasoning of significance testingCalculate and interpret the P-valuesDefine statistical significance and significance level α. Be sure to distinguish between “significance” and “importance”. A result may be statistically significant, but the difference may be so small that it has no practical importance. Reiterate that “failing to reject Ho” does not mean “accepting the Ho.” Using the example of criminal court cases, juries weigh evidence against a defendant’s claim of innocence (Ho). At the conclusion of the trial, if the jury is convinced beyond a reasonable doubt (i.e. the significance level of a test), they conclude the defendant is guilty (reject Ho). If there is not enough evidence, the jury does not conclude the defendant is “innocent”, instead they conclude “not guilty” (fail to reject Ho). The truth about a defendant’s innocence may never be known, just as the truth about a claim may never be known. We can only examine evidence or results from a sample, which is another reason to ensure our data collection methods are sound.Explore the uses and misuses of tests of significance2.Understand the relationship between Type I errors, Type II errors, and power of a testDefine Type I and Type II errorsThe following graphic relates these errors to the jury trial exampleThe probability of a Type I Error (α) is the same as the significance level αThe probability of a Type II Error (β) can only be determined when there is a specific alternative in mind. This probability is a more difficult concept for students to understand. Practice: Type I and Type II Errors provides real life applications and interpretations of the errorsExplore the concept of power of a test as a way of measuring its sensitivity to alternatives.Calculating the power of a test or the probability of a Type II error from alternatives is not as important as understanding the relationship between α, β, and power. Students are expected to describe how a change to one value affects the others.Pages 565-569 discuss the relationship between power and Type II errorsUDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Statistical Inference ComicOption #2:Against All Odds: Tests of Significance video, lesson resources, and interactive tools: #3:“Alpha, Beta, and Power” PowerPoint Option #4:TI Activity: Type I and Type II errorsTextbook Resources:Section 9.1Section 9.2 pages 565-569 Literary Connections: Other General Resources:Summary of Inference Procedures – A completed template is attached; however this document can be filled-in by students as procedures are developed throughout the unit. This document includes intervals and tests for one- and two-sample proportions and means, chi-square, and slope of a regression line.AMSCO’s AP Statistics (student resource)Section 7.4 pp. 199-201 Section 7.6 pp. 223-230Barron’s AP Statistics practice assessment problems – teacher resource Overview of Significance Tests on pp. 363-364, 375-376Assessment items pp. 379-386 #1-2, 6-7, 10, 13, 16, 21-22, 24 and p. 388 #4Assessment EOL 11.“Check Your Understanding” page 5412.“Check Your Understanding” page 5493.“Check Your Understanding” page 5694.Exit Ticket: Alpha, Beta, Power & Significance - True/FalseAssessment EOL 21.Identify three ways to decrease the margin of error in a confidence interval for a population mean.2.Management asks the laboratory in the previous example to produce results accurate to within ±.005 with 95% confidence. How many measurements must be averaged to comply with this request?3.A botanist just received a grant that allows him to hire more people to help him with monitoring and surveying his experimental fertilizer on plants. In the past the botanist was only able to experiment on 100 plants. Now he will able to use 500 plants. Which of the following will be true about the botanist’s future hypothesis tests now that he will be using 500 plants?I.The probability of a Type I Error will decrease.II.The Power of the test increasesIII.Beta decreasesA. I onlyB. II onlyC. III onlyD. II and III onlyE. All three are trueAssessment EOL 3AP FRQ 2009 #6a – sample responseAP FRQ 2009 #5 – sample responseAP FRQ 2009 #1c – sample responseAP FRQ 2008B #1b – sample responseAP FRQ 2006B #6ade – sample responseAP FRQ 2007 #5bd – sample responseAP FRQ 2006 #6abd – sample responseAP FRQ 2004 #6cd – sample responseAP FRQ 2003 #1c – sample responseAP FRQ 2003 #2ab – sample responseAP FRQ 2009B #4b – sample responseAP FRQ 2008B #4b – sample responseDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 9Testing a ClaimSection 9.2Tests about a Population Proportion(1 day)Learning Outcomes:State and check conditions and assumptions for inference for proportionsPerform a significance test about a population proportion Interpret the power of a test and describe factors that affect the powerVocabulary: test statistic, one-sample z-testCritical Content and Skills:Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): Lesson Seeds (Essential Lesson Components) 1.Large sample test for proportionState the hypothesesDiscuss conditions that must be met for inference about the population proportion p. Review sampling distribution of pCalculate test statisticDetermine the P-value for one- and two-sided alternatives using knowledge of sampling distribution as well as technologyInterpret P-value and make a decision about the hypotheses in the context of the problemStudents must also be able to interpret output from statistical software to perform tests of significanceMake connection between confidence intervals and two-sided tests, and explain why the confidence interval provides more information than the test. This is explained on pages 563-564Interpret the power of a test and relate it to the probability of a Type II error. “Activity: What Affects the Power of a Test?” on pages 567-568 refers to an applet for one-sample tests for proportions to help students understand the concept of powerUDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1:Against All Odds: Inference for Proportions video, lesson resources, and interactive tools: #2: TI Activity: Testing Claims about a ProportionTextbook Resources:Section 9.2 pages 554-564 Literary Connections: Other General Resources:Review One Proportion z Inference – template and answer sheetAMSCO’s AP Statistics (student resource)Sections 7.5 pp. 202-222 (includes test of significance for one- and two-sample proportions and means) Barron’s AP Statistics practice assessment problems – teacher resource Overview of Hypothesis Tests for a Proportion on pp. 365-366Assessment items pp. 383-384 #14, 17 and p. 387 #2, 5Assessment EOL 11.“Check Your Understanding” page 5602.“Check Your Understanding” page 5633.“Check Your Understanding” page 564Assessment EOL 2A union spokesperson claims 75% of union members will support a strike if their basic demands are not met. A company negotiator believes the true percent is lower and runs a hypothesis test at the 10% significance level. What is the conclusion if 87 out of an SRS of 125 union members say they will strike?Assessment EOL 3AP FRQ 2006B #6ab – sample responseAP FRQ 2005 #4 – sample responseAP FRQ 2004 #3c – sample responseDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 9:Testing a ClaimSection 9.3Tests about a Population Mean(3 days, including review and assessment)Learning Outcomes:State and check conditions and assumptions for inference for meansPerform a significance test about a population mean Use a confidence interval to draw conclusion for a two-sided test about a population parameterPerform a significance test about a mean difference using paired dataVocabulary: t test statistic, standard error, one-sample t test, paired versus unpaired data, paired t procedures, effect sizeCritical Content and Skills:Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): Lesson Seeds (Essential Lesson Components) 1.Test for a meanState the hypothesesDiscuss conditions that must be met for inference about the population mean μ Review sampling distribution of x when σ is known. Remind students that this is an unusual case, and more likely σ is unknown, requiring the use of the t distribution.Calculate z test statistic when σ is known (optional) or the t test statistic when σ is unknown. Be sure to state degrees of freedom.Determine the P-value for one- and two-sided alternatives using knowledge of sampling distribution (standard normal or t distribution depending on test statistic) as well as technologyInterpret P-value and make a decision about the hypotheses in the context of the problemStudents must also be able to interpret output from statistical software to perform tests of significanceMake connection between confidence intervals and two-sided tests, and explain why the confidence interval provides more information than the test. This is explained on pages 583-585Distinguish between paired and unpaired data. Page 586 provides an explanation of situations that yield paired data - when two observations are made on the same individual or one observation is made on each of two similar individuals. This situation often arises in a matched-pairs experiment or study, and one sample t procedures (or paired t procedures) can be used on the differences in each pair. Unpaired data results from two independent samples or treatment groups, and two-sample procedures may be used, this will be covered in Chapter 10Using tests wiselyRelate sample size, error probabilities, effect size, and power. “Example: Developing Stronger Bones” on page 590 provides an example of how to use a significance test wisely.“Activity: Investigating Power” on pages 590-591 refers to an applet to illustrate the concept of power with significance tests about the population meanReview relationship between statistical significance and practical importanceInvestigate dangers of multiple analysis2.Review and assess Chapter 9 topicsUDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1:Against All Odds: Small Sample Inference for One Mean video, lesson resources, and interactive tools: #2: Snickers DI Activity for One-Sample t Tests: Low, Med, HighOption #3: Matched Pairs Skittles Activity includes confidence interval and paired t testOption #4:TI Activity: Hypothesis Testing - MeansTextbook Resources:Section 9.3 Literary Connections: Other General Resources:1-Sample t Test Review – template and answer sheetMatched Pairs t Test Review – template and answer sheetAMSCO’s AP Statistics (student resource)Sections 7.5 pp. 202-222 (includes test of significance for one- and two-sample proportions and means), Section 8.4 pp. 256-257Barron’s AP Statistics practice assessment problems – teacher resource Overview of Hypothesis Tests for a Mean on pp. 369-371Assessment items pp. 379-384 #3, 4, 8, 9, 11, 18, 19, 23 and p. 387-388 #1, 3, 6Assessment EOL 11.Inference about Means – Fill In the Blank2.An article in a local newspaper stated that students who attended University of Maryland take an average of 4.5 years to finish their undergraduate degrees. Suppose you believe the average is longer. You conduct a survey of 49 students and obtain a sample mean of 5.1 with a sample standard deviation of 1.2. What is the value of the test statistic?3.“Check Your Understanding” page 5794.“Check Your Understanding” page 5835.“Check Your Understanding” page 5846.“Check Your Understanding” page 589Assessment EOL 21.Weekly sales of regular ground coffee at a supermarket have in the recent past varied according to a normal distribution with mean μ=354 units per week. The store reduces the price by 5%. Sales in the next three weeks are 405, 378, and 411 units. Is this good evidence that average sales are now higher?2.Which of the following is a matched pairs design?paring men and women’s GPAs from a paring a school’s average SAT score to the national SAT score.paring two school’s average SAT scores.paring first semester grades and second semester math grades for a school’s students.E.None of these.Assessment EOL 3AP FRQ 2009B #5a – sample responseAP FRQ 2006B #4 – sample responseAP FRQ 2005B #6a – sample responseAP FRQ 2004 #6b – sample responseAP FRQ 2008B #6c – sample responseAP FRQ 2007 #4 – sample responseDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 10Comparing Two Populations or GroupsSection 10.1 Comparing Two Proportions(2 days)Learning Outcomes:State and check conditions and assumptions for inference for proportionsDescribe sampling distribution for difference between two proportionsConstruct and interpret confidence interval to compare two proportionsPerform a significance test to compare two proportionsVocabulary: two-sample z interval, test statistic, two-sample z-test, pooled sample proportionCritical Content and Skills:Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): Lesson Seeds (Essential Lesson Components) 1.Sampling distribution for the difference between two proportions“Activity: Is Yawning Contagious?” on pages 610-611 provides a hands-on simulation activity related to evaluating evidence against a claim regarding the difference between two proportionsDetermine sampling distribution for the difference between two proportions p1-p2 Discuss conditions that must be met for sampling distributionsDetermine the mean of the sampling distribution μp1-p2Determine the standard deviation σp1-p2 and standard error SEp1-p22.Large-sample confidence interval for difference between two population proportions p1-p2Discuss conditions for inference. Students must state the conditions and show evidence that they are met for the AP Exam. If conditions are not met, say so, and proceed with caution.Calculate two-sample z interval for the difference between two proportions using formulas and technology. Refer to the “Technology Corner” on pages 618-619 for tips on using the graphing calculator.Interpret the confidence interval. Interpreting confidence intervals for the difference in proportions can be confusing to students. Typically, we would state, “We are ___% confident that the difference in the two proportions is between ____ and ____”, but it is often unclear which proportion is larger when stated this way. It makes more sense to state, “We are ___% confident that proportion A is between ____% and ____% larger than (or smaller than) proportion B.”When 0 is contained in the interval, then we can’t be confident that a difference exists between the two proportions. The results of an hypothesis test would not be significant.Students should also be able to use a confidence interval estimate to determine whether a claim made about a population is valid.3.Large-sample test for difference between two proportions p1-p2Discuss conditions for inference. Students must state the conditions and show evidence that they are met for the AP Exam. If conditions are not met, say so, and proceed with caution.State hypotheses using either format, Ho: p1-p2=0 or Ho: p1=p2Define and calculate the pooled sample proportionPerform two-sample z test for the difference between two proportions using results from two random samples as well as from experimental data. Students should be able to perform test using technology and by interpreting output from statistical softwareInterpret the P-value and state the conclusion in the context of the problemUDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1:TI Activity: Claims about Two Proportions orTI Activity: NUMB3RS – Two Proportion TestTextbook Resources:Section 10.1 Literary Connections: Other General Resources:Review Two-Proportion z Inference – template and answer sheetAMSCO’s AP Statistics (student resource)Sections 7.2 & 7.3 pp. 188-199 (includes one- and two-sample intervals for proportions and means) Section 7.5 pp. 202-222 (includes test of significance for one- and two-sample proportions and means)Barron’s AP Statistics practice assessment problems – teacher resource Overview Sampling Distribution for Difference between Two Proportions pp. 296-298 Sample assessment items p. 307 #12 Overview of Confidence Intervals for a Difference between Two Proportion pp. 324-326Assessment items pp. 343-344 #20, 21, 22 (optional) and p. 349 #3Overview of Hypothesis Tests for Difference between Two Proportions pp. 366-368Assessment items pp. 379-383 #5, 15 and p. 388-389 #7, 9Assessment EOL 11.“Check Your Understanding” page 6192.“Check Your Understanding” page 6283.A researcher created a 90% confidence interval from data about the difference in the proportion of females and males ( ) that suffer from depression. It was: (0.052, 0.340). Which of the following conclusions can be made?A.The data suggests there is no difference in the proportions of males and females that have depression.B.The true difference in the proportion of females and males that have depression is between 5.2% and 34%.C.The data suggests that males are more likely than females to have depressionD.The data suggests that females are more likely than males to have depression.4.A research project conducted in 2009 wanted to estimate the difference in the proportion of teens and adults who use social-networking websites. Which of the following inferential procedures would be most appropriate to analyze the data they collect?A.1 – Proportion Z – Interval B.1 – Proportion Z – Test C.2 - Proportion Z – IntervalD. 2 – Proportion Z – TestAssessment EOL 21.In the 2001 regular baseball season, the World Series Champion Arizona Diamondbacks played 81 games at home and 81 games away. They won 48 of their home games and 44 of the games played away. We can consider these games as samples from potentially large populations of games played at home and away.a)Determine the standard error needed to compute a confidence interval for the difference in the proportions of home games and away games that would be won by the Diamondbacks.b)Construct and interpret a 90% confidence interval for the difference between the proportion of games that the Diamondbacks win at home and the proportion that they win when on the road.c)Many people believe in a home-field advantage, thinking it’s easier to win at home than away. Would the confidence interval for the Arizona Diamondbacks support this belief? Explain.2.A study of “adverse symptoms” in users of over-the-counter pain relief medications assigned subjects at random to one of two common pain relievers: acetaminophen and ibuprofen. In all, 650 subjects took acetaminophen, and 44 experienced some adverse symptom. Of the 347 subjects who took ibuprofen, 49 had an adverse symptom. How strong is the evidence that the two pain relievers differ in the proportion of people who experience an adverse symptom?Assessment EOL 3AP FRQ 2009B #6b – sample responseAP FRQ 2009 #4 – sample responseAP FRQ 2007 #5c – sample responseAP FRQ 2006B #2a – sample responseAP FRQ 2004B #6a – sample responseAP FRQ 2007B 6a – sample responseAP FRQ 2003B #3b – sample responseDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 10Comparing Two Populations or GroupsSection 10.2Comparing Two Means(4 days, including review and assessment)Learning Outcomes:State and check conditions and assumptions for inference for meansDescribe sampling distribution for difference between two meansConstruct and interpret confidence interval to compare two meansUse a confidence interval to draw conclusions about a population parameterPerform a significance test to compare two meansDetermine when it is appropriate to use two-sample t procedures versus paired t proceduresVocabulary: t-distribution, degrees of freedom, two-sample t proceduresCritical Content and Skills:Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): Lesson Seeds (Essential Lesson Components) 1.Define sampling distribution for the difference between two population means μ1-μ2Determine sampling distribution for the difference between two means x1-x2 Discuss conditions that must be met for sampling distributionsDetermine the mean of the sampling distribution μx1-x2Determine the standard deviation σx1-x2 and z test statistic when σ for each population is known, and determine the standard error SEx1-x2 and t test statistic when σ is unknown2.Construct a confidence interval for the difference in population means μ1-μ2Discuss conditions for inference. Students must state the conditions and show evidence that they are met for the AP Exam. If conditions are not met, say so, and proceed with caution.Calculate the two-sample t statistic Calculate a two-sample t interval for the difference between two means given summary statistics or given data. Students must be able to use formulas and technology to construct confidence intervals. Refer to the “Technology Corner” on page 643 for tips on using the graphing calculator.Interpret confidence interval in the context of the problem. Interpreting confidence intervals for the difference in means can be confusing to students. Typically, we would state, “We are ___% confident that the difference in the two means is between ____ and ____”, but it is often unclear which mean is larger when stated this way. It makes more sense to state, “We are ___% confident that mean A is between ____ and ____ units larger than (or smaller than) mean B.”When 0 is contained in the interval, then we can’t be confident that a difference exists between the two means. The results of a significance test would not be significant.Students should also be able to use a confidence interval estimate to determine whether a claim made about a population is valid.3.Test for a difference between two means μ1-μ2Review difference between paired and unpaired data. Two sample t procedures should be used only with data from two independent samples“Example: Comparing Tires and Comparing Workers” on page 650 provides several scenarios for students to distinguish between paired and unpaired data and identify which type of t procedures should be used in each caseDiscuss conditions for inference.Discuss “pooling”. Emphasize that students should only pool data when dealing with sample proportions, not means. Refer to explanation on pages 650-651.Perform two-sample t test for the difference between two means using results from two independent random samples as well as from experimental data. Students should be able to perform test using technology and by interpreting output from statistical softwareCalculate and interpret P-value in the context of the problem4.Review and assess Chapter 10 topicsUDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1:Against All Odds: Comparing Two Means video, lesson resources, and interactive tools: #2: TI Activity: Comparing Two MeansTextbook Resources:Section 10.2 Literary Connections: Other General Resources:Review 2-Sample t Test – template and answer sheetAMSCO’s AP Statistics (student resource)Sections 7.2 & 7.3 pp. 188-199 (includes one- and two-sample intervals for proportions and means) Section 7.5 pp. 202-222 (includes test of significance for one- and two-sample proportions and means), Section 8.5 pp. 257-259Barron’s AP Statistics practice assessment problems – teacher resource Overview Sampling Distribution for Difference between Two Means pp. 299-300 Sample assessment items p. 308 #13Overview of Confidence Intervals for a Difference between Two Means pp. 331-334 (includes paired and unpaired)Assessment items pp. 344-345 #23, 24, 25 (optional), 26, 27 and p. 351 #8Overview of Hypothesis Tests for Difference between Two Means pp. 371-375 (includes paired and unpaired)Assessment items pp. 382-385 #12, 20 and p. 389 #8, 10Assessment EOL 11.“Check Your Understanding” page 6442.“Check Your Understanding” page 649Assessment EOL 21.Having done poorly on their math final exams in June, six students repeat the course in summer school, and they retake another exam in August. If we consider these students representative of all students, who might attend with this summer school in other years, do these results provide evidence that the program is worthwhile?Student123456June544968666262August5065746468722.An experiment was performed to see whether sensory deprivation over an extended period of time has any effect on the alpha-wave patterns produced by the brain. To determine this, 20 subjects, inmates in a Canadian prison, were randomly split into two groups. Members of one group were placed in solitary confinement. Those in the other group were allowed to remain in their own cells. Seven days later, alpha-wave frequencies were measured for all subjects, as shown in the table.Non-ConfinedConfined10.79.610.710.410.49.710.910.310.59.210.39.39.69.911.19.511.29.010.410.9Is there evidence to conclude that sensory deprivation causes a change to alpha-brain wave patterns?Assessment EOL 3AP FRQ 2004B #5c – sample responseAP FRQ 2005B #4a – sample responseAP FRQ 2006 #4a – sample responseAP FRQ 2004B #4ab – sample responseAP FRQ 2001 #5 – sample responseAP FRQ 2004 #6b – sample responseAP FRQ 2007B #5 – sample responseAP FRQ 2010 #5 – sample responseAP FRQ 2004B #5ac – sample responseAP FRQ 2003B #4c – sample responseAP FRQ 2008 #6a – sample responseDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 11Inference for Distributions of Categorical DataSection 11.1Chi-Square Tests for Goodness of Fit(2 days)Learning Outcomes:State appropriate hypotheses and compute expected counts State and check conditions and assumptions for inferenceCalculate the chi-square statistic, degrees of freedom, and P-value for chi-square test for goodness of fitConduct follow-up analysis when the results of a chi-square test are statistically significantVocabulary: chi-square test for goodness of fit, observed counts, expected counts, chi-square statistic, chi-square distribution, componentsCritical Content and Skills:Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): In this chapter students will explore inference procedures for distributions of categorical data.Lesson Seeds (Essential Lesson Components) 1.Explore the Chi-square distribution Refer to “Case Study: Do Dogs Resemble Their Owners?” on page 677 for an activity that will engage students and introduce them to the idea of evaluating evidence against a claim about a distribution of categorical data. “Activity: The Candy Man Can” on pages 678-679 provides a hands-on activity using M&M candies where students can explore the relationship between the color distribution they observe in their bag of M&Ms and the expected distribution as defined by the company.State hypotheses in wordsCompute expected counts and compare to observed counts to determine the chi-square test statistic χ2Explore the χ2 distribution and degrees of freedomCalculate the P-value using technology or χ2 distribution table2.Perform a test of significance for goodness of fit.State conditions for inferenceState hypotheses in wordsDetermine expected counts and calculate the chi-square test statistic χ2Perform the chi-square test for goodness of fit by hand and using technology, as some calculators do not perform the goodness of fit test.State the P-value and interpret the results in the context of the problemPerform follow-up analysis when results of test are significantUDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: TI Activity: Chi-Square Test for Goodness of FitOption #2: TI Activity: Candy Pieces – Test for Goodness of FitTextbook Resources:Section 11.1 Literary Connections: Other General Resources:Barron’s AP Statistics practice assessment problems – teacher resource Overview of Hypothesis Tests for Chi-Square Goodness of Fit pp. 405-409 Assessment items pp. 420-427 #2, 5, 8, p. 425 #1, and p. 428 #1 Assessment EOL 11.“Check Your Understanding” page 6842.“Check Your Understanding” page 6873.“Check Your Understanding” page 691Assessment EOL 2In a recent year, at the 6 P.M. time slot, television channels 2, 3, 4, and 5 captured the entire audience with 30%, 25%, 20%, and 25%, respectively. During the first week of the next season, 500 viewers are interviewed.Suppose that the observed numbers of viewers are as follows:Channel2345Observed # Viewers139138112111A network executive believes that with his network’s new line-up, the viewing preferences are different in this next season. Conduct an appropriate test to check his claim.Assessment EOL 3AP FRQ 2003B #5c – sample responseAP FRQ 2008 #5ab – sample responseDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 11Inference for Distributions of Categorical DataSection 11.2Inference for Two-Way Tables(3 days, including review and assessment)Learning Outcomes:State appropriate hypotheses and compute expected counts for all types of chi-square tests Compare conditional distributions for data in a two-way tableCalculate the chi-square statistic, degrees of freedom, and P-value for all types of chi-square testsChoose and perform appropriate chi-square test: for goodness of fit, homogeneity, or independenceConduct follow-up analysis when the results of a chi-square test are statistically significantVocabulary: chi-square test for homogeneity, chi-square test for independence, multiple comparisonsCritical Content and Skills:Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): Lesson Seeds (Essential Lesson Components) 1.Chi-square test for homogeneity of proportions and independence between two categorical variablesCompare conditional distributions of a categorical variable (two-way tables)State hypotheses in wordsCompute expected counts and the chi-square statistic using formulas and technology. It may be necessary to provide a brief overview of using matrix features on the graphing calculator. Students will need to enter observed counts into matrices, but no other operations will be performed.State conditions for performing tests for homogeneity and independence. Suggest strategies for “collapsing” cells in the table, so expected counts are greater than 5. This strategy is suggested on page 721 in the text.Perform a chi-square test using formulas and technology by computing the P-value and stating the conclusion in the context of the problemStudents often struggle trying to identify which type of chi-square test to perform, so it will help to provide some examplesDiscuss the appropriate use of these tests.Perform follow-up analysis when results are significant2.Review and assess Chapter 11 topicsUDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Against All Odds: Inference for Two-Way Tables video, lesson resources, and interactive tools: #2:M&M Inference Activity – provides a review of inference procedures for single proportions as well as chi-square testsOption #3:TI Activity: Chi-Square Tests for Independence & Homogeneity orTI Activity: Inference for Two-Way TablesTextbook Resources:Section 11.2 Literary Connections: Other General Resources:Review Chi-Squared – template and answer sheetAMSCO’s AP Statistics (student resource)Section 7.8 pp. 237-244 (includes test of significance for all chi-square tests)Section 8.6 pp. 259-262Barron’s AP Statistics practice assessment problems – teacher resource Overview of Chi-Square Test for Homogeneity and Independence pp. 409-415 Assessment items pp. 420-427 #3, 4, 6, 7, 9, 10 and pp. 425-427 #2-5Assessment EOL 11.“Check Your Understanding” page 4992.“Check Your Understanding” page 7053.“Check Your Understanding” page 7114.“Check Your Understanding” page 717Assessment EOL 2In a nationwide telephone poll of 1000 adults representing Democrats, Republicans, and Independents, respondents were asked if their confidence in the U.S. banking system had been shaken by the savings and loan crisis. The answers, cross-classified by party affiliation, are given in the following table.YesNoNo OpinionDemocrats17522055Republicans15016535Independents7510520Test the null hypothesis that shaken confidence in the banking system is independent of party affiliation. Use a 10% significance level.Assessment EOL 3AP FRQ 2006 #6abc – sample responseAP FRQ 2004 #5a – sample responseAP FRQ 2003 #5 – sample responseDesired Student Learning Outcomes Instructional Delivery and Resources Chapter 12More about RegressionSection 12.1Inference for Linear Regression(5 days, including review and assessment)Learning Outcomes:State and check conditions and assumptions for inference about the slope of a regression lineInterpret the values of a, b, s, SEb, and r2 in context, and determine these values from computer outputConstruct and interpret a confidence interval for the slope of the population regression linePerform a significance test about the slope of the population (true) regression lineVocabulary: population regression line, sample regression line, normal probability plot, standard error of the slope, t test for the slopeCritical Content and Skills:Can Know Content and Skills:Background Information for Teachers (should NOT be used with students): The standard error formula for the slope of the regression line is very complicated. In most cases the standard error is provided via output from statistical software. Emphasis should be placed on interpreting the output from a variety of sources rather than on computing the standard error by hand. This is particularly important for calculating the confidence interval, since some of the older graphing calculators do not have this capacity.Lesson Seeds (Essential Lesson Components) 1.Define the sampling distribution for the slope b of the sample regression lineDescribe the difference between the population regression line and the regression line produced from sample data, called the sample (or estimated) regression lineDescribe the sampling distribution of the slope b of the sample regression line. The formula for the standard error of the slope can be quite confusing to students, as it relies on the standard deviation of the residuals. Emphasis should be placed on interpreting output from various statistical software packages, not computing the value by hand.Discuss conditions that must be met to estimate the slope β of the population regression line. The text references an acronym LINER, which is explained on page 743Students must state the conditions and show evidence that they are met for the AP Exam. When the conditions require examination of a graph, students must sketch a graph to verify the condition. If conditions are not met, say so, and proceed with caution.2.Calculate and interpret confidence intervals for slope of a regression lineVerify conditions for inferenceCalculate the numerical value of the statistic b that estimates βDetermine the standard error of the slope SEb from output from statistical softwareDetermine critical t* values for various confidence levels using the table or technology. Note that the formula for degrees of freedom is df=n-2.Construct a confidence interval from summary statistics (“Example: The Helicopter Experiment” on pages 748-749) and from data (“Example: How Much is that Truck Worth” on pages 749-751) using formulas and technology. It’s also a good idea to provide examples of various computer outputs, so students understand how to interpret results from other technology sources (“Check for Understanding” on pages 752-753).Interpret a confidence interval for the slope of the regression line in the context of the problem. In addition to stating the interval, students should also be able to state the slope as the rate of change between two variables. When interpreting the confidence interval, we can state, “We are ___% confident that as the independent variable increases by 1 unit, the dependent variable increases (or decreases, if the interval is negative) between ___ and ___ units.”When 0 is contained in the interval, we cannot be confident that a relationship exists, so the results would not be statistically significant.Students should also be able to use a confidence interval estimate to determine whether a claim made about a population is valid.3.Test for the slope of a least-squares regression lineState hypotheses. For the slope of the regression line, we are often trying to determine if a relationship between two variables exists and/or whether the relationship has a positive or negative association.The null hypothesis will be Ho: β=0 for the purposes of this courseVerify conditions for inferenceConduct significance test using data and technology or by interpreting summary statistics as output from various statistical software packages. “Example: Crying and IQ” on pages 754-756 provides an example that includes actual data as well as output from statistical software.Calculate the P-value and interpret the results of the test in the context of the problem.4.Review and assess Section 12.1 topics5.Review All Inference ProceduresStudents need practice recognizing which test to perform in a variety of contexts. The Unit 4 Review provides several examples for students to match real-life scenarios to the appropriate test. There are also several applications of significance tests with an emphasis on chi-square and slope of a regression line.UDL/DI -Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1:Against All Odds: Inference for Regression video, lesson resources, and interactive tools: #2: TI Activity: Inference for Regression and CorrelationTextbook Resources:Section 12.1 Literary Connections: Other General Resources:Review Regression-Slope t Inference – template and answer sheetAMSCO’s AP Statistics (student resource)Section 7.7 pp. 230-236, Section 8.2 pp. 250-254Barron’s AP Statistics practice assessment problems – teacher resource Overview of Confidence Intervals for Slope on pp. 334-337Assessment items pp. 347-348 #35 and p. 351#9-10Overview of Significance Tests for Slope pp. 415-419 Assessment items pp. 420-427 #1, 11, 12, p. 427 #6, 7 and pp. 428-429 #2Assessment EOL 11.State the conditions for inference for the slope of a regression line.2.“Check Your Understanding” page 7523.“Check Your Understanding” page 757Assessment EOL 2Sixteen students collected data on their height and length of arm span measured in inches. The scatter plot of the data showed there was a strong, positive, linear association between the two variables. Linear regression analysis was performed, and the computer output is shown below.Independent Variable = HeightPredictorCoefStdErrt-ratiopConstant16.4365.6182.930.01Arm span0.76720.08129.450.000s = 1.618R-sq = 84.8%R-sq(adj) = 83.9%a)What is the equation for the line of best fit?b)Construct a 95% confidence interval estimate for the slope of the regression line. Interpret the interval.c)If there were no association between height and arm span, what would be the slope of the regression equation? Write the null hypothesis.d)What type of association do you expect between height and arm span: positive or negative? Write the alternative hypothesis.e)Assuming all conditions for inference are met, conduct a test of significance against your null hypothesis. Is the result significant? Interpret the result in the context of the problem.Assessment EOL 3AP FRQ 2010B #6d – sample responseAP FRQ 2007B #6b – sample responseAP FRQ 2005B #5c – sample response AP FRQ 2008 #6cd – sample responseAP FRQ 2007 #6c – sample responseAP FRQ 2006 #2c – sample response Click to Return to Course NavigationStudent Reflection: 1. What methods or strategies helped you learn and understand the content the best?2. What would help you learn the math concepts better?Teacher Reflection:1. How do you know that you were effective? What questions, connected to the lesson standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?2.What strategies were highly effective for students’ success?3.What resources might be helpful to assist in instruction?Teacher Feedback and SharingTeachers: Please contribute feedback, questions, or comments to the Curriculum Resource Office.Unit V OverviewUnit Title: AP ReviewReturn to Unit NavigationStandards for Unit: (Click to view standards brace map) type alignment hereWrite the standards out here…Maryland Content Standards: N/ACommon Core State Standards:All items in red refer to a link between the CCSS and the lessons, activities and/or assessment that correlate to the CCSS. Actual link to CC6 and / or CC7 ….depending on the unitReading Standards: Writing Standards:Speaking and Listening Standards (opportunities to address speaking and listening standards are dependent upon which instructional strategies you choose and are noted in the center column):Indicators for Unit:Vertical Alignment: skip for nowClick Here to View Vertical Alignment Multi-Flow MapEssential Questions:The Big Idea:Students will prepare to demonstrate mastery on the AP exam.Globalization / Relevance:In all fields and disciplines, students must synthesize learned content and apply it in order to demonstrate mastery. Desired Student Learning Outcomes Instructional Delivery and Resources Learning Outcomes:Demonstrate mastery of skills required for success on the AP Statistics examVocabulary:Critical Content and Skills: Can Know Content and Skills:Background Information for Teachers (should NOT be used with students):These lesson seeds are suggestions for AP review. Teachers should choose which best suits their students’ needs and the time available.Lesson SeedsDiscuss the breakdown of the AP Statistics exam:Multiple choice40 questions90 minutesNo penalty for incorrect answers50% of total scoreFree response6 questions90 minutes50% of total scoreAll questions are given a whole number score from 1 to 4.Questions 1-5 are weighted equally, each worth 15% of the free response score.AP recommends 65 total minutes on questions 1-5.Question 6 is worth 25% of the free response score.AP recommends 25 minutes on question 6.Sample scoring worksheet is available via AP Central Scoring WorksheetSpend a class period taking a practice multiple choice test or free response test.Free response tests are available via AP Central FRQs.The full 1997 test (MC and FRQ) is available via AP Central Released 1997 TestThe 2008 practice exam is available via BlackboardPractice exams are available via College BoardBoth AMSCO and Barron’s have practice exams.Review the formula sheets and tables provided on the AP exam. These are available in any free response test posted on AP Central FRQs. (Formula sheets available in both the AMSCO and Barron’s books are very similar.)Have students complete practice FRQs from AP Central FRQs. Use scoring guidelines to help students review and assess their peers.Review the exam tips posted on AP Central Exam Tips, or download the Word document from BlackboardReview each of the four major units: data analysis, data collection, probability, inference.If you have several days available, spend a whole day on each unit.If you only have one day available, set up review stations so students may choose which units to review.Use the FRQ Topic Breakdown to select questions specific to concepts that students need to review.Host after-school AP Statistics review get-togethers.UDL/DI-Representation & Engagement Choices (Purposefully choose one or more of the following based upon student needs): Option #1: Jeopardy ReviewOption #2: Option #3: Textbook Resources:AMSCO’s AP Statistics (student resource), Pages 264-313Barron’s AP Statistics (teacher resource), Pages 439-585Literary Connections: Other General Resources:Exam Tips PowerPoint Click to Return to Course NavigationStudent Reflection: What methods or strategies helped you learn and understand the content the best?What would help you learn the math concepts better?Teacher Reflection:How do you know that you were effective? What questions, connected to the lesson standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?What strategies were highly effective for students’ success?What resources might be helpful to assist in instruction?Teacher Feedback and SharingTeachers: Please contribute feedback, questions, or comments to the Curriculum Resource Office. ................
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