• ADVANCED PLACEMENT STATISTICS

Primary Textbook

Yates, Daniel S., David S. Moore and Daren S. Starnes.  The Practice of Statistics, second edition.  W.H. Freeman and Company, New York.

Technology

• All students are required to have a TI-83/TI-83+ or TI-84 graphing calculator for use in class, at home and on examinations.  The use of the graphing calculator is infused into the curriculum on an almost daily basis.  All students are expected to be completely familiar with the use of the graphing calculator in each area of the curriculum.
• Students have learned the use of Microsoft Excel and Minitab statistics software.  Software is available on the school’s computer labs, which are used on occasion at teacher’s discretion.

Students are also introduced to various websites and applets to demonstrate statistical concepts graphically.

Chapter 1 – Exploring Data [1 ½ weeks]

 Content Homework Additional Resources Assessments Displaying Distributions with Graphs Using graphs to describe both categorical and quantitative variables.  Collecting information on a number of individuals.  For each individual, the data gives values of one or more variables.  Creating bar graphs, pie-charts, dotplots, steam and leaf plots and histograms. Examining graphs for an overall pattern, describing center, spread, shape, gaps, clusters and outliers. Describing distributions as symmetric or skewed. Ex. 1.10, 1.16, 1.17, 1.23, 1.25, 1.27, 1.36, 1.39 Group project on constructing statistical graphs Describing Distributions with Numbers Calculating the mean, median, quartiles.  Creating a box-and-whisker plot from a five-number summary.  Using the interquartile range to determine outliers.  Calculating variance and standard deviation.  Determining the effects of linear transformations on measures of center and spread. Ex. 1.41, 1.45, 1.46, 1.52, 1.54, 1.55 Calculating statistics and producing graphs using Minitab and Excel Test 1-1 Additional activities: Students will use real world data to construct bar graphs, pie charts, dotplots, stem and leaf plots and histograms. Students will compare and explain the differences between different graphical displays and differences among statistical measures. Students will compare means and medians of distributions and the impact on skewness of a distribution. Students will use real world data (ages of people in classroom, including teacher) to explain resistance to outliers. Students will understand that “in context” means that students must explain their answers in the context of the problem, citing specific statistical evidence as it relates to the problem.

Chapter 2 – The Normal Distribution [2 weeks]

 Content Homework Additional Resources Assessments Density Curves and Normal Distributions The area under a density curve is equal to one.  A histogram can show the shape of a density curve.  The mean, median quartiles can be located on a density curve.  The normal distributions are a family of bell shaped curves.  The empirical rule states that 65%, 95%, 99.7% of data is within 1, 2 or 3 standard deviations of the mean. Ex. 2.1, 2.3, 2.4, 2.6, 2.7, 2.8, Website based applet on normal curves. Standard Normal Calculations The z-score of results in a standardized score from the standard normal distribution N(0,1).  Normality can be assessed by constructing a histogram, stem-and-leaf plot or a normal probability plot. Ex. 2.19, 2.20, 2.21, 2.23, 2.24, 2.31, 2.32, 2.36 Normal probability plot of baseball statistics   Test 1-2 Additional activities: Students will create density curves with an area of 1. Approximate mean and median on density curves of various shapes. Be able to compare scores on different scales (i.e. test scores) by using standardized scores. Be able to communicate the connection between mean, standard deviation and z-score. Use a normal probability plot to determine if a distribution is approximately normal. Use histogram, stem-and-leaf plots and boxplots along with normal probability plots to assess the normality of a distribution and be able to justify their conclusions using statistical evidence.

Chapter 3 – Relationships Among Bivariate Data [1 ½ weeks]

 Content Homework Additional Resources Assessments Scatter plots and Correlation Explanatory and response variables, analyzing patterns in scatterplots, time series, correlation and linearity.  Discussion of outliers vs. influential points; Ex. 3.6, 3.7, 3.9, 3.10, 3.12, 3.30, 3.32, 3.34 Constructing fitted line plots on Mintab. Group quiz on correlation. Bivariate Data Least squares regression line, residual plots, outliers, influential points, and transformations to achieve linearity. Approximately two days are spent in instruction with the TI-83 calculator. Two days are spent in the classroom for a data collection and curve-fitting lab. Pairs of students work together to generate three sets of bivariate data. Students use calculators and computers to write equations of the curves that best model their data. Ex. 3.38, 3.39, 3.42, 3.44, 3.45, 3.46, 3.52, 3.55 Test 1-3 Additional activities: Explain the differences between quantitative and categorical variables and recognize, identify and create types of variables that may cause confusion between the two. Identify the explanatory and response variables in bivariate data and situations where the two are indistinguishable. Create, interpret and explain scatter plots. Be able to use correlation coefficient and a scatter plot to explain the nature of the association between two variables. Create a line of best fit and be able to explain the meaning of the slope and intercept in context. Be able to analyze the value of coefficient of determination and its connection to correlation coefficient, including the direction of the association. Recognize the extrapolation of a line of best fit and situations in which extrapolation is dangerous. Recognize the difference between outliers and influential points and be able to explain the impact of each in context. Calculate residuals and explain the manner in which a pattern in the residual plot describes the nature of the association and the viability of the line of best fit.

Chapter 4 – More on Bivariate Data [2 weeks]

 Content Homework Additional Resources Assessments Transforming Relationships Transforming bivariate data to achieve linearity; identify the ladder of power transformations; identify exponential growth and decay models; identify and apply the natural base e; use the TI-83/84 calculator for exponential, logarithmic, power and sine regression. Ex. 4.21, 4.24, 4.27, 4.28, 4.30 Transformations on Mintab, using fitted line plots. Power regression lab Applications of Correlation and Regression Apply the regression model to understand the meaning of slope and intercept; make predictions using regression model; extrapolation and its dangers; the connection between lurking variables, common response and confounding; correlation does not imply causation; causation can best be found through a controlled experiment. Ex. 4.33, 4.34, 4.37, 4.41, 4.43, 4.53, 4.54 Write a short essay (1/2 page-1 page) comparing causation, confounding and common response Relationships in Categorical Data Create a cross tabulation and calculate marginal distributions and marginal frequencies; find conditional distributions using cross tabulation; present categorical data in bar graphs and pie charts; understand Simpson’s paradox. Ex. 4.59, 4.60, 4.61 Creating cross tabulation on Minitab. Test 1-4 Additional activities: Students will be able to explain the type of transformation that is needed and explain the reasons for using a specific type of transformation in context. Students will be able to explain the situations that result in models of exponential growth/decay and power functions in context. Be able to interpret coefficient of determination and explain why correlation coefficient only measures the linear relationship between two variables. Understand and explain the difference between correlation, causation and confounding in context. Be able to recognize when Simpson’s paradox occurs and explain its influence in context

Chapter 5 – Sampling Design [2 ½ weeks]

 Content Homework Additional Resources Assessments Sampling Design Explore relationships among variables; sampling from a population; experiments versus observational studies; designing a sample using probability based sampling; performing a simple random sample (SRS); conducting a stratified random sample, a multi-stage cluster sample; using a random digits table; avoiding bias in sampling by avoiding judgment samples or voluntary samples; avoiding response bias from undercoverage or nonresponse. Ex. 5.1, 5.3, 5.4, 5.6, 5.20, 5.23, 5.25, 5.26, 5.30 Experimental Design Treatments applied to experimental units or subjects; levels of the explanatory variable are called factors; basic principles of experimental design are control, randomization and replication; avoid confounding by comparing two or more treatments; use randomization to assign subjects to treatments; identify double-blind experiments; identify methods of block design to control variables; identify matched pairs designs. Ex. 5.32, 5.33, 5.39, 5.41, 5.42, 5.43, 5.44, 5.46, 5.47 The Pepsi Challenge Class project – constructing a sample to predict results of school government elections. Simulation of Experiments Simulate experiments using TI-83/84 calculator using random integer functions; perform a Monte Carlo simulation. Ex. 5.51, 5.53, 5.55, 5.56, 5.59, 5.60, 5.62, 5.63 Performing simulations on Microsoft Excel Test 2-1 Additional activities: Be able to explain the difference between samples and populations and identify the measures (parameters/statistics) used for each. Be able to explain and discuss the differences among various types of sampling and explain the potential biases for each type of sampling in context. Be able to describe situations exemplary of various types of sampling. Recognize and be able to describe the difference between an observational study and an experiment and be able to give examples of each. Recognize and be able to describe the placebo effect in context, also be able to describe when a relationship is a double blind experiment. Be able to create block design experiments and comment on the appropriateness of the model. Be able to conduct a matched pairs experiment and describe the statistical validity of such an experiment. Be able to comment on the conditions that can lead us to conclude that a cause-and-effect relationship exits.

Chapter 6 – Probability [2 weeks]

 Content Homework Additional Resources Assessments Basics of Probability Understand the idea of randomness; independence of trials; understand the difference between theoretical and empirical probability. 6.2, 6.4, 6.5, 6.9, 6.34, 6.35, 6.40, 6.41, 6.44 Probability Lab – dice, cards, coins, folded card. Probability Models Identify the sample size of a random event; find the compliment of an event; identify mutually exclusive and independent events; calculating conditional probability; use of a tree diagram to answer probability problems. Ex. 6.46, 6.47, 6.52, 6.54, 6.55, 6.58, 6.61 Monty’s Dilemma Simulation of Probability Simulate experiments using TI-83/84 calculator using random integer functions; perform a Monte Carlo simulation. Ex. 6.62, 6.64, 6.68, 6.69, 6.73, 6.78, 6.79, 6.82, 6.85, 6.87 Performing simulations on Microsoft Excel Test 2-2 Additional activities: Be able to communicate the connection between probability and a density function (both have an area of 1) Determine if a probability assignment is valid and communicate the reasons if it is not. Understand and explain when events are disjoint or independent. Be able to determine when to use the different formulas for probability depending upon disjointedness or independence. Recognize the connection between Venn diagrams cross tabulations to set up probability models. Students will be able to calculate conditional probabilities and convert word problems into conditional probabilities and explain the answers in context. Set up tree diagrams to calculate the probabilities of multiple stage events and calculate probabilities of each outcome and use those outcomes to calculate conditional probabilities. Students will be able to communicate answers to probability questions in context.

Chapter 7 – Random Variables [2 weeks]

 Content Homework Additional Resources Assessments Discrete and Continuous Random Variables A random variable is the possible outcomes of a random phenomenon; discrete random variable has a countable number of possible values; a continuous random variable takes on all values over and interval and can be described by a density curve; a normal distribution is a mound-shaped, symmetric continuous probability distribution. Ex. 7.10, 7.11, 7.14, 7.15, 7.20, 7.22, 7.24, 7.25, 7.29, 7.31, 7.32, 7.33 Probability function extra credit package Means and variances of random variables. The mean is the expected value and is the sum of the average values of x, multiplied by the probability.  The law of large numbers states that the average values of X observed over many trials will approach the population mean. Ex. 7.34, 7.36, 7.37, 7.41, 7.54, 7.60, 7.63, 7.66, 7.67 Estimating expected values using computer simulations. Test 2-3 Additional activities: Be able to recognize and describe discrete random variables and set up probability tables. Be able to recognize and define continuous random variables and connect their probabilities to density curves, explaining the meaning of the probability in context. Calculate expected value of events, including games of chance. Use simulation techniques to predict the probabilities of random events and explain how simulations will relate to theoretical probabilities using the law of large numbers. Calculate mean and standard deviation of random events and transform those probabilities.

Chapter 8 – Binomial and Geometric Distributions [1 ½ weeks]

 Content Homework Additional Resources Assessments Binomial Distribution Each observation in a binomial setting is independent; formulas association with the binomial distribution and its mean and standard deviation; the normal approximation of the binomial under given circumstances. Calculating probabilities for the exact number of successes or problems with at least or at most a given number of successes. 8.1, 8.2, 8.3, 8.4, 8.6, 8.7, 8.8, 8.19, 8.20, 8.27, 8.29, 8.32, 8.33, 8.34 Binomial Distribution Group Quiz Geometric Distribution Identify a geometric variable by verifying four conditions: two outcomes (success and failure); the same probability for each trial; trials are independent; the variable of interest is the number of trials required to achieve the first success.  Determining the mean and standard deviation of a geometric random variable. Ex. 8.37, 8.38, 8.39, 8.40, 8.49, 8.50, 8.54 Test 2-4 Additional activities: Calculate probabilities of binomial random variables and connect those probabilities to the context of the problem. Use TI-83/84 to calculate exact and cumulative binomial and geometric probabilities. Use the normal approximation of the binomial and recognize the conditions that need to be met in order to use the approximation. Identify and describe the conditions necessary for a binomial or geometric situation.

Chapter 9 – Sampling Distributions [2 ½ weeks]

 Content Homework Additional Resources Assessments Sampling Distributions Students will identify statistics and parameters in a sample or experiment.  Statistic will take on different values in repeated sampling.  Bias and variability of a statistic is measured in terms of the mean and spread of the sampling distribution.  The variability of a sample decreases as the sample size increases. Ex. 9.5, 9.6, 9.7, 9.8, 9.9, 9.10, 9.12, 9.13, 9.15, 9.17 Group project using dice to simulate central limit theorem Sample Proportions Recognizing a sampling proportion.  Finding the mean and standard deviation of a sample proportion from a sample of size n from a given population.  The standard deviation decreases by a factor of the square root of n.  Normal approximation to calculate probabilities similar to the binomial distribution. Ex. 9.19, 9.22, 9.23 a & b, 9.31, 9.32, 9.34, 9.35, Central Limit Theorem Applet Sample means Recognizing a sampling mean.  Finding the mean and standard deviation of a sample proportion from a sample of size n from a given population mean.  The standard deviation decreases by a factor of the square root of n.  Using the central limit theorem and using the normal approximation to calculate probabilities concerning the sample mean. Ex. 9.36, 9.37 9.39, 9.40, 9.41 Test 2-5 Additional activities: Recognize that a statistic will take on different values dependent upon the members of the sample. Interpret a sampling distribution as the values taken on by the statistic. Understand and be able to communicate the concept that the standard deviation of a sampling distribution will be decreased by a factor of , and that the same will be true of the standard deviation of sampling distribution of proportions. Recognize and be able to identify problems that involve a sample mean or a sample proportion. Understand that the central limit theorem indicates that a sampling distribution of sample means will approximate a normal distribution. Be able to explain the application of the central limit theorem in the context of a given problem.

Chapter 10 – Tests of Significance [2 weeks]

 Content Homework Additional Resources Assessments Confidence Intervals Understanding the meaning of a confidence interval.  Calculate a confidence interval from a mean with a normal population and a known standard deviation.  Recognize that the sample size must be large enough if the distribution is non-normal to construct an appropriate confidence interval. Understand how the margin error changes based on the sample size and the level of confidence.  Determining the sample size for a specified margin of error. Ex. 10.5, 10.6, 10.7, 10.8, 10.9, 10.12, 10.13, 10.16, 10.19, 10.22, 10.23, 10.27, 10.28, 10.33, Significance Testing Able to state the null and alternative hypothesis concerning a population mean.  Ability to calculate the one-sample z-statistic for one-sided and two-sided tests.  Determine whether or not the null hypothesis should be rejected based on a given level of significance.  Explain Type I and Type II error and power of a significance test. Ex. 10.36, 10.37, 10.38, 10.39, 10.43, 10.46, 10.54, 10.62, 10.63, 10.64, 10.66, 10.67, 10.68, 10.70, 10.71, 10.72 Test 3-1 Additional activities: Be able to explain the meaning of a confidence interval to someone who does not necessarily understand statistics. Be able to calculate a confidence interval and explain the meaning of the interval in context. Be able to explain the change in the confidence interval as sample size and level of confidence change. Find the sample size necessary for a given margin of error and round the result correctly, as well as explain the sample size in context. State the null and alternative hypothesis in symbols and words.   Explain the meaning of p-value in context. Draw conclusions in context using specific statistical evidence from the data given. Explain the meaning of Type I and Type II error and power in context.

Chapter 11 – Inference for Distributions [1 ½ weeks]

 Content Homework Additional Resources Assessments Inference for the Mean of a Population Understanding the meaning of a confidence interval.  Be able to read and interpret a t-table and determine the appropriate number of degrees of freedom.  Calculate a confidence interval from a mean with a normal population and an unknown standard deviation.  Using the tprocedure to obtain a confidence interval and conduct a hypothesis test.  Recognize that matched pairs data uses a one-sample tprocedure comparing the difference between the two means. Ex. 11.2, 11.5, 11.7, 11.10, 11.12, 11.24, 11.26, 11.32, 11.34 Using student responses in before an after tasks to simulate a matched pairs experiment. Hypothesis Testing Group Quiz. Comparing Two Means Construct a confidence interval for the difference between two means, using either a conservative estimate of degrees of freedom or using a TI-83/84 calculator.  Test the hypothesis that two populations have equal means versus either a one-sided or two-sided alternative. Ex. 11.40 & 11.42, 11.50, 11.53, 11.54 Test 3-2 Additional activities: Recognize when a problem requires the use of a tdistribution rather than a z distribution. Students will be able to completely communicate the hypothesis testing procedure for a t-test.

Chapter 12 – Inference for Proportions [1 ½ weeks]

 Content Homework Additional Resources Assessments Inference for a Population Proportion When the sample size is large enough to meet the conditions of a normal distribution, a zconfidence interval can be constructed. Determine the minimum sample size needed for a given margin of error when the sample proportion can be estimated or when the sample proportion must be estimated as 0.5. Ex. 12.2, 12.4, 12.5,12.6, 12.8, 12.10, 12.12 Comparing Two Proportions Compare the difference between two proportions based on the difference of the sample proportion of successes.  Construct a confidence interval for the difference of two proportions.  Perform a test of significance with the null hypothesis that the population proportions are equal, using the pooled sample proportion. Ex. 12.22, 12.24, 12.26 Test 3-3 Additional activities: Students will be able to recognize from the design of a study whether a one-sample, matched pairs or two-sample procedure are needed. Students will be able to completely communicate the hypothesis testing procedure for a proportion.

Chapter 13 – Inference for Tables [1 week]

 Content Homework Additional Resources Assessments Test for Goodness of Fit Perform a chi-square test for goodness of fit that test the null hypothesis that a population distribution is the same as a reference distribution.  Find the expected count for each variable category.  Determine the chi-square statistic and compare with the critical value for the appropriate number of degrees of freedom.  Test the null hypothesis that the population proportions equal the hypothesized values. Ex. 13.1, 13.2, 13.10, 13.11, 13.12 Inference for Two-Way Tables Be able to conduct a chi-square test for homogeneity of populations and a chi-square test for independence.  Calculate the expected count for each cell in a contingency table. Calculate the chi-square statistic from a contingency table and compare with the critical value for the appropriate number of degrees of freedom.  Test the null hypothesis that the categorical variables are independent. Ex. 13.26, 13.28, 13.30 Test 3-4 Additional activities: Students will be able to distinguish between test of homogeneity of populations and tests of association/independence. Be able to perform chi-squared tests and explain the results in context, including the value of the chi-square statistic and the associated p-value. Students will be able to identify from a table where the largest deviation from expected values occurs and explain this in context.

Chapter 14 – Inference for Regression [1 ½ weeks]

 Content Homework Additional Resources Assessments Inference About the Model Understand that there is a true regression line that describes how the mean response varies as xchanges.  Determine the number of degrees of freedom as n-2.  Construct a confidence interval for the slope of the true regression line.  Test the hypothesis that the true slope is zero using a tstatistic. Ex.  14.1, 14.2, 14.4, 14.6, 14.7, 14.8 Predictions and Conditions Construct a confidence interval of a prediction interval for any response variable.  We must have independent observations, a true linear relationship, the true line must have a constant standard deviation which varies normally around the true regression line. Ex. 14.12, 14.14, 14.15, 14.18 Test 4-1 Additional activities: Students will be able to identify the type of inference needed in a particular regression setting. Identify and communicate situations where inference may not be appropriate. Students will be able to perform the inference on the TI-83/84 and draw conclusions with citing specific statistical evidence.

QUARTERLY PROJECTS

First Quarter – Univariate data – Students will choose their own set of univariate data and perform a statistical analysis on the data, including measures of central tendency and spread; at least two graphical displays; mention of potential outliers. Students will communicate and create a professional display of their data.

Scoring Rubric

 0 points 1 point 2 points 3 points Data Collection Data is not presented adequately at all. Data is not clearly labeled and it is not stated where data came from. Data is presented well, but not clearly labeled or not stated where data came from. Data is presented clearly and it is clearly indicated where the data came from. Statistical Analysis Data is not analyzed adequately at all. Data is analyzed with insufficient number of statistics and several mistakes are made. Data is analyzed, but not a sufficient number of statistics are used or mistakes are made. Data is analyzed with at least 3 measures of central tendency and at least 3 measures of spread. Interpretation Student displays no understanding of the concepts behind the statistical measures. Several mistakes are made; student shows minimal understanding of the concepts behind the statistical measures. Three or fewer mistakes are made in interpretation of data; student shows an acceptable understanding of the concepts behind the statistical measures. Student indicates through data interpretation an understanding of statistical measures. Presentation Presentation is unacceptable.  No organization, numerous errors. Presentation is unorganized; statistics or graphs not labeled. Presentation is adequate, statistics or graphs are not clearly labeled. Presentation is professional and easy to read; appropriate graphs are included, statistics a

Second Quarter – Bivariate data – Students will choose their own set of bivariate data and perform a regression analysis on the data.  Students will communicate and create a professional display of their data.

Scoring Rubric

 0 points 1 point 2 points 3 points Data Collection Data is not presented adequately at all. Data is not clearly labeled and it is not stated where data came from. Data is presented well, but not clearly labeled or not stated where data came from. Data is presented clearly and it is clearly indicated where the data came from. Statistical Analysis Regression analysis is not conducted adequately at all. Regression analysis is conducted insufficiently and several mistakes are made. Regression analysis is conducted , but with one or two mistakes. Regression analysis is conducted completely, including correct identification of the dependent and independent variables. Interpretation Student displays no understanding of the concepts behind regression analysis. Several mistakes are made; student shows minimal understanding of the concepts behind regression analysis. Three or fewer mistakes are made in interpretation of data; student shows an acceptable understanding of the concepts behind the regression analysis. Student indicates through data interpretation an understanding of regression analysis. Presentation Presentation is unacceptable.  No organization, numerous errors. Presentation is unorganized; statistics or graphs not labeled. Presentation is adequate, statistics or graphs are not clearly labeled. Presentation is professional and easy to read; appropriate graphs are included, statistics are clearly labeled.

Third Quarter – Hypothesis Testing – Students will choose their own set of data upon which to perform a hypothesis test.  Students will communicate and create a professional display of their data.

Scoring Rubric

 0 points 1 point 2 points 3 points Data Collection Data is not presented adequately at all. Data is not clearly labeled and it is not stated where data came from. Data is presented well, but not clearly labeled or not stated where data came from. Data is presented clearly and it is clearly indicated where the data came from. Statistical Analysis Hypothesis test is not conducted adequately at all. Hypothesis test is conducted insufficiently and several mistakes are made. Hypothesis test is conducted , but with one or two mistakes. Hypothesis test is conducted completely, including correct identification of test and statement of hypotheses and conditions. Conclusion Interpretation Student displays no understanding of the concepts behind hypothesis testing. Several mistakes are made; student shows minimal understanding of the concepts behind hypothesis testing. Three or fewer mistakes are made in interpretation of data; student shows an acceptable understanding of the concepts behind hypothesis testing. Student indicates through data interpretation an understanding of hypothesis testing. Presentation Presentation is unacceptable.  No organization, numerous errors. Presentation is unorganized; statistics or graphs not labeled. Presentation is adequate, statistics or graphs are not clearly labeled. Presentation is professional and easy to read; appropriate graphs are included, statistics are clearly labeled.