AP Statistics

College Board CED for AP Statistics

• VAR-1.A Identify questions to be answered, based on variation in one-variable data.Identify variables in a set of data.
• VAR-1.B Identify variables in a set of data.Classify types of variables.
• VAR-1.C Classify types of variables.
• UNC-1.A Represent categorical data using frequency or relative frequency tables.
• UNC-1.B Describe categorical data represented in frequency or relative tables.
• UNC-1.C Represent categorical data graphically.
• UNC-1.D Describe categorical data represented graphically.
• UNC-1.E Compare multiple sets of categorical data.
• UNC-1.F Classify types of quantitative variables.
• UNC-1.G Represent quantitative data graphically.
• UNC-1.H Describe the characteristics of quantitative data distributions.
• UNC-1.I Calculate measures of center and position for quantitative data.
• UNC-1.J Calculate measures of variability for quantitative data.
• UNC-1.K Explain the selection of a particular measure of center and/or variability for describing a set of quantitative data.
• UNC-1.L Represent summary statistics for quantitative data graphically.
• UNC-1.M Describe summary statistics of quantitative data represented graphically.
• UNC-1.N Compare graphical representations for multiple sets of quantitative data.
• UNC-1.O Compare summary statistics for multiple sets of quantitative data.
• VAR-2.A Compare a data distribution to the normal distribution model.
• VAR-2.B Determine proportions and percentiles from a normal distribution.
• VAR-2.C Compare measures of relative position in data sets.
   

A variable is a characteristic that changes from one individual to another.

Variables can be categorical or quantitative.

Percentages, relative frequencies, and rates all provide the same information as proportions.

A discrete variable can take on a countable number of values. 

A continuous variable can take on infinitely many values.

A cumulative graph represents the number or proportion of a data set less than or equal to a given number.

Outliers for one-variable data are data points that are unusually small or large relative to the rest of the data.

A distribution is skewed to the right (positive skew) if the right tail is longer than the left. 

A distribution is skewed to the left (negative skew) if the left tail is longer than the right. 

A distribution is symmetric if the left half is the mirror image of the right half.

Univariate graphs with one main peak are known as unimodal.

A gap is a region of a distribution between two data values where there are no observed data.

Clusters are concentrations of data usually separated by gaps.

A statistic is a numerical summary of sample data.

The mean is the sum of all the data values divided by the number of values.

The median of a data set is the middle value when data are ordered. 

The first quartile, Q1, is the median of the half of the ordered data set from the minimum to the position of the median. 

The third quartile, Q3, is the median of the half of the ordered data set from the position of the median to the maximum. 

Range = maximum – minimum. 

The interquartile range (IQR) = Q3 − Q1.

Standard deviation is a way to measure variability of the distribution of a quantitative variable.

The sample variance is the square of the sample standard deviation, s2.

If a distribution is relatively symmetric, then the mean and median are relatively close to one another.

If a distribution is skewed right, then the mean is usually to the right of the median. 

If the distribution is skewed left, then the mean is usually to the left of the median.

The mean, standard deviation, and range are considered nonresistant (or non-robust).

The median and IQR are considered resistant (or robust).

The five-number summary is the minimum data value, the first quartile (Q1), the median, the third quartile (Q3), and the maximum data value.

A parameter is a numerical summary of a population.

A normal curve is mound-shaped and symmetric. 

The parameters of a normal distribution are the population mean, μ, and the population standard deviation, σ.

The empirical rule for a normal distribution.

 

• Given that variation may be random or not, conclusions are uncertain.
• Graphical representations and statistics allow us to identify and represent key features of data.
• The normal distribution can be used to represent some population distributions.
   

• Summarize the distribution of data with a frequency table and/or a relative frequency table.
• Display the distribution of categorical data with a bar chart (or bar graph) of  frequencies (counts) or relative frequencies.
• Justify claims about data in context using counts and relative frequencies of categorical data.
• Compare two or more data sets in terms of the same categorical variable using frequency tables, bar graphs, or other representations.
• Display the distribution of a quantitative data with a dotplot, a stem and leaf plot, and/or a histogram.
• Describe the distribution of quantitative data: shape, center, and variability (spread), as well as any unusual features.
• Compute the mean and the median of quantitative data to measure the center of the values.
• Compute the range, interquartile range (IQR), and standard deviation of quantitative data to measure the variability of the values.
• Calculate and interpret the percentile for a value.
• Determine how changing units of measurement affects the values of calculated statistics.
• Determine outliers in quantitative data using the 1.5 × IQR rule.
• Determine outliers in quantitative data using the 2 standard deviations rule.
• Display the distribution of quantitative data with a boxplot.
• Summary statistics of quantitative data, or of sets of quantitative data, can be used to justify claims about the data in context.
• Compare two or more independent samples on center, variability, clusters, gaps, outliers, and other features using graphs.
• Compare two or more independent samples using numerical summaries.
• Calculate and interpret a z-score (one type of standardized score) for a value in a set of quantitative data.
• Find the proportion of data values located on a given interval of a normally distributed random variable using technology, such as a calculator, a standard normal table, or computer-generated output.
• Use technology to estimate parameters for some populations given the area of a region under the graph of the normal distribution curve.
• Compare relative positions of points within a data set or between data sets using percentiles and/or z-scores.

• How can I represent data visually and numerically?
• How can I compare two groups of data for the same variable?
• Where do values fall, relative to other values?
• How can mathematics model distributions of unimodal, symmetric data?

• What are the two types of variables?
• How can we summarize the distribution of a categorical variable?
• How can we visualize the distribution of a categorical variable?
• How can we visualize the distribution of a quantitative variable?
• How can we describe the distribution of a quantitative variable?
• What is the center of the distribution of a quantitative variable?
• How much do the values of a quantitative variable vary?
• How are the distributions of these variables different and the same?
• Where is the relative location of a value in a distribution? 
• Where is the relative location of a value in a normal distribution? 

• Textbook
• Graphing calculator
• Online statistical applets (e.g. https://www.stapplet.com/)

• Data
• Observational UnitVariability
• Categorical variable
• Quantitative variable
• Discrete variable
• Continuous variable
• Distribution
• Bar graph
• Dotplot
• Histogram
• Stemplot
• Boxplot
• Skewed left
• Skewed right
• Symmetric
• Unimodal
• Statistic
• Mean
• Median
• Standard deviation
• Range
• Interquartile range (IQR)
• Outliers
• Parameter 
• Normal distribution
• Standardized score
• Z-score

• Unit quizzes
• Unit tests
• Informal and formal writing assignments
• Presentations
• Informal check of daily assignments
• Progress check (College Board)
   

College Board CED for AP Statistics

• VAR-1.D Identify questions to be answered about possible relationships in data.
• UNC-1.P Compare numerical and graphical representations for two categorical variables.
• UNC-1.Q Calculate statistics for two categorical variables.
• UNC-1.R Compare statistics for two categorical variables.
• UNC-1.S Represent bivariate quantitative data using scatterplots.
• DAT-1.A Describe the characteristics of a scatter plot.
• DAT-1.B Determine the correlation for a linear relationship.
• DAT-1.C Interpret the correlation for a linear relationship.
• DAT-1.D Calculate a predicted response value using a linear regression model.
• DAT-1.E Represent differences between measured and predicted responses using residual plots.
• DAT-1.F Describe the form of association of bivariate data using residual plots.
• DAT-1.G Estimate parameters for the least-squares regression line model.
• DAT-1.H Interpret coefficients for the least-squares regression line model.
• DAT-1.I Identify influential points in regression.
• DAT-1.J Calculate a predicted response using a least-squares regression line for a transformed data set.

• Apparent patterns and associations in data may be random or not.
• Determine if two categorical variables are associated with graphical representations of two categorical variables.
• In a two-way table, a joint relative frequency is a cell frequency divided by the total for the entire table.
• Summary statistics for two categorical variables can be used to compare distributions and/or determine if variables are associated.
• A bivariate quantitative data set consists of observations of two different quantitative variables made on individuals in a sample or population.
• An explanatory variable is a variable whose values are used to explain or predict corresponding values for the response variable.
• The direction of the association shown in a scatterplot, if any, can be described as positive or negative.
• A positive association means that as values of one variable increase, the values of the other variable tend to increase. 
• A negative association means that as values of one variable increase, values of the other variable tend to decrease.
• The form of the association shown in a scatterplot, if any, can be described as linear or non-linear to varying degrees.
• The strength of the association is how closely the individual points follow a specific pattern, e.g., linear, and can be shown in a scatterplot.
• Strength can be described as strong, moderate, or weak.
• Unusual features of a scatter plot include clusters of points or points with relatively large discrepancies between the value of the response variable and a predicted value for the response variable.
• The correlation, r, gives the direction and quantifies the strength of the linear association between two quantitative variables.
• A correlation coefficient close to 1 or −1 does not necessarily mean that a linear model is appropriate.
• The correlation, r, is unit-free.
• The correlation, r, is between −1 and 1, inclusive. 
• A value of r = 0 indicates that there is no linear association.
• A value of r = 1 or r = –1 indicates that there is a perfect linear association.
• A perceived or real relationship between two variables does not mean that changes in one variable cause changes in the other. 
• Correlation does not necessarily imply causation.
• A simple linear regression model is an equation that uses an explanatory variable, x, to predict the response variable, y.
• The predicted response value, denoted by y, is calculated as y = a +bx.
• a is the y-intercept of the regression line.
• b is the slope of the regression line.
• Extrapolation is predicting a response value using a value for the explanatory variable that is beyond the interval of x-values used to determine the regression line.
• The residual is the difference between the actual value and the predicted value.
• A residual plot is a plot of residuals versus explanatory variable values or predicted response values.
• Apparent randomness in a residual plot for a linear model is evidence of a linear form to the association between the variables.
• The least-squares regression model minimizes the sum of the squares of the residuals.
• The least-squares regression model contains the point (x, y).
• The slope is the amount that the predicted y-value changes for every unit increase in x.
• The y-intercept value is the predicted value of the response variable when the explanatory variable is equal to 0.
• An outlier in regression is a point that does not follow the general trend shown in the rest of the data and has a large residual when the Least Squares Regression Line (LSRL) is calculated.
• A high-leverage point in regression has a substantially larger or smaller x-value than the other observations have.
• An influential point in regression is any point that, if removed, changes the relationship substantially.
• Transformations of variables can be used to create transformed data sets, which may be more linear in form than the untransformed data.

• Given that variation may be random or not, conclusions are uncertain.
• Graphical representations and statistics allow us to identify and represent key features of data.
• Regression models may allow us to predict responses to changes in an explanatory variable.

• Display the relationship between two categorical variables with side-by-side bar graphs.
• Display the relationship between two categorical variables with segmented bar graphs.
• Display the relationship between two categorical variables with mosaic plots.
• Organize data from two categorical variables with a two-way table.
• Compute marginal relative frequencies in two-way tables.
• Compute conditional relative frequencies in two-way tables.
• Construct a scatterplot.
• Describe the association between two variables in a scatterplot.
• Interpret a correlation coefficient r.
• Compute a least-squares regression equation.
• Predict a value of the response variable using a least-squares regression equation.
• Compute and interpret a residual.
• Interpret the slope b of a least-squares regression equation.
• Interpret the y-intercept a of a least-squares regression equation.
• Sometimes, the y-intercept of the line does not have a logical interpretation in context.
• Investigate the appropriateness of a selected model using a residual plot.
• Interpret the coefficient of determination r2.
• Interpret the standard deviation of the residuals s..
• Determine if a transformation of data results in an improved  least-squares regression model. 

• Is there an association between two variables?
• How can the association between two variables be quantified?
• How can associations between variables be used to make predictions?
   

• How are data from two categorical variables organized?
• Is there an association between two categorical variables?
• How are data from two quantitative variables organized?
• Is there an association between two quantitative variables?
• Can the relationship between two quantitative variables be modeled?
• Is the least-squares regression model adequate?
• How is the least-squares regression model interpreted?
• How can departures from linearity be interpreted?
   

• Textbook
• Graphing calculator
• Online statistical applets (e.g. https://www.stapplet.com/)
   

• Explanatory variable
• Response variable
• Two-way table (contingency table)
• Marginal distribution
• Conditional distribution
• Association
• Statistical tendency
• Scatterplot
• Direction
• Form
• Strength
• Unusual features
• Influential point
• Outlier
• High leverage point
• Correlation coefficient
• Least squares regression line
• Slope coefficient
• Intercept coefficient
• Residual
• Residual plot
• Coefficient of determination
• Standard deviation of residuals

• Unit quizzes
• Unit tests
• Informal and formal writing assignments
• Presentations
• Informal check of daily assignments
• Progress check (College Board)
   

College Board CED for AP Statistics

• VAR-1.E Identify questions to be answered about data collection methods.
• DAT-2.A Identify the type of a study.
• DAT-2.B Identify appropriate generalizations and determinations based on observational studies.
• DAT-2.C Identify a sampling method, given a description of a study.
• DAT-2.D Explain why a particular sampling method is or is not appropriate for a given situation.
• DAT-2.E Identify potential sources of bias in sampling methods.
• VAR-3.A Identify the components of an experiment.
• VAR-3.B Describe elements of a well-designed experiment.
• VAR-3.C Compare experimental designs and methods.
• VAR-3.D Explain why a particular experimental design is appropriate.
• VAR-3.E Interpret the results of a well-designed experiment.
   

• Methods for data collection that do not rely on chance result in untrustworthy conclusions.
• A population consists of all items or subjects of interest.
• A sample selected for study is a subset of the population.
• In an observational study, treatments are not imposed. 
• In a retrospective study, investigators examine data for a sample of individuals.
• In a prospective study, investigators follow a sample of individuals into the future, collecting data.
• A sample survey collects data from a sample in an attempt to learn about the population from which the sample was taken.
• In an experiment, different conditions (treatments) are assigned to experimental units (participants or subjects).
• It is only appropriate to make generalizations about a population based on samples that are randomly selected or otherwise representative of that population.
• A sample is only generalizable to the population from which the sample was selected.
• It is not possible to determine causal relationships between variables using data collected in an observational study
• When an item from a population can be selected only once, this is called sampling without replacement. 
• When an item from the population can be selected more than once, this is called sampling with replacement.
• A simple random sample (SRS) is a sample in which every group of a given size has an equal chance of being chosen.
• A stratified random sample involves the division of a population into separate groups, called strata, based on shared attributes or characteristics (homogeneous grouping).
• A cluster sample involves the division of a population into smaller groups, called clusters.
• A systematic random sample is a method in which sample members from a population are selected according to a random starting point and a fixed, periodic interval.
• A census selects all items/subjects in a population.
• There are advantages and disadvantages for each sampling method depending upon the question that is to be answered and the population from which the sample will be drawn.
• Bias occurs when certain responses are systematically favored over others.
• Voluntary response bias may occur when a sample consists of volunteers or people who choose to participate.
• Undercoverage bias may occur when part of the population has a reduced chance of being included in the sample.
• Nonresponse bias may occur when individuals chosen for the sample for whom data cannot be obtained differ from those for whom data can be obtained .
• Problems in the data gathering instrument or process result in response bias.
• Non-random sampling methods introduce potential for bias because they do not use chance to select the individuals.
• The experimental units are the individuals that are assigned treatments. 
• An explanatory variable (or factor) in an experiment is a variable whose levels are
• manipulated intentionally. 
• The levels or combination of levels of the explanatory variable(s) are called treatments.
• A response variable in an experiment is an outcome from the experimental units that is measured after the treatments have been administered.
• A confounding variable in an experiment is a variable that is related to the explanatory variable and influences the response variable.
• A confounding variable may create a false perception of association between explanatory and response variables
• A well-designed experiment should include the following: comparison, random assignment/allocation of treatments, replication, and control.
• There are advantages and disadvantages for each experimental design.
• In a completely randomized design, treatments are assigned to experimental units completely at random. 
• Random assignment tends to balance the effects of uncontrolled (confounding) variables so that differences in responses can be attributed to the treatments.
• In a single-blind experiment, subjects do not know which treatment they are receiving, but members of the research team do, or vice versa.
• In a double-blind experiment neither the subjects nor the members of the research team who interact with them know which treatment a subject is receiving.
• A control group is a collection of experimental units either not given a treatment of interest or given a treatment with an inactive substance (placebo).
• The placebo effect occurs when experimental units have a response to a placebo.
• For randomized block designs, treatments are assigned completely at random within each block.
• Blocking ensures that at the beginning of the experiment the units within each block are similar to each other with respect to at least one blocking variable.
• A randomized block design helps to separate natural variability from differences due to the blocking variable.
• A matched pairs design is a special case of a randomized block design.
• There are advantages and disadvantages for each experimental design.
• Statistical inference attributes conclusions based on data to the distribution from which the data were collected.
• Statistically significant differences between or among experimental treatment groups are evidence that the treatments caused the effect.
• Random assignment of treatments to experimental units allows researchers to conclude that some observed changes are so large as to be unlikely to have occurred by chance.
• Statistically significant differences between or among experimental treatment groups are evidence that the treatments caused the effect.
• If the experimental units used in an experiment are representative of some larger group of units, the results of an experiment can be generalized to the larger group.

• Given that variation may be random or not, conclusions are uncertain.
• The way we collect data influences what we can and cannot say about a population.
• Well-designed experiments can establish evidence of causal relationships.

• Determine if a statistical study is an observational study or an experiment. 
• Classify observational studies.
• Describe how to select a simple random sample.
• Describe how to select a stratified random sample.
• Describe how to select a cluster random sample.
• Describe how to select a systematic random sample.
• Determine the largest population to which results in a sample can be safely generalized.
• Explain the problems in poor data collection protocols.
• Identify the experimental units, the explanatory variable(s), the response variable, and the treatments in an experiment.
• Determine the composition of the treatments in a multi-factor experiment.
• Describe a method to randomly allocate/assign treatments to experimental units.
• Design a completely randomized experiment.
• Design a randomized block experiment.
• Design a matched pairs experiment.
• Identify blinding in an experiment.
• Describe the placebo effect in an experiment.
• Explain the purpose of blocking in an experiment.
• Explain the purpose of pairing in an experiment. 
• Determine if differences in groups are statistically significant using a simulation.
• Determine if cause and effect inferences are justified based on the results and methodology of an experiment.

• What are poor data collection practices?
• Why is randomness important in data collection?
• How does the design of a statistical study relate to the inferences that can be drawn from the study?

• Which type of statistical study is being used?
• How is a representative sample obtained?
• What problems can occur when sampling?
• What are the principles of good experiments?
• How are experiments designed?
• What inferences can be drawn from statistical studies?

• Textbook
• Graphing calculator
• Online statistical applets (e.g. https://www.stapplet.com/)
   

• Population
• Sample
• Parameter
• Statistic
• Statistical inference
• Confounding variable
• Observational study
• Prospective
• Retrospective
• Experiment
• Census
• Sampling variability
• Bias
• Simple random sample
• Stratified random sample
• Cluster random sample
• Systematic sample
• Unbiased estimators
• Convenience sample
• Voluntary response sample
• Undercoverage bias
• Nonresponse
• Response bias
• Experimental units
• Factor
• Level
• Treatment
• Random assignment
• Comparison
• Replication
• Control
• Control group
• Placebo
• Placebo effect
• Blinding
• Completely randomized experiment
• Blocking
• Randomized block experiment
• Matched pairs experiment
• Statistically significant

• Unit quizzes
• Unit tests
• Informal and formal writing assignments
• Presentations
• Informal check of daily assignments
• Progress check (College Board)
   

College Board CED for AP Statistics

• VAR-1.F Identify questions suggested by patterns in data.
• UNC-2.A Estimate probabilities using simulation.
• VAR-4.A Calculate probabilities for events and their complements.
• VAR-4.B Interpret probabilities for events.
• VAR-4.C Explain why two events are (or are not) mutually exclusive.
• VAR-4.D Calculate conditional probabilities.
• VAR-4.E Calculate probabilities for independent events and for the union of two events.
• VAR-5.A Represent the probability distribution for a discrete random variable.
• VAR-5.B Interpret a probability distribution.
• VAR-5.C Calculate parameters for a discrete random variable.
• VAR-5.D Interpret parameters for a discrete random variable.
• VAR-5.E Calculate parameters for linear combinations of random variables.
• VAR-5.F Describe the effects of linear transformations of parameters of random variables.
• UNC-3.A Estimate probabilities of binomial random variables using data from a simulation.
• UNC-3.B Calculate probabilities for a binomial distribution.
• UNC-3.C Calculate parameters for a binomial distribution.
• UNC-3.D Interpret probabilities and parameters for a binomial distribution.
• UNC-3.E Calculate probabilities for geometric random variables.
• UNC-3.F Calculate parameters of a geometric distribution.
• UNC-3.G Interpret probabilities and parameters for a geometric distribution.

• Patterns in data do not necessarily mean that variation is not random.
• A random process generates results that are determined by chance.
• An outcome is the result of a trial of a random process.
• An event is a collection of outcomes.
• Simulation is a way to model random events, such that simulated outcomes closely match
• real-world outcomes.
• The relative frequency of an outcome or event in simulated or empirical data can be used to
• estimate the probability of that outcome or event.
• The law of large numbers states that simulated (empirical) probabilities tend to get closer to the true probability as the number of trials increases.
• The sample space of a random process is the set of all possible non-overlapping outcomes.
• The probability of an event is a number between 0 and 1, inclusive.
• Probabilities of events in repeatable situations can be interpreted as the relative frequency
• with which the event will occur in the long run.
• The probability that events A and B both will occur, sometimes called the joint probability, is the probability of the intersection of A and B, denoted P(A∩B).
• Two events are mutually exclusive or disjoint if they cannot occur at the same time. In other words, P(A∩B) = 0.
• The probability that event A will occur given that event B has occurred is called a conditional probability and denoted P(A | B) = P(A∩B)/P(B).
• The multiplication rule states that the probability that events A and B both will occur is P(A∩B) = P(A)⋅P(B |A).
• Events A and B are independent if, and only if, knowing whether event A has occurred (or will occur) does not change the probability that event B will occur.
• For independent events, P(A∩B) = P(A)⋅P(B).
• The probability that event A or event B (or both) will occur is the probability of the union of
• A and B, denoted P(A∪B).
• The addition rule states that the probability that event A or event B or both will occur is  P(A∪B) = P(A) + P(B) − P(A∩B).
• The values of a random variable are the numerical outcomes of random behavior.
• A discrete random variable is a variable that can only take a countable number of values.
• The sum of the probabilities over all of the possible values of a random variable must be 1.
• A probability distribution can be represented as a graph, table, or function showing the
• probabilities associated with values of a random variable.
• A cumulative probability distribution can be represented as a table or function showing the
• probability of being less than or equal to each value of the random variable.
• An interpretation of a probability distribution provides information about the shape, center, and spread of a population.
• A numerical value measuring a characteristic of a population or the distribution of a random variable is known as a parameter, which is a single, fixed value.
• Parameters for a discrete random variable should be interpreted using appropriate units and within the context of a specific population.
• A probability distribution can be constructed using the rules of probability or estimated with a simulation using random number generators.
• Two random variables are independent if knowing information about one of them does not change the probability distribution of the other.
• A binomial random variable, X, counts the number of successes in n repeated independent trials, each trial having two possible outcomes (success or failure), with the probability of success p and the probability of failure 1 – p.
• The binomial probability function is P(X = x) = nCx(p)x(1 – p)n – x.
• If a random variable is binomial, its mean μx = np and its standard deviation σx = np(1− p).
• Probabilities and parameters for a binomial distribution should be interpreted using appropriate units and within the context of a specific population or situation.
• For a sequence of independent trials, a geometric random variable, X, gives the number of the trial on which the first success occurs. Each trial has two possible outcomes (success or failure) with the probability of success p and the probability of failure 1 – p.
• The geometric  probability function is P(X = x) =(1 – p)x – 1p.
• If a random variable is geometric, its mean μx = 1/p and its standard deviation σx = (1− p)/p.
• Probabilities and parameters for a geometric distribution should be interpreted using appropriate units and within the context of a specific population or situation.

• Given that variation may be random or not, conclusions are uncertain.
• Simulation allows us to anticipate patterns in data.
• The likelihood of a random event can be quantified.
• Probability distributions may be used to model variation in populations.
• Probabilistic reasoning allows us to anticipate patterns in data.

• Calculate the probability of an event given equally likely outcomes.
• Find the probability of the complement of an event E.
• Calculate the joint probability of the event (A and B) for independent events.
• Calculate the conditional probability of the event (A | B).
• Calculate the joint probability of the event (A and B) for dependent events.
• Justify whether two events are independent.
• Calculate the probability of the event (A or B).
• Represent the probability distribution of a discrete random variable in a table.
• Represent the probability distribution of a discrete random variable with a histogram.
• Find the probability of a value of a random variable.
• Find the probability of an event defined for a random variable.
• Compute and interpret the mean of a discrete random variable.
• Compute and interpret the standard deviation of a discrete random variable.
• Find and interpret the mean of a linear combination of random variables.
• Find and interpret the standard deviation of a linear combination of independent random variables.
• Find and interpret the mean of a linear transformation of a random variable.
• Find and interpret the standard deviation of a linear transformation of a random variable.
• Justify whether a random variable is a binomial random variable.
• Calculate the probability of an event for a binomial random variable.
• Calculate and interpret the mean of a binomial random variable.
• Calculate and interpret the standard deviation of a binomial random variable.
• Justify whether a random variable is a geometric random variable with the geometric probability function and with technology.
• Calculate the probability of an event for a geometric random variable.
• Calculate and interpret the mean of a geometric random variable.
• Calculate and interpret the standard deviation of a geometric random variable.

• How certain can we be about our conclusions?
• How can simulation be used to model chance in our world?
• How can mathematics be used to model chance in our world?

• What does “random” mean?
• How is chance quantified?
• What are mutually exclusive events?
• What is the probability of an event occurring when it depends on the outcome of another event?
• What is the probability of independent events and unions of events?
• What is a random variable?
• How much do the values of random variables vary?
• How can random variables be combined?
• How can likelihoods be quantified in binomial settings?
• How can likelihoods be quantified in geometric settings?
   

• Textbook
• Graphing calculator
• Online statistical applets (e.g. https://www.stapplet.com/)
   

• Random process
• Outcome
• Event
• Relative frequency
• Law of large numbers
• Probability
• Simulation
• Complement
• Sample space
• Mutually exclusive
• Disjoint
• Joint probability
• Intersection
• Conditional probability
• The multiplication rule
• The addition rule
• Union
• Independent events
• Random variable
• Discrete random variable
• Probability distribution
• Cumulative probability distribution
• Parameter
• Mean of a random variable
• Standard deviation of a random variable
• Linear combination
• Independent random variables
• Linear transformation
• Binomial random variable
• Geometric random variable

• Unit quizzes
• Unit tests
• Informal and formal writing assignments
• Presentations
• Informal check of daily assignments
• Progress check (College Board)
   

College Board CED for AP Statistics

• VAR-6.A Calculate the probability that a particular value lies in a given interval of a normal distribution.
• VAR-6.B Determine the interval associated with a given area in a normal distribution.
• VAR-6.C Determine the appropriateness of using the normal distribution to approximate probabilities for unknown distributions.
• UNC-3.H Estimate sampling distributions using simulation.
• UNC-3.I Explain why an estimator is or is not unbiased.
• UNC-3.J Calculate estimates for a population parameter.
• UNC-3.K Determine parameters of a sampling distribution for sample proportions.
• UNC-3.L Determine whether a sampling distribution for a sample proportion can be described as approximately normal.
• UNC-3.M Interpret probabilities and parameters for a sampling distribution for a sample proportion.
• UNC-3.N Determine parameters of a sampling distribution for a difference in sample proportions.
• UNC-3.O Determine whether a sampling distribution for a difference of sample proportions can be described as approximately normal.
• UNC-3.P Interpret probabilities and parameters for a sampling distribution for a difference in proportions.
• UNC-3.Q Determine parameters for a sampling distribution for sample means.
• UNC-3.R Determine whether a sampling distribution of a sample mean can be described as approximately normal.
• UNC-3.S Interpret probabilities and parameters for a sampling distribution for a sample mean.
• UNC-3.T Determine parameters of a sampling distribution for a difference in sample means.
• UNC-3.U Determine whether a sampling distribution of a difference in sample means can be described as approximately normal.
• UNC-3.V Interpret probabilities and parameters for a sampling distribution for a difference in sample means.

• Variation in statistics for samples taken from the same population may be random or not.
• A continuous random variable is a variable that can take on any value within a specified domain.
• A continuous random variable with a normal distribution is commonly used to describe populations. 
• The distribution of a normal random variable can be described by a normal, or “bell-shaped,” curve.
• The area under a normal curve over a given interval represents the probability that a particular value lies in that interval.
• The boundaries of an interval associated with a given area in a normal distribution can be determined using z-scores or technology.
• Intervals associated with a given area in a normal distribution can be determined by assigning appropriate inequalities to the boundaries of the intervals.
• Normal distributions can be used to approximate distributions with similar characteristics.
• A sampling distribution of a statistic is the distribution of values for the statistic for all possible samples of a given size from a given population.
• The central limit theorem (CLT) states that when the sample size is sufficiently large, a sampling distribution of the mean of a random variable will be approximately normally distributed.
• The central limit theorem requires that the sample values are independent of each other and that n is sufficiently large.
• A randomization distribution is a collection of statistics generated by simulation assuming known values for the parameters.
• The sampling distribution of a statistic can be simulated by generating repeated random
• samples from a population.
• When estimating a population parameter, an estimator is unbiased if, on average, the value of the estimator is equal to the population parameter.
• When estimating a population parameter, an estimator exhibits variability that can be modeled using probability.
• A sample statistic is a point estimator of the corresponding population parameter.
• The sampling distribution of the sample proportion, p, has a mean p and a standard deviation p(1-p)n.
• If sampling without replacement, the standard deviation of the sample proportion is smaller than what is given by the formula above. 
• If the sample size is less than 10% of the population size, the difference between the formula and the actual standard deviation is negligible.
• The sampling distribution of the sample proportion, p, will have an approximate normal distribution, provided np ≥10 and n(1− p) ≥10.
• Probabilities and parameters for a sampling distribution for a sample proportion should be interpreted using appropriate units and within the context of a specific population.
• The sampling distribution of the difference in sample proportions p1-p2  has mean p1 – p2 and standard deviation p1(1-p1)n1+p2(1-p2)n2
• If sampling without replacement, the standard deviation of the difference in proportions is smaller than what is given by the formula above. 
• If the sample size is less than 10% of the population size, the difference between the formula and the actual standard deviation is negligible.
• The sampling distribution of the difference in proportions  will have an approximate normal distribution, provided n1p1 ≥10 and n1(1− p1) ≥10 and n2p2 ≥10 and n2(1− p2) ≥10.
• Parameters for a sampling distribution for a difference of proportions should be interpreted using appropriate units and within the context of specific populations.
• The sampling distribution of the sample mean x has mean μ and standard deviation σ/n.
• If sampling without replacement, the standard deviation of the sample mean is smaller than what is given by the formula above. 
• If the sample size is less than 10% of the population size, the difference between the formula and the actual standard deviation is negligible.
• The sampling distribution of the sample mean x, will have an approximate normal distribution, provided the population is approximately normal or the sample size is large (e.g n ≥ 30).
• Probabilities and parameters for a sampling distribution for a sample mean should be interpreted using appropriate units and within the context of a specific population.
• The sampling distribution of the difference in sample means x1-x2  has mean μ1 – μ2 and standard deviation 12n1+22n2
• If sampling without replacement, the standard deviation of the difference in means smaller than what is given by the formula above. 
• If the sample size is less than 10% of the population size, the difference between the formula and the actual standard deviation is negligible.
• The sampling distribution of the difference in means will have an approximate normal distribution, provided each sample comes from a normal population or the sample size is large (e.g n ≥ 30).
• Parameters for a sampling distribution for a difference of means should be interpreted using appropriate units and within the context of specific populations.

• Given that variation may be random or not, conclusions are uncertain.
• The normal distribution may be used to model variation.
• Probabilistic reasoning allows us to anticipate patterns in data.
   

• Represent the probability of an event using inequalities and symbolic notation.
• Compute the probability of an event for a normal random variable.
• Compute the values of a random variable that correspond to a percentile. 
• Explain the Central Limit Theorem in the context of a given population.
• Determine whether an estimator of a population parameter is biased or unbiased.
• Verify whether the conditions for the sampling distribution of the sample proportion p have been met.
• Compute the mean and standard deviation of the sampling distribution of the sample proportion p.
• Compute probabilities of events given the sampling distribution of a sample proportion  p.
• Verify whether the conditions for the sampling distribution of a difference in proportions p1-p2 have been met.
• Compute the mean and standard deviation of the sampling distribution of a difference in proportions p1-p2.
• Compute probabilities of events given the sampling distribution of a difference in proportions  p1-p2.
• Verify whether the conditions for the sampling distribution of the sample mean x have been met.
• Compute the mean and standard deviation of the sampling distribution of the sample mean x.
• Compute probabilities of events given the sampling distribution of a sample mean  x.
• Verify whether the conditions for the sampling distribution of a difference in means x1-x2  have been met.
• Compute the mean and standard deviation of the sampling distribution of a difference in means x1-x2 .
• Compute probabilities of events given the sampling distribution of a difference in means 
• x1-x2 .

• What is an unusual event when sampling from a population?
• What patterns occur when sampling from a population when measuring a categorical variable?
• What patterns occur when sampling from two populations when measuring categorical variables?
• What patterns occur when sampling from a population when measuring a quantitative variable?
• What patterns occur when sampling from two populations when measuring quantitative variables?
   

• How do normal curves model outcomes of random variables?
• What happens to sampling distributions when sample size increases?
• What is an unbiased estimator?
• What is the sampling distribution of a sample proportion?
• What is the sampling distribution of a difference in sample proportions?
• What is the sampling distribution of a sample mean?
• What is the sampling distribution of a difference in sample means?

• Textbook
• Graphing calculator
• Online statistical applets (e.g. https://www.stapplet.com/)

• Continuous random variable
• Normal random variable
• Sampling distribution
• Randomization distribution
• Biased estimator
• Unbiased estimator
• Sampling distribution of the sample proportion
• Random condition
• 10% condition
• Large Counts condition
• Sampling distribution of a difference in proportions
• Sampling distribution of the sample mean
• Random condition
• 10% condition
• Central Limit Theorem
• Normal/large sample condition
• Sampling distribution of a difference in means

• Unit quizzes
• Unit tests
• Informal and formal writing assignments
• Presentations
• Informal check of daily assignments
• Progress check (College Board)
   

College Board CED for AP Statistics

• VAR-1.H Identify questions suggested by variation in the shapes of distributions of samples taken from the same population.
• UNC-4.A Identify an appropriate confidence interval procedure for a population proportion.
• UNC-4.B Verify the conditions for calculating confidence intervals for a population proportion.
• UNC-4.C Determine the margin of error for a given sample size and an estimate for the sample size that will result in a given margin of error for a population proportion.
• UNC-4.D Calculate an appropriate confidence interval for a population proportion.
• UNC-4.E Calculate an interval estimate based on a confidence interval for a population proportion.
• UNC-4.F Interpret a confidence interval for a population proportion.
• UNC-4.G Justify a claim based on a confidence interval for a population proportion.
• UNC-4.H Identify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population proportion.
• VAR-6.D Identify the null and alternative hypotheses for a population proportion.
• VAR-6.E Identify an appropriate testing method for a population proportion.
• VAR-6.F Verify the conditions for making statistical inferences when testing a population proportion.
• VAR-6.G Calculate an appropriate test statistic and p-value for a population proportion.
• DAT-3.A Interpret the p-value of a significance test for a population proportion.
• DAT-3.B Justify a claim about the population based on the results of a significance test for a population proportion.
• UNC-5.A Identify Type I and Type II errors.
• UNC-5.B Calculate the probability of a Type I and Type II errors.
• UNC-5.C Identify factors that affect the probability of errors in significance testing.
• UNC-5.D Interpret Type I and Type II errors.
• UNC-4.I Identify an appropriate confidence interval procedure for a comparison of population proportions.
• UNC-4.J Verify the conditions for calculating confidence intervals for a difference between population proportions.
• UNC-4.K Calculate an appropriate confidence interval for a comparison of population proportions.
• UNC-4.L Calculate an interval estimate based on a confidence interval for a difference of proportions.
• UNC-4.M Interpret a confidence interval for a difference of proportions.
• UNC-4.N Justify a claim based on a confidence interval for a difference of proportions.
• VAR-6.H Identify the null and alternative hypotheses for a difference of two population proportions.
• VAR-6.I Identify an appropriate testing method for the difference of two population proportions.
• VAR-6.J Verify the conditions for making statistical inferences when testing a difference of two population proportions.
• VAR-6.K Calculate an appropriate test statistic for the difference of two population proportions.
• DAT-3.C Interpret the p-value of a significance test for a difference of population proportions.
• DAT-3.D Justify a claim about the population based on the results of a significance test for a difference of population proportions

• The appropriate confidence interval procedure for a one-sample proportion for one categorical variable is a one sample z-interval for a proportion.
• In order to make assumptions necessary for inference on population proportions, means, and slopes, we must check for independence in data collection methods and for selection of the appropriate sampling distribution.
• The standard error of p is p(1-p)n.
• A margin of error gives how much a value of a sample statistic is likely to vary from the value of the corresponding population parameter.
• The margin of error of  p is z*p(1-p)n.
• The formula for margin of error can be rearranged to solve for n, the minimum sample size needed to achieve a given margin of error.
• In general, an interval estimate can be constructed as a point estimate ± (margin of error).
• For a one sample z-interval for a proportion, the interval estimate is p ± z*p(1-p)n.
• Critical values represent the boundaries encompassing the middle C% of the standard normal distribution, where C% is an approximate confidence level for a proportion.
• Confidence intervals for population proportions can be used to calculate interval estimates with specified units.
• Confidence intervals are based on data from random samples, which vary from sample to sample.
• A confidence interval for a population proportion either contains the population proportion or it does not.
• We are C% confident that the confidence interval for a population proportion captures the population proportion.
• In repeated random sampling with the same sample size, approximately C% of confidence intervals created will capture the population proportion.
• Interpreting a confidence interval for a one-sample proportion should include a reference to the sample taken and details about the population it represents.
• When all other things remain the same, the width of the confidence interval for a population proportion tends to decrease as the sample size increases. 
• For a population proportion, the width of the confidence interval is proportional to 1/n.
• For a given sample, the width of the confidence interval for a population proportion increases as the confidence level increases.
• The width of a confidence interval for a population proportion is exactly twice the margin of error.
• The null hypothesis is the situation that is assumed to be correct unless evidence suggests otherwise, and the alternative hypothesis is the situation for which evidence is being collected.
• For a one-sample z-test for a population proportion, the null hypothesis specifies a value for the population proportion, usually one indicating no difference or effect.
• For a single categorical variable, the appropriate testing method for a population proportion is a one-sample z-test for a population proportion.
• The distribution of the test statistic assuming the null hypothesis is true (null distribution) can be either a randomization distribution or when a probability model is assumed to be true, a theoretical distribution (z).
• When using a z-test, the standardized test statistic can be written as standardized test statistic =sample statistic-null value of the parameterstandard deviation of the statistic .
• The test statistic for a population proportion is z=p-p0p0(1-p0)n.
• A p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed test statistic when the null hypothesis and probability model are assumed to be true.
• The p-value is the proportion of values for the null distribution that are as extreme or more extreme than the observed value of the test statistic.
• An interpretation of the p-value of a significance test for a one-sample proportion should recognize that the p-value is computed by assuming that the true population proportion is equal to the particular value stated in the null hypothesis.
• The significance level ɑ is the predetermined probability of rejecting the null hypothesis given that it is true.
• A formal decision explicitly compares the p-value to the significance level ɑ. 
• If the p-value ≤  ɑ, reject the null hypothesis. If the p-value >  ɑ, fail to reject the null hypothesis.
• Rejecting the null hypothesis means there is sufficient statistical evidence to support the alternative hypothesis. 
• Failing to reject the null means there is insufficient statistical evidence to support the alternative hypothesis.
• The conclusion about the alternative hypothesis must be stated in context.
• A significance test can lead to rejecting or not rejecting the null hypothesis, but can never lead to concluding or proving that the null hypothesis is true.
• Small p-values indicate that the observed value of the test statistic would be unusual if the null hypothesis and probability model were true. 
• The lower the p-value, the more convincing the statistical evidence for the alternative hypothesis.
• p-values that are not small indicate that the observed value of the test statistic would not be unusual if the null hypothesis and probability model were true.
• p-values that are not small do not provide evidence that the null hypothesis is true.
• A Type I error occurs when the null hypothesis is true and is rejected (false positive).
• A Type II error occurs when the null hypothesis is false and is not rejected (false negative).
• The significance level ɑ is the probability of making a Type I error, if the null hypothesis is true.
• The power of a test is the probability that a test will correctly reject a false null hypothesis.
• The probability of making a Type II error = 1− power.
• Whether a Type I or a Type II error is more consequential depends upon the situation.
• Since the significance level ɑ is the probability of a Type I error, the consequences of a Type I error influence decisions about a significance level.
• The appropriate confidence interval procedure for a two-sample comparison of proportions for one categorical variable is a two-sample z-interval for a difference between population proportions.
• In order to calculate confidence intervals to estimate a difference between proportions, we must check for independence and that the sampling distribution is approximately normal.
• For a comparison of proportions, the interval estimate is p1-p2 ± z*p1(1-p1)n1+p2(1-p2)n2
• Confidence intervals for a difference in proportions can be used to calculate interval estimates with specified units.
• In repeated random sampling with the same sample size, approximately C% of confidence intervals created will capture the difference in population proportions.
• A confidence interval for difference in population proportions provides an interval of values that may provide sufficient evidence to support a particular claim in context.
• For a single categorical variable, the appropriate testing method for the difference of two population proportions is a two-sample z-test for a difference between two population proportions.
• In order to make statistical inferences when testing a difference between population proportions, we must check for independence and that the sampling distribution is approximately normal.
• The test statistic for a difference in proportions is z=p1-p2-0pc1-pc1n1+1n2 where pc=n1p1+ n2p2n1+ n2.
• An interpretation of the p-value of a significance test for a difference of two population proportions should recognize that the p-value is computed by assuming that the null hypothesis is true.
• The results of a significance test for a difference of two population proportions can serve as the statistical reasoning to support the answer to a research question about the two populations that were sampled.

• Given that variation may be random or not, conclusions are uncertain.
• An interval of values should be used to estimate parameters, in order to account for uncertainty.
• The normal distribution may be used to model variation.
• Significance testing allows us to make decisions about hypotheses within a particular context.
• Probabilities of Type I and Type II errors influence inference.

• Check for independence (random and 10% conditions) and that the sampling distribution is  approximately normal (Large Counts condition) before constructing a confidence interval to estimate a population proportion p.
• Compute the margin of error for a population proportion p.
• Calculate the minimum sample size required to estimate p given a required margin of error.
• Check for independence (random and 10% conditions) and that the sampling distribution is  approximately normal (Large Counts condition) before constructing a confidence interval for a population proportion p.
• Calculate and interpret a one-sample z interval for p.
• Interpret the confidence level of a confidence interval.
• Evaluate a claim about a parameter using a confidence interval.
• Describe how the margin of error relates to sample size and confidence level.
• State hypotheses for a significance test about a population proportion.
• Check for independence (random and 10% conditions) and that the sampling distribution is  approximately normal (Large Counts condition) before performing a significance test about a population proportion p.
• Calculate the standardized test statistic and p-value for a one-sample z test for p.
• State a conclusion for a one-sample z test for p.
• Interpret the p-value for a one-sample z test for p.
• Describe a Type I error and give a possible consequence of such an error.
• Describe a Type II error and give a possible consequence of such an error.
• Explain how sample size, significance level, standard error, and the true (alternative) value of the parameter influence the power of a test.
• Check for independence (random and 10% conditions) and that the sampling distribution is  approximately normal (Large Counts condition) before constructing a confidence interval for a difference in two population proportions p1 – p2.
• Calculate and interpret a two-sample z interval for p1 – p2.
• State hypotheses for a significance test about a difference in two population proportions p1 – p2.
• Check for independence (random and 10% conditions) and that the sampling distribution is  approximately normal (Large Counts condition) before performing a significance test about a difference in two population proportions p1 – p2.
• Calculate the standardized test statistic and p-value for a two-sample z test for p1 – p2.
• State a conclusion for a two-sample z test for p1 – p2.

• How is an unknown population parameter estimated?
• How is an unknown difference in population parameters estimated?
• How is a claim about an unknown population parameter tested? 
• How is a claim about an unknown difference in population parameters tested?
   

• How can a population proportion be estimated?
• What affects margin of error?
• How is a claim about a population proportion tested?
• What does a P-value mean?
• What is the conclusion for a test about a population proportion? 
• What errors could occur when testing a population proportion?
• How can a difference in two population proportions be estimated?
• How can a claim about a difference in two population proportions be tested?
   

• Textbook
• Graphing calculator
• Online statistical applets (e.g. https://www.stapplet.com/)

• Point estimate
• Standard error
• Confidence level
• Critical value
• Margin of error
• One-sample z interval for p
• Null hypothesis
• Alternative hypothesis
• One-sided hypothesis
• Two-sided hypothesis
• One-sample z test for p
• Standardized test statistic
• P-value
• Significance level
• Type I error
• Type II error
• Power of a test
• Difference in proportions
• Independent samples
• Two-sample z interval for p1 – p2 
• Two-sample z test for p1 – p2

• Unit quizzes
• Unit tests
• Informal and formal writing assignments
• Presentations
• Informal check of daily assignments
• Progress check (College Board)
   

• How can a population mean be estimated?
• How can a population mean difference be estimated?
• What is the connection between confidence intervals and significance tests? 
• How is a claim about a population mean tested?
• How is a claim about a population mean difference tested?
• How can a difference in two population means be estimated?
• How can a claim about a difference in two population means be tested?
   

College Board CED for AP Statistics

• VAR-1.I Identify questions suggested by probabilities of errors in statistical inference.
• VAR-7.A Describe t-distributions.
• UNC-4.O Identify an appropriate confidence interval procedure for a population mean, including the mean difference between values in matched pairs.
• UNC-4.P Verify the conditions for calculating confidence intervals for a population mean, including the mean difference between values in matched pairs.
• UNC-4.Q Determine the margin of error for a given sample size for a one-sample t-interval.
• UNC-4.R Calculate an appropriate confidence interval for a population mean, including the mean difference between values in matched pairs.
• UNC-4.S Interpret a confidence interval for a population mean, including the mean difference between values in matched pairs.
• UNC-4.T Justify a claim based on a confidence interval for a population mean, including the mean difference between values in matched pairs.
• UNC-4.U Identify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population mean.
• VAR-7.B Identify an appropriate testing method for a population mean with unknown σ, including the mean difference between values in matched pairs.
• VAR-7.C Identify the null and alternative hypotheses for a population mean with unknown σ, including the mean difference between values in matched pairs.
• VAR-7.D Verify the conditions for the test for a population mean, including the mean difference between values in matched pairs.
• VAR-7.E Calculate an appropriate test statistic for a population mean, including the mean difference between values in matched pairs.
• DAT-3.E Interpret the p-value of a significance test for a population mean, including the mean difference between values in matched pairs.
• DAT-3.F Justify a claim about the population based on the results of a significance test for a population mean.
• UNC-4.V Identify an appropriate confidence interval procedure for a difference of two population means.
• UNC-4.W Verify the conditions to calculate confidence intervals for the difference of two population means.
• UNC-4.X Determine the margin of error for the difference of two population means.
• UNC-4.Y Calculate an appropriate confidence interval for a difference of two population means.
• UNC-4.Z Interpret a confidence interval for a difference of population means.
• UNC-4.AA Justify a claim based on a confidence interval for a difference of population means.
• UNC-4.AB Identify the effects of sample size on the width of a confidence interval for the difference of two means.
• VAR-7.F Identify an appropriate selection of a testing method for a difference of two population means.
• VAR-7.G Identify the null and alternative hypotheses for a difference of two population means.
• VAR-7.H Verify the conditions for the significance test for the difference of two population means.
• VAR-7.I Calculate an appropriate test statistic for a difference of two means.
• DAT-3.G Interpret the p-value of a significance test for a difference of population means.
• DAT-3.H Justify a claim about the population based on the results of a significance test for a difference of two population means in context.

• Random variation may result in errors in statistical inference.
• When s is used instead of σ to calculate a test statistic, the corresponding distribution, known as the t-distribution.
• The t-distribution varies from the normal distribution in shape, in that more of the area is allocated to the tails of the density curve than in a normal distribution.
• As the degrees of freedom increase, the area in the tails of a t-distribution decreases.
• Because σ is typically not known for distributions of quantitative variables, the appropriate confidence interval procedure for estimating the population mean of one quantitative variable for one sample is a one-sample t-interval for a mean.
• For one quantitative variable X that is normally distributed, the distribution of t=x - 0sn is a t-distribution with n − 1 degrees of freedom.
• Matched pairs can be thought of as one sample of pairs. Once differences between pairs of values are found, inference for confidence intervals proceeds as for a population mean.
• In order to calculate confidence intervals to estimate a population mean, we must check for independence and that the sampling distribution is approximately normal.
• The standard error of x is sn.
• The margin of error of x is t*sn.
• The point estimate for a population mean is the sample mean x.
• For the population mean for one sample with unknown population standard deviation, the confidence interval is xt*sn.
• A confidence interval for a population mean either contains the population mean or it does not.
• We are C% confident that the confidence interval for a population mean captures the population mean.
• An interpretation of a confidence interval for a population mean includes a reference to the sample taken and details about the population it represents.
• A confidence interval for a population mean provides an interval of values that may provide sufficient evidence to support a particular claim in context.
• When all other things remain the same, the width of a confidence interval for a population mean tends to decrease as the sample size increases.
• For a population mean, the width of the confidence interval is proportional to 1/n.
• For a given sample, the width of the confidence interval for a population mean increases as the confidence level increases.
• The appropriate test for a population mean with unknown  is a one-sample t-test for a population mean.
• Matched pairs can be thought of as one sample of pairs. Once differences between pairs of values are found, inference for significance testing proceeds as for a population mean.
• When finding the mean difference μd between values in a matched pair, it is important to define the order of subtraction.
• In order to make statistical inferences when testing a population mean, we must check for independence and that the sampling distribution is approximately normal.
• For a single quantitative variable when random sampling with replacement from a population that can be modeled with a normal distribution with mean μ and standard deviation σ, the sampling distribution of t=x-0sn has a t-distribution with n – 1 degrees of freedom.
• An interpretation of the p-value of a significance test for a population mean should recognize that the p-value is computed by assuming that the null hypothesis is true.
• A formal decision explicitly compares the p-value to the significance level ɑ. 
• If the p-value ≤ ɑ, reject the null hypothesis. If the p-value > ɑ, fail to reject the null hypothesis.
• The results of a significance test for a population mean can serve as the statistical reasoning to support the answer to a research question about the population that was sampled.
• The appropriate confidence interval procedure for one quantitative variable for two independent samples is a two-sample t- interval for a difference between population means.
• In order to calculate confidence intervals to estimate a difference of population means, we must check for independence and that the sampling distribution is approximately normal.
• For the difference of two sample means, the margin of error is the critical value ( t*) times the standard error (SE) of the difference of two means.
• The standard error of x1-x2 is s12n1+s22n2.
• The margin of error of x1-x2 is t*s12n1+s22n2.
• The point estimate for the difference of two population means is the difference in sample means x1-x2.
• For a difference of two population means where the population standard deviations are not known, the confidence interval is x1-x2t*s12n1+s22n2 where ± t* are the critical values for the central C% of a t-distribution with appropriate degrees of freedom that can be found using technology.
• In repeated random sampling with the same sample size, approximately C% of confidence intervals created will capture the difference of population means.
• An interpretation for a confidence interval for the difference of two population means should include a reference to the samples taken and details about the populations they represent.
• A confidence interval for a difference of population means provides an interval of values that may provide sufficient evidence to support a particular claim in context.
• When all other things remain the same, the width of the confidence interval for the difference of two means tends to decrease as the sample sizes increase.
• For a quantitative variable, the appropriate test for a difference of two population means is a two-sample t-test for a difference of two population means μ1 – μ2.
• In order to make statistical inferences when testing a difference between population means, we must check for independence and that the sampling distribution is approximately normal.
• For a single quantitative variable, data collected using independent random samples or a randomized experiment from two populations, each of which can be modeled with a normal distribution, the sampling distribution of x1-x2 is an approximate t-distribution with degrees of freedom that can be found using technology.
• An interpretation of the p-value of a significance test for a two-sample difference of population means should recognize that the p-value is computed by assuming that the null hypothesis is true.
• The results of a significance test for a two-sample test for a difference between two population means can serve as the statistical reasoning to support the answer to a research question about the populations that were sampled.

• Random variation may result in errors in statistical inference.
• The t-distribution may be used to model variation.
• An interval of values should be used to estimate parameters, in order to account for uncertainty.
• Significance testing allows us to make decisions about hypotheses within a particular context.

• Check for independence (random and 10% conditions) and that the sampling distribution is  approximately normal (Normal/large sample condition) before constructing a confidence interval to estimate a population mean μ.
• Determine the critical value t* using a table or technology.
• Calculate and interpret a one-sample t interval for μ.
• Check for independence (random and 10% conditions) and that the sampling distribution is  approximately normal (Normal/large sample condition) before constructing a confidence interval to estimate a population mean difference μDiff.
• Calculate and interpret a paired t interval for μDiff.
• Interpret the confidence level of a confidence interval.
• Evaluate a claim about a parameter using a confidence interval.
• Describe how the margin of error relates to sample size and confidence level.
• State hypotheses for a significance test about a population mean μ.
• Check for independence (random and 10% conditions) and that the sampling distribution is  approximately normal (Large Counts condition) before performing a significance test about a population mean μ.
• Calculate the standardized test statistic and p-value for a one-sample t test for μ.
• State a conclusion for a paired t test for μDiff.
• State hypotheses for a significance test about a population mean difference μDiff.
• Check for independence (random and 10% conditions) and that the sampling distribution is  approximately normal (Large Counts condition) before performing a significance test about a population mean difference μDiff.
• Calculate the standardized test statistic and p-value for a paired t test for μDiff
• State a conclusion for a paired t test for μDiff.
• Explain the relationship between a C% confidence interval and two-sided significance test with a significance level of ɑ.
• Interpret the p-value of a significance test. 
• Describe a Type I error and give a possible consequence of such an error.
• Describe a Type II error and give a possible consequence of such an error.
• Explain how sample size, significance level, standard error, and the true (alternative) value of the parameter influence the power of a test.
• Check for independence (random and 10% conditions) and that the sampling distribution is  approximately normal (Large Counts condition) before constructing a confidence interval for a difference in two population means μ1 – μ2.
• Calculate and interpret a two-sample t interval for μ1 – μ2.
• State hypotheses for a significance test about a difference in two population means μ1 – μ2.
• Check for independence (random and 10% conditions) and that the sampling distribution is  approximately normal (Large Counts condition) before performing a significance test about a difference in two population means μ1 – μ22.
• Calculate the standardized test statistic and p-value for a two-sample t test for μ1 – μ2.
• State a conclusion for a two-sample t test for μ1 – μ2.

• How is an unknown population parameter estimated?
• How is an unknown difference in population parameters estimated?
• How is a claim about an unknown population parameter tested? 
• How is a claim about an unknown difference in population parameters tested? 

• Textbook
• Graphing calculator
• Online statistical applets (e.g. https://www.stapplet.com/)

• t-distribution
• Degrees of freedom
• One-sample t interval for μ
• Paired data
• Difference in means
• Paired t interval for μDiff
• One-sample t test for μ
• Paired t test for μDiff
• Mean difference Two-sample t interval for μ1 – μ2 
• Two-sample t test for μ1 – μ2

• Unit quizzes
• Unit tests
• Informal and formal writing assignments
• Presentations
• Informal check of daily assignments
• Progress check (College Board)
   

College Board CED for AP Statistics

• VAR-1.J Identify questions suggested by variation between observed and expected counts in categorical data.
• VAR-8.A Describe chi-square distributions.
• VAR-8.B Identify the null and alternative hypotheses in a test for a distribution of proportions in a set of categorical data.
• VAR-8.C Identify an appropriate testing method for a distribution of proportions in a set of categorical data.
• VAR-8.D Calculate expected counts for the chi-square test for goodness of fit.
• VAR-8.E Verify the conditions for making statistical inferences when testing goodness of fit for a chi-square distribution.
• VAR-8.F Calculate the appropriate statistic for the chi-square test for goodness of fit.
• VAR-8.G Determine the p-value for chi-square test for goodness of fit significance test.
• DAT-3.I Interpret the p-value for the chi-square test for goodness of fit.
• DAT-3.J Justify a claim about the population based on the results of a chi-square test for goodness of fit.
• VAR-8.H Calculate expected counts for two-way tables of categorical data.
• VAR-8.I Identify the null and alternative hypotheses for a chi-square test for homogeneity or independence.
• VAR-8.J Identify an appropriate testing method for comparing distributions in two-way tables of categorical data.
• VAR-8.K Verify the conditions for making statistical inferences when testing a chi-square distribution for independence or homogeneity.
• VAR-8.L Calculate the appropriate statistic for a chi-square test for homogeneity or independence.
• VAR-8.M Determine the p-value for a chi-square significance test for independence or homogeneity.
• DAT-3.K Interpret the p-value for the chi-square test for homogeneity or independence.
• DAT-3.L Justify a claim about the population based on the results of a chi-square test for homogeneity or independence.

• Variation between what we find and what we expect to find may be random or not.
• Expected counts of categorical data are counts consistent with the null hypothesis. In general, an expected count is a sample size times a probability.
• The chi-square statistic measures the distance between observed and expected counts relative to expected counts.
• Chi-square distributions have positive values and are skewed right. Within a family of density curves, the skew becomes less pronounced with increasing degrees of freedom.
• For a chi-square goodness-of-fit test, the null hypothesis specifies null proportions for each category, and the alternative hypothesis is that at least one of these proportions is not as specified in the null hypothesis.
• When considering a distribution of proportions for one categorical variable, the appropriate test is the chi-square test for goodness of fit.
• Expected counts for a chi-square goodness-of-fit test are (sample size)✕(null proportion).
• In order to make statistical inferences for a chi-square test for goodness of fit we must check for independence (Random and 10% condition) and that the sampling distribution has an approximately Chi-square distribution (Large Counts condition).
• The test statistic for the chi-square test for goodness of fit is x²=observed count - expected count2expected count with degrees of freedom = (number of categories – 1).
• The distribution of the test statistic assuming the null hypothesis is true (null distribution) can be either a randomization distribution or, when a probability model is assumed to be true, a theoretical distribution (chi-square).
• The p-value for a chi-square test for goodness of fit for a number of degrees of freedom is found using the appropriate table or computer generated output.
• An interpretation of the p-value for the chi-square test for goodness of fit is the probability, given the null hypothesis and probability model are true, of obtaining a test statistic as, or more, extreme than the observed value.
• A decision to either reject or fail to reject the null hypothesis is based on comparison of the p-value to the significance level ɑ.
• The results of a chi-square test for goodness of fit can serve as the statistical reasoning to support the answer to a research question about the population that was sampled.
• The expected count in a particular cell of a two-way table of categorical data can be
• calculated using the formula expected count = row totalcolumn totaltable total.
• The appropriate hypotheses for a chi-square test for homogeneity are:
• H0: There is no difference in distributions of a categorical variable across populations or treatments.
• Ha: There is a difference in distributions of a categorical variable across populations or treatments.
• The appropriate hypotheses for a chi-square test for independence are:
• H0: There is no association between two categorical variables in a given population or the two categorical variables are independent.
• Ha: Two categorical variables in a population are associated or dependent.
• When comparing distributions to determine whether proportions in each category for
• categorical data collected from different populations are the same, the appropriate test
• is the chi-square test for homogeneity.
• To determine whether row and column variables in a two-way table of categorical data might be associated in the population from which the data were sampled, the appropriate test is the chi-square test for independence.
• In order to make statistical inferences for a chi-square test for homogeneity or independence we must check for independence (Random and 10% condition) and that the sampling distribution has an approximately Chi-square distribution (Large Counts condition).
• The appropriate test statistic for a chi-square test for homogeneity or independence is the chi-square statistic 2=observed count - expected count2expected count with degrees of freedom = (number of rows – 1) ✕ (number of columns – 1).
• The p-value for a chi-square test for independence or homogeneity for a number of degrees of freedom is found using the appropriate table or technology.
• For a test of independence or homogeneity for a two-way table, the p-value is the proportion of values in a chi-square distribution with appropriate degrees of freedom that are equal to or larger than the test statistic.
• An interpretation of the p-value for the chi-square test for homogeneity or independence is the probability, given the null hypothesis and probability model are true, of obtaining a test statistic as, or more, extreme than the observed value.
• A decision to either reject or fail to reject the null hypothesis for a chi-square test for homogeneity or independence is based on comparison of the p-value to the significance level ɑ.
• The results of a chi-square test for homogeneity or independence can serve as the statistical reasoning to support the answer to a research question about the population that was sampled (independence) or the populations that were sampled (homogeneity).

• Given that variation may be random or not, conclusions are uncertain.
• The chi-square distribution may be used to model variation.
   

• Compute expected counts for categories of one categorical variable given a null hypothesis.
• State hypotheses for a chi-square test for goodness of fit.
• Check for independence (random and 10% conditions) and that the sampling distribution is  approximately chi-square before performing a chi-square test for goodness of fit.
• Calculate the χ2 test statistic and p-value for a chi-square test for goodness of fit.
• State a conclusion for a chi-square test for goodness of fit.
• Interpret the p-value of a significance test. 
• Describe a Type I error and give a possible consequence of such an error.
• Describe a Type II error and give a possible consequence of such an error.
• Explain how sample size, significance level, standard error, and the true (alternative) value of the parameter influence the power of a test.
• State hypotheses for a chi-square test for independence.
• Compute expected counts for a two-way table given a null hypothesis.
• Check for independence (random and 10% conditions) and that the sampling distribution is  approximately chi-square before performing a chi-square test for independence.
• Calculate the χ2 test statistic and p-value for a chi-square test for independence.
• State a conclusion for a chi-square test for independence.
• State hypotheses for a chi-square test for homogeneity.
• Check for independence (random and 10% conditions) and that the sampling distribution is  approximately chi-square before performing a chi-square test for homogeneity.
• Calculate the χ2 test statistic and p-value for a chi-square test for homogeneity.
• State a conclusion for a chi-square test for homogeneity.

• How can inferences about distributions of categorical data be made?
• How can inferences about relationships between categorical variables be made?
   

• How is a claim about the distribution of a categorical variable in one population tested?
• How is a claim about independence between two categorical variables in one population tested?
• How is a claim about the distribution of a categorical variable in multiple populations tested?
   

• Textbook
• Graphing calculator
• Online statistical applets (e.g. https://www.stapplet.com/)

• Observed counts
• Expected counts
• χ2-distribution
• Chi-square test for goodness of fit
• Two-way table
• Association
• Independence
• Chi-square test for independence
• Chi-square test for homogeneity

• Unit quizzes
• Unit tests
• Informal and formal writing assignments
• Presentations
• Informal check of daily assignments
• Progress check (College Board)
   

College Board CED for AP Statistics

• VAR-1.K Identify questions suggested by variation in scatter plots.
• UNC-4.AC Identify an appropriate confidence interval procedure for a slope of a regression model.
• UNC-4.AD Verify the conditions to calculate confidence intervals for the slope of a regression model.
• UNC-4.AE Determine the given margin of error for the slope of a regression model.
• UNC-4.AF Calculate an appropriate confidence interval for the slope of a regression model.
• UNC-4.AG Interpret a confidence interval for the slope of a regression model.
• UNC-4.AH Justify a claim based on a confidence interval for the slope of a regression model.
• UNC-4.AI Identify the effects of sample size on the width of a confidence interval for the slope of a regression model.
• VAR-7.J Identify the appropriate selection of a testing method for a slope of a regression model.
• VAR-7.K Identify appropriate null and alternative hypotheses for a slope of a regression model.
• VAR-7.L Verify the conditions for the significance test for the slope of a regression model.
• VAR-7.M Calculate an appropriate test statistic for the slope of a regression model.
• DAT-3.M Interpret the p-value of a significance test for the slope of a regression model.
• DAT-3.N Justify a claim about the population based on the results of a significance test for the slope of a regression model.

• Variation in points’ positions relative to a theoretical line may be random or non-random.
• The mean of the sampling distribution of the sample slope b is the slope 𝛃 of the population regression line.
• The standard deviation of the sampling distribution of the sample slope b is bxn.
• The shape of the sampling distribution of the sample slope b is approximately normal for sufficiently large sample sizes (e.g. n ≥ 30).
• The appropriate confidence interval for the slope of a regression model is a t-interval for the slope.
• In order to calculate a confidence interval to estimate the slope of a regression line, we must check the Random condition, the 10% condition, the Linear condition, the Equal standard deviation condition, and the Normal condition.
• The standard error of the sample slope b is ssxn-1.
• For the slope of a population regression line, the margin of error is t*ssxn-1.
• The point estimate for the slope of a regression model is the slope b of the line of best fit.
• For the slope of a regression model, the interval estimate is bt*ssxn-1.
• In repeated random sampling with the same sample size, approximately C% of confidence intervals created will capture the true slope of the population regression model.
• An interpretation for a confidence interval for the slope of a regression line should include a reference to the sample taken and details about the population it represents.
• A confidence interval for the slope of a regression model provides an interval of values that may provide sufficient evidence to support a particular claim in context.
• When all other things remain the same, the width of the confidence interval for the slope of a regression model tends to decrease as the sample size increases.
• The appropriate test for the slope of a regression model is a t-test for a slope.
• In order to test a claim about the slope of a regression line, we must check the Random condition, the 10% condition, the Linear condition, the Equal standard deviation condition, and the Normal condition.
• The distribution of the slope of a regression model assuming all conditions are satisfied and the null hypothesis is true (null distribution) is a t-distribution.
• For simple linear regression when random sampling from a population for the response that can be modeled with a normal distribution for each value of the explanatory variable, the sampling distribution of t=b-SEb has a t-distribution with n – 2 degrees of freedom.
• An interpretation of the p-value of a significance test for the slope of a regression model should recognize that the p-value is computed by assuming that the true population slope is equal to the particular value stated in the null hypothesis.
• A formal decision explicitly compares the p-value to the significance level ɑ. 
• If the p-value ≤ ɑ, reject the null hypothesis. If the p-value > ɑ, fail to reject the null hypothesis.
• The results of a significance test for the slope of a regression model can serve as the statistical reasoning to support the answer to a research question about that sample.

• Given that variation may be random or not, conclusions are uncertain.
• An interval of values should be used to estimate parameters, in order to account for uncertainty.
• The t-distribution may be used to model variation.
• Significance testing allows us to make decisions about hypotheses within a particular context.

• Describe the sampling distribution of the sample slope b, given characteristics of the population.
• Determine the critical value t* using a table or technology.
• Check the appropriate conditions before constructing a confidence interval to estimate a slope 𝛃 of a population regression line.
• Calculate and interpret a t interval for 𝛃.
• Interpret the confidence level of a confidence interval.
• Evaluate a claim about a parameter using a confidence interval.
• Describe how the margin of error relates to sample size and confidence level.
• State hypotheses for a significance test about the slope 𝛃 of a population regression line.
• Check the appropriate conditions before performing a significance test about the slope 𝛃 of a population regression line.
• Calculate the standardized test statistic and p-value for a t test for 𝛃.
• State a conclusion for a significance test about the slope 𝛃 of a population regression line.
• Interpret the p-value of a significance test. 
• Describe a Type I error and give a possible consequence of such an error.
• Describe a Type II error and give a possible consequence of such an error.
• Explain how sample size, significance level, standard error, and the true (alternative) value of the parameter influence the power of a test.

• How is an unknown parameter estimated?
• How is a claim about an unknown parameter tested? 

• How do sample slopes vary in random sampling?
• How is an unknown population slope estimated?
• How is a claim about an unknown population slope tested? 
• What is the connection between confidence intervals and significance tests? 
   

• Textbook
• Graphing calculator
• Online statistical applets (e.g. https://www.stapplet.com/)

• Sample regression line
• Population regression line
• Population slope 𝛃
• Sample slope b
• Standard deviation of the residuals
• Standard error of the sample slope
• Point estimate
• t-distribution
• Margin of error
• Confidence level
• t interval for slope
• Confidence interval
• Null hypothesis
• Alternative hypothesis
• One-sided hypothesis
• Two-sided hypothesis
• t test for slope
• Standardized test statistic
• P-value
• Significance level
• Type I error
• Type II error
• Power of a test

• Unit quizzes
• Unit tests
• Informal and formal writing assignments
• Presentations
• Informal check of daily assignments
• Progress check (College Board)