AP Statistics

Enduring Understanding (VAR-1): Given that variation may be random or not, conclusions are uncertain.

Learning Objective VAR-1.A: Identify questions to be answered, based on variation in one-variable data. [Skill 1.A]

• VAR-1.A.1 Numbers may convey meaningful information, when placed in context.

Enduring Understanding (VAR-1): Given that variation may be random or not, conclusions are uncertain.

Learning Objective VAR-1.B: Identify variables in a set of data. [Skill 2.A]

• VAR-1.B.1 A variable is a characteristic that changes from one individual to another.

Learning Objective VAR-1.C: Classify types of variables. [Skill 2.A]

• VAR-1.C.1 A categorical variable takes on values that are category names or group labels.
• VAR-1.C.2 A quantitative variable is one that takes on numerical values for a measured or counted quantity.
  • Illustrative examples for VAR-1.C:
    • Categorical variables:
      • Dominant hand
      • Age group (young or old)
      • Highest degree earned
    • Quantitative variables:
      • Age of a structure
      • Height of a child
      • Concentration of a sample

Enduring Understanding (UNC-1): Graphical representations and statistics allow us to identify and represent key features of data.

Learning Objective UNC-1.A: Represent categorical data using frequency or relative frequency tables. [Skill 2.B]

• UNC-1.A.1 A frequency table gives the number of cases falling into each category. A relative frequency table gives the proportion of cases falling into each category.

Learning Objective UNC-1.B: Describe categorical data represented in frequency or relative tables. [Skill 2.A]

• UNC-1.B.1 Percentages, relative frequencies, and rates all provide the same information as proportions.
• UNC-1.B.2 Counts and relative frequencies of categorical data reveal information that can be used to justify claims about the data in context.

Enduring Understanding (UNC-1): Graphical representations and statistics allow us to identify and represent key features of data.

Learning Objective UNC-1.C: Represent categorical data graphically. [Skill 2.B]

• UNC-1.C.1 Bar charts (or bar graphs) are used to display frequencies (counts) or relative frequencies (proportions) for categorical data.
• UNC-1.C.2 The height or length of each bar in a bar graph corresponds to either the number or proportion of observations falling within each category.
• UNC-1.C.3 There are many additional ways to represent frequencies (counts) or relative frequencies (proportions) for categorical data.

Learning Objective UNC-1.D: Describe categorical data represented graphically. [Skill 2.A]

• UNC-1.D.1 Graphical representations of a categorical variable reveal information that can be used to justify claims about the data in context.

Learning Objective UNC-1.E: Compare multiple sets of categorical data. [Skill 2.D]

• UNC-1.E.1 Frequency tables, bar graphs, or other representations can be used to compare two or more data sets in terms of the same categorical variable.

Enduring Understanding (UNC-1): Graphical representations and statistics allow us to identify and represent key features of data.

Learning Objective UNC-1.F: Classify types of quantitative variables. [Skill 2.A]

• UNC-1.F.1 A discrete variable can take on a countable number of values. The number of values may be finite or countably infinite, as with the counting numbers.
• UNC-1.F.2 A continuous variable can take on infinitely many values, but those values cannot be counted. No matter how small the interval between two values of a continuous variable, it is always possible to determine another value between them.
  • Illustrative examples for UNC-1.F:
    • A discrete variable:
      • Number of students in a class
    • A continuous variable:
      • Height of a child

Learning Objective UNC-1.G: Represent quantitative data graphically. [Skill 2.B]

• UNC-1.G.1 In a histogram, the height of each bar shows the number or proportion of observations that fall within the interval corresponding to that bar. Altering the interval widths can change the appearance of the histogram.
• UNC-1.G.2 In a stem and leaf plot, each data value is split into a "stem" (the first digit or digits) and a "leaf" (usually the last digit).
• UNC-1.G.3 A dotplot represents each observation by a dot, with the position on the horizontal axis corresponding to the data value of that observation, with nearly identical values stacked on top of each other.
• UNC-1.G.4 A cumulative graph represents the number or proportion of a data set less than or equal to a given number.
• UNC-1.G.5 There are many additional ways to graphically represent distributions of quantitative data.

Enduring Understanding (UNC-1): Graphical representations and statistics allow us to identify and represent key features of data.

Learning Objective UNC-1.H: Describe the characteristics of quantitative data distributions. [Skill 2.A]

• UNC-1.H.1 Descriptions of the distribution of quantitative data include shape, center, and variability (spread), as well as any unusual features such as outliers, gaps, clusters, or multiple peaks.
• UNC-1.H.2 Outliers for one-variable data are data points that are unusually small or large relative to the rest of the data.
• UNC-1.H.3 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.
• UNC-1.H.4 Univariate graphs with one main peak are known as unimodal. Graphs with two prominent peaks are bimodal. A graph where each bar height is approximately the same (no prominent peaks) is approximately uniform.
• UNC-1.H.5 A gap is a region of a distribution between two data values where there are no observed data.
• UNC-1.H.6 Clusters are concentrations of data usually separated by gaps.
• UNC-1.H.7 Descriptive statistics does not attribute properties of a data set to a larger population, but may provide the basis for conjectures for subsequent testing.

Enduring Understanding (VAR-2): The normal distribution can be used to represent some population distributions.

Learning Objective VAR-2.A: Compare a data distribution to the normal distribution model. [Skill 2.D]

• VAR-2.A.1 A parameter is a numerical summary of a population.
• VAR-2.A.2 Some sets of data may be described as approximately normally distributed. A normal curve is mound-shaped and symmetric. The parameters of a normal distribution are the population mean, $\mu$, and the population standard deviation, $\sigma$.
• VAR-2.A.3 For a normal distribution, approximately 68% of the observations are within 1 standard deviation of the mean, approximately 95% of observations are within 2 standard deviations of the mean, and approximately 99.7% of observations are within 3 standard deviations of the mean. This is called the empirical rule.
• VAR-2.A.4 Many variables can be modeled by a normal distribution.
  • Illustrative examples for VAR-2.A:
    • Variables that can be modeled by a normal distribution:
      • Body temperature
      • Weight of a loaf of bread

Learning Objective VAR-2.B: Determine proportions and percentiles from a normal distribution. [Skill 3.A]

• VAR-2.B.1 A standardized score for a particular data value is calculated as (data value − mean)/(standard deviation), and measures the number of standard deviations a data value falls above or below the mean.
• VAR-2.B.2 One example of a standardized score is a $z$-score, which is calculated as $z\text{-score} = \left(\dfrac{x_i - \mu}{\sigma}\right)$. A $z$-score measures how many standard deviations a data value is from the mean.
• VAR-2.B.3 Technology, such as a calculator, a standard normal table, or computer-generated output, can be used to find the proportion of data values located on a given interval of a normally distributed random variable.
• VAR-2.B.4 Given the area of a region under the graph of the normal distribution curve, it is possible to use technology, such as a calculator, a standard normal table, or computer-generated output, to estimate parameters for some populations.

Learning Objective VAR-2.C: Compare measures of relative position in data sets. [Skill 2.D]

• VAR-2.C.1 Percentiles and $z$-scores may be used to compare relative positions of points within a data set or between data sets.

Enduring Understanding (UNC-1): Graphical representations and statistics allow us to identify and represent key features of data.

Learning Objective UNC-1.Q: Calculate statistics for two categorical variables. [Skill 2.C]

• UNC-1.Q.1 The marginal relative frequencies are the row and column totals in a two-way table divided by the total for the entire table.
• UNC-1.Q.2 A conditional relative frequency is a relative frequency for a specific part of the contingency table (e.g., cell frequencies in a row divided by the total for that row).

Learning Objective UNC-1.R: Compare statistics for two categorical variables. [Skill 2.D]

• UNC-1.R.1 Summary statistics for two categorical variables can be used to compare distributions and/or determine if variables are associated.

Enduring Understanding (UNC-1): Graphical representations and statistics allow us to identify and represent key features of data.

Learning Objective UNC-1.S: Represent bivariate quantitative data using scatterplots. [Skill 2.B]

• UNC-1.S.1 A bivariate quantitative data set consists of observations of two different quantitative variables made on individuals in a sample or population.
• UNC-1.S.2 A scatterplot shows two numeric values for each observation, one corresponding to the value on the $x$-axis and one corresponding to the value on the $y$-axis.
• UNC-1.S.3 An explanatory variable is a variable whose values are used to explain or predict corresponding values for the response variable.

Enduring Understanding (DAT-1): Regression models may allow us to predict responses to changes in an explanatory variable.

Learning Objective DAT-1.A: Describe the characteristics of a scatter plot. [Skill 2.A]

• DAT-1.A.1 A description of a scatter plot includes form, direction, strength, and unusual features.
• DAT-1.A.2 The direction of the association shown in a scatterplot, if any, can be described as positive or negative.
• DAT-1.A.3 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.
• DAT-1.A.4 The form of the association shown in a scatterplot, if any, can be described as linear or non-linear to varying degrees.
• DAT-1.A.5 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.
• DAT-1.A.6 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.

Enduring Understanding (DAT-1): Regression models may allow us to predict responses to changes in an explanatory variable.

Learning Objective DAT-1.B: Determine the correlation for a linear relationship. [Skill 2.C]

• DAT-1.B.1 The correlation, $r$, gives the direction and quantifies the strength of the linear association between two quantitative variables.
• DAT-1.B.2 The correlation coefficient can be calculated by: $r = \dfrac{1}{n-1} \sum \left( \dfrac{x_i - \bar{x}}{s_x} \right) \left( \dfrac{y_i - \bar{y}}{s_y} \right)$. However, the most common way to determine $r$ is by using technology.
• DAT-1.B.3 A correlation coefficient close to 1 or $-1$ does not necessarily mean that a linear model is appropriate.

Learning Objective DAT-1.C: Interpret the correlation for a linear relationship. [Skill 4.B]

• DAT-1.C.1 The correlation, $r$, is unit-free, and always 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.
• DAT-1.C.2 A perceived or real relationship between two variables does not mean that changes in one variable cause changes in the other. That is, correlation does not necessarily imply causation.

Enduring Understanding (DAT-1): Regression models may allow us to predict responses to changes in an explanatory variable.

Learning Objective DAT-1.D: Calculate a predicted response value using a linear regression model. [Skill 2.C]

• DAT-1.D.1 A simple linear regression model is an equation that uses an explanatory variable, $x$, to predict the response variable, $y$.
• DAT-1.D.2 The predicted response value, denoted by $\hat{y}$, is calculated as $\hat{y} = a + bx$, where $a$ is the $y$-intercept and $b$ is the slope of the regression line, and $x$ is the value of the explanatory variable.
• DAT-1.D.3 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 predicted value is less reliable as an estimate the further we extrapolate.

Enduring Understanding (DAT-1): Regression models may allow us to predict responses to changes in an explanatory variable.

Learning Objective DAT-1.E: Represent differences between measured and predicted responses using residual plots. [Skill 2.B]

• DAT-1.E.1 The residual is the difference between the actual value and the predicted value: $\text{residual} = y - \hat{y}$.
• DAT-1.E.2 A residual plot is a plot of residuals versus explanatory variable values or predicted response values.

Learning Objective DAT-1.F: Describe the form of association of bivariate data using residual plots. [Skill 2.A]

• DAT-1.F.1 Apparent randomness in a residual plot for a linear model is evidence of a linear form to the association between the variables.
• DAT-1.F.2 Residual plots can be used to investigate the appropriateness of a selected model.

Enduring Understanding (DAT-1): Regression models may allow us to predict responses to changes in an explanatory variable.

Learning Objective DAT-1.G: Estimate parameters for the least-squares regression line model. [Skill 2.C]

• DAT-1.G.1 The least-squares regression model minimizes the sum of the squares of the residuals and contains the point $(\bar{x}, \bar{y})$.
• DAT-1.G.2 The slope, $b$, of the regression line can be calculated as $b = r \left( \dfrac{s_y}{s_x} \right)$ where $r$ is the correlation between $x$ and $y$, $s_y$ is the sample standard deviation of the response variable, $y$, and $s_x$ is the sample standard deviation of the explanatory variable, $x$.
• DAT-1.G.3 Sometimes, the $y$-intercept of the line does not have a logical interpretation in context.
• DAT-1.G.4 In simple linear regression, $r^2$ is the square of the correlation, $r$. It is also called the coefficient of determination. $r^2$ is the proportion of variation in the response variable that is explained by the explanatory variable in the model.

Learning Objective DAT-1.H: Interpret coefficients for the least-squares regression line model. [Skill 4.B]

• DAT-1.H.1 The coefficients of the least-squares regression model are the estimated slope and $y$-intercept.
• DAT-1.H.2 The slope is the amount that the predicted $y$-value changes for every unit increase in $x$.
• DAT-1.H.3 The $y$-intercept value is the predicted value of the response variable when the explanatory variable is equal to $0$. The formula for the $y$-intercept, $a$, is $a = \bar{y} - b\bar{x}$.

Enduring Understanding (DAT-1): Regression models may allow us to predict responses to changes in an explanatory variable.

Learning Objective DAT-1.I: Identify influential points in regression. [Skill 2.A]

• DAT-1.I.1 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.
• DAT-1.I.2 A high-leverage point in regression has a substantially larger or smaller $x$-value than the other observations have.
• DAT-1.I.3 An influential point in regression is any point that, if removed, changes the relationship substantially. Examples include much different slope, $y$-intercept, and/or correlation. Outliers and high leverage points are often influential.

Learning Objective DAT-1.J: Calculate a predicted response using a least-squares regression line for a transformed data set. [Skill 2.C]

• DAT-1.J.1 Transformations of variables, such as evaluating the natural logarithm of each value of the response variable or squaring each value of the explanatory variable, can be used to create transformed data sets, which may be more linear in form than the untransformed data.
• DAT-1.J.2 Increased randomness in residual plots after transformation of data and/or movement of $r^2$ to a value closer to 1 offers evidence that the least-squares regression line for the transformed data is a more appropriate model to use to predict responses to the explanatory variable than the regression line for the untransformed data.

Enduring Understanding (VAR-1): Given that variation may be random or not, conclusions are uncertain.

Learning Objective VAR-1.E: Identify questions to be answered about data collection methods. [Skill 1.A]

• VAR-1.E.1 Methods for data collection that do not rely on chance result in untrustworthy conclusions.

Enduring Understanding (DAT-2): The way we collect data influences what we can and cannot say about a population.

Learning Objective DAT-2.A: Identify the type of a study. [Skill 1.C]

• DAT-2.A.1 A population consists of all items or subjects of interest.
• DAT-2.A.2 A sample selected for study is a subset of the population.
• DAT-2.A.3 In an observational study, treatments are not imposed. Investigators examine data for a sample of individuals (retrospective) or follow a sample of individuals into the future collecting data (prospective) in order to investigate a topic of interest about the population. A sample survey is a type of observational study that collects data from a sample in an attempt to learn about the population from which the sample was taken.
• DAT-2.A.4 In an experiment, different conditions (treatments) are assigned to experimental units (participants or subjects).

Learning Objective DAT-2.B: Identify appropriate generalizations and determinations based on observational studies. [Skill 4.A]

• DAT-2.B.1 It is only appropriate to make generalizations about a population based on samples that are randomly selected or otherwise representative of that population.
• DAT-2.B.2 A sample is only generalizable to the population from which the sample was selected.
• DAT-2.B.3 It is not possible to determine causal relationships between variables using data collected in an observational study.

Enduring Understanding (DAT-2): The way we collect data influences what we can and cannot say about a population.

Learning Objective DAT-2.C: Identify a sampling method, given a description of a study. [Skill 1.C]

• DAT-2.C.1 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.
• DAT-2.C.2 A simple random sample (SRS) is a sample in which every group of a given size has an equal chance of being chosen. This method is the basis for many types of sampling mechanisms. A few examples of mechanisms used to obtain SRSs include numbering individuals and using a random number generator to select which ones to include in the sample, ignoring repeats, using a table of random numbers, or drawing a card from a deck without replacement.
• DAT-2.C.3 A stratified random sample involves the division of a population into separate groups, called strata, based on shared attributes or characteristics (homogeneous grouping). Within each stratum a simple random sample is selected, and the selected units are combined to form the sample.
• DAT-2.C.4 A cluster sample involves the division of a population into smaller groups, called clusters. Ideally, there is heterogeneity within each cluster, and clusters are similar to one another in their composition. A simple random sample of clusters is selected from the population to form the sample of clusters. Data are collected from all observations in the selected clusters.
• DAT-2.C.5 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.
• DAT-2.C.6 A census selects all items/subjects in a population.

Learning Objective DAT-2.D: Explain why a particular sampling method is or is not appropriate for a given situation. [Skill 1.C]

• DAT-2.D.1 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.

Enduring Understanding (DAT-2): The way we collect data influences what we can and cannot say about a population.

Learning Objective DAT-2.E: Identify potential sources of bias in sampling methods. [Skill 1.C]

• DAT-2.E.1 Bias occurs when certain responses are systematically favored over others.
• DAT-2.E.2 When a sample is comprised entirely of volunteers or people who choose to participate, the sample will typically not be representative of the population (voluntary response bias).
• DAT-2.E.3 When part of the population has a reduced chance of being included in the sample, the sample will typically not be representative of the population (undercoverage bias).
• DAT-2.E.4 Individuals chosen for the sample for whom data cannot be obtained (or who refuse to respond) may differ from those for whom data can be obtained (nonresponse bias).
• DAT-2.E.5 Problems in the data gathering instrument or process result in response bias. Examples include questions that are confusing or leading (question wording bias) and self-reported responses.
• DAT-2.E.6 Non-random sampling methods (for example, samples chosen by convenience or voluntary response) introduce potential for bias because they do not use chance to select the individuals.

Enduring Understanding (VAR-3): Well-designed experiments can establish evidence of causal relationships.

Learning Objective VAR-3.A: Identify the components of an experiment. [Skill 1.C]

• VAR-3.A.1 The experimental units are the individuals (which may be people or other objects of study) that are assigned treatments. When experimental units consist of people, they are sometimes referred to as participants or subjects.
• VAR-3.A.2 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.
• VAR-3.A.3 A response variable in an experiment is an outcome from the experimental units that is measured after the treatments have been administered.
• VAR-3.A.4 A confounding variable in an experiment is a variable that is related to the explanatory variable and influences the response variable and may create a false perception of association between the two.

Learning Objective VAR-3.B: Describe elements of a well-designed experiment. [Skill 1.B]

• VAR-3.B.1 A well-designed experiment should include the following:
  • a. Comparisons of at least two treatment groups, one of which could be a control group.
  • b. Random assignment/allocation of treatments to experimental units.
  • c. Replication (more than one experimental unit in each treatment group).
  • d. Control of potential confounding variables where appropriate.

Learning Objective VAR-3.C: Compare experimental designs and methods. [Skill 1.C]

• VAR-3.C.1 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.
• VAR-3.C.2 Methods for randomly assigning treatments to experimental units in a completely randomized design include using a random number generator, a table of random values, drawing chips without replacement, etc.
• VAR-3.C.3 In a single-blind experiment, subjects do not know which treatment they are receiving, but members of the research team do, or vice versa.
• VAR-3.C.4 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.
• VAR-3.C.5 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) in order to determine if the treatment of interest has an effect.
• VAR-3.C.6 The placebo effect occurs when experimental units have a response to a placebo.
• VAR-3.C.7 For randomized complete block designs, treatments are assigned completely at random within each block.
• VAR-3.C.8 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.
• VAR-3.C.9 A matched pairs design is a special case of a randomized block design. Using a blocking variable, subjects (whether they are people or not) are arranged in pairs matched on relevant factors. Matched pairs may be formed naturally or by the experimenter. Every pair receives both treatments by randomly assigning one treatment to one member of the pair and subsequently assigning the remaining treatment to the second member of the pair. Alternately, each subject may get both treatments.

Enduring Understanding (VAR-4): The likelihood of a random event can be quantified.

Learning Objective VAR-4.A: Calculate probabilities for events and their complements. [Skill 3.A]

• VAR-4.A.1 The sample space of a random process is the set of all possible non-overlapping outcomes.
• VAR-4.A.2 If all outcomes in the sample space are equally likely, then the probability an event E will occur is defined as the fraction: $\dfrac{\text{number of outcomes in event E}}{\text{total number of outcomes in sample space}}$
• VAR-4.A.3 The probability of an event is a number between 0 and 1, inclusive.
• VAR-4.A.4 The probability of the complement of an event E, $E'$ or $E^{C}$, (i.e., not E) is equal to $1 - P(E)$.

Learning Objective VAR-4.B: Interpret probabilities for events. [Skill 4.B]

• VAR-4.B.1 Probabilities of events in repeatable situations can be interpreted as the relative frequency with which the event will occur in the long run.

Enduring Understanding (VAR-4): The likelihood of a random event can be quantified.

Learning Objective VAR-4.C: Explain why two events are (or are not) mutually exclusive. [Skill 4.B]

• VAR-4.C.1 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 $P(A \cap B)$.
• VAR-4.C.2 Two events are mutually exclusive or disjoint if they cannot occur at the same time. So $P(A \cap B) = 0$.

Enduring Understanding (VAR-4): The likelihood of a random event can be quantified.

Learning Objective VAR-4.D: Calculate conditional probabilities. [Skill 3.A]

• VAR-4.D.1 The probability that event $A$ will occur given that event $B$ has occurred is called a conditional probability and denoted $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$.
• VAR-4.D.2 The multiplication rule states that the probability that events $A$ and $B$ both will occur is equal to the probability that event $A$ will occur multiplied by the probability that event $B$ will occur, given that $A$ has occurred. This is denoted $P(A \cap B) = P(A) \cdot P(B \mid A)$.

Enduring Understanding (VAR-4): The likelihood of a random event can be quantified.

Learning Objective VAR-4.E: Calculate probabilities for independent events and for the union of two events. [Skill 3.A]

• VAR-4.E.1 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.
• VAR-4.E.2 If, and only if, events $A$ and $B$ are independent, then $P(A \mid B) = P(A)$, $P(B \mid A) = P(B)$, and $P(A \cap B) = P(A) \cdot P(B)$.
• VAR-4.E.3 The probability that event $A$ or event $B$ (or both) will occur is the probability of the union of $A$ and $B$, denoted $P(A \cup B)$.
• VAR-4.E.4 The addition rule states that the probability that event $A$ or event $B$ or both will occur is equal to the probability that event $A$ will occur plus the probability that event $B$ will occur minus the probability that both events $A$ and $B$ will occur. This is denoted $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Enduring Understanding (VAR-5): Probability distributions may be used to model variation in populations.

Learning Objective VAR-5.A: Represent the probability distribution for a discrete random variable. [Skill 2.B]

• VAR-5.A.1 The values of a random variable are the numerical outcomes of random behavior.
• VAR-5.A.2 A discrete random variable is a variable that can only take a countable number of values. Each value has a probability associated with it. The sum of the probabilities over all of the possible values must be 1.
• VAR-5.A.3 A probability distribution can be represented as a graph, table, or function showing the probabilities associated with values of a random variable.
• VAR-5.A.4 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.
  • Illustrative examples for VAR-5.A: Outcomes of trials of a random process:
    • The sum of the outcomes for rolling two dice
    • The number of puppies in a randomly selected litter for a certain breed of dog

Learning Objective VAR-5.B: Interpret a probability distribution. [Skill 4.B]

• VAR-5.B.1 An interpretation of a probability distribution provides information about the shape, center, and spread of a population and allows one to make conclusions about the population of interest.

Enduring Understanding (VAR-5): Probability distributions may be used to model variation in populations.

Learning Objective VAR-5.C: Calculate parameters for a discrete random variable. [Skill 3.B]

• VAR-5.C.1 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.
• VAR-5.C.2 The mean, or expected value, for a discrete random variable $X$ is $\mu_X = \sum x_i \cdot P(x_i)$.
• VAR-5.C.3 The standard deviation for a discrete random variable $X$ is $\sigma_X = \sqrt{\sum (x_i - \mu_x)^2 \cdot P(x_i)}$.

Learning Objective VAR-5.D: Interpret parameters for a discrete random variable. [Skill 4.B]

• VAR-5.D.1 Parameters for a discrete random variable should be interpreted using appropriate units and within the context of a specific population.

Enduring Understanding (VAR-5): Probability distributions may be used to model variation in populations.

Learning Objective VAR-5.E: Calculate parameters for linear combinations of random variables. [Skill 3.B]

• VAR-5.E.1 For random variables $X$ and $Y$ and real numbers $a$ and $b$, the mean of $aX + bY$ is $a\mu_x + b\mu_y$.
• VAR-5.E.2 Two random variables are independent if knowing information about one of them does not change the probability distribution of the other.
• VAR-5.E.3 For independent random variables $X$ and $Y$ and real numbers $a$ and $b$, the mean of $aX + bY$ is $a\mu_x + b\mu_y$, and the variance of $aX + bY$ is $a^2\sigma^2_x + b^2\sigma^2_y$.

Learning Objective VAR-5.F: Describe the effects of linear transformations of parameters of random variables. [Skill 3.C]

• VAR-5.F.1 For $Y = a + bX$, the probability distribution of the transformed random variable, $Y$, has the same shape as the probability distribution for $X$, so long as $a > 0$ and $b > 0$. The mean of $Y$ is $\mu_y = a + b\mu_x$. The standard deviation of $Y$ is $\sigma_y = |b|\sigma_x$.

Enduring Understanding (VAR-1): Given that variation may be random or not, conclusions are uncertain.

Learning Objective VAR-1.G: Identify questions suggested by variation in statistics for samples collected from the same population. [Skill 1.A]

• VAR-1.G.1 Variation in statistics for samples taken from the same population may be random or not.

Enduring Understanding (VAR-6): The normal distribution may be used to model variation.

Learning Objective VAR-6.A: Calculate the probability that a particular value lies in a given interval of a normal distribution. [Skill 3.A]

• VAR-6.A.1 A continuous random variable is a variable that can take on any value within a specified domain. Every interval within the domain has a probability associated with it.
• VAR-6.A.2 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.
• VAR-6.A.3 The area under a normal curve over a given interval represents the probability that a particular value lies in that interval.
  • Illustrative examples for VAR-6.A: Continuous random variable: If one looks at a clock at a random time, the probability that the minute hand is between the 3 and the 6 is one fourth.

Learning Objective VAR-6.B: Determine the interval associated with a given area in a normal distribution. [Skill 3.A]

• VAR-6.B.1 The boundaries of an interval associated with a given area in a normal distribution can be determined using $z$-scores or technology, such as a calculator, a standard normal table, or computer-generated output.
• VAR-6.B.2 Intervals associated with a given area in a normal distribution can be determined by assigning appropriate inequalities to the boundaries of the intervals:
  • a. $P(X < x_a) = \dfrac{p}{100}$ means that the lowest $p\%$ of values lie to the left of $x_a$.
  • b. $P(x_a < X < x_b) = \dfrac{p}{100}$ means that $p\%$ of values lie between $x_a$ and $x_b$.
  • c. $P(X > x_b) = \dfrac{p}{100}$ means that the highest $p\%$ of values lie to the right of $x_b$.
  • d. To determine the most extreme $p\%$ of values requires dividing the area associated with $p\%$ into two equal areas on either extreme of the distribution: $P(X < x_a) = \dfrac{1}{2}\dfrac{p}{100}$ and $P(X > x_b) = \dfrac{1}{2}\dfrac{p}{100}$ means that half of the $p\%$ most extreme values lie to the left of $x_a$ and half of the $p\%$ most extreme values lie to the right of $x_b$.

Learning Objective VAR-6.C: Determine the appropriateness of using the normal distribution to approximate probabilities for unknown distributions. [Skill 3.C]

• VAR-6.C.1 Normal distributions are symmetrical and "bell-shaped." As a result, normal distributions can be used to approximate distributions with similar characteristics.

Enduring Understanding (UNC-3): Probabilistic reasoning allows us to anticipate patterns in data.

Learning Objective UNC-3.H: Estimate sampling distributions using simulation. [Skill 3.C]

• UNC-3.H.1 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.
• UNC-3.H.2 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.
• UNC-3.H.3 The central limit theorem requires that the sample values are independent of each other and that $n$ is sufficiently large.
• UNC-3.H.4 A randomization distribution is a collection of statistics generated by simulation assuming known values for the parameters. For a randomized experiment, this means repeatedly randomly reallocating/reassigning the response values to treatment groups.
• UNC-3.H.5 The sampling distribution of a statistic can be simulated by generating repeated random samples from a population.

Enduring Understanding (UNC-3): Probabilistic reasoning allows us to anticipate patterns in data.

Learning Objective UNC-3.I: Explain why an estimator is or is not unbiased. [Skill 4.B]

• UNC-3.I.1 When estimating a population parameter, an estimator is unbiased if, on average, the value of the estimator is equal to the population parameter.

Learning Objective UNC-3.J: Calculate estimates for a population parameter. [Skill 3.B]

• UNC-3.J.1 When estimating a population parameter, an estimator exhibits variability that can be modeled using probability.
• UNC-3.J.2 A sample statistic is a point estimator of the corresponding population parameter.

Enduring Understanding (UNC-3): Probabilistic reasoning allows us to anticipate patterns in data.

Learning Objective UNC-3.K: Determine parameters of a sampling distribution for sample proportions. [Skill 3.B]

• UNC-3.K.1 For independent samples (sampling with replacement) of a categorical variable from a population with population proportion, $p$, the sampling distribution of the sample proportion, $\hat{p}$, has a mean, $\mu_{\hat{p}} = p$ and a standard deviation, $\sigma_{\hat{p}} = \sqrt{\dfrac{p(1-p)}{n}}$.
• UNC-3.K.2 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 is negligible.

Learning Objective UNC-3.L: Determine whether a sampling distribution for a sample proportion can be described as approximately normal. [Skill 3.C]

• UNC-3.L.1 For a categorical variable, the sampling distribution of the sample proportion, $\hat{p}$, will have an approximate normal distribution, provided the sample size is large enough: $np \geq 10$ and $n(1-p) \geq 10$

Learning Objective UNC-3.M: Interpret probabilities and parameters for a sampling distribution for a sample proportion. [Skill 4.B]

• UNC-3.M.1 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.

Enduring Understanding (UNC-3): Probabilistic reasoning allows us to anticipate patterns in data.

Learning Objective UNC-3.N: Determine parameters of a sampling distribution for a difference in sample proportions. [Skill 3.B]

• UNC-3.N.1 For a categorical variable, when randomly sampling with replacement from two independent populations with population proportions $p_1$ and $p_2$, the sampling distribution of the difference in sample proportions $\hat{p}_1 - \hat{p}_2$ has mean, $\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2$ and standard deviation, $\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\dfrac{p_1(1-p_1)}{n_1} + \dfrac{p_2(1-p_2)}{n_2}}$.
• UNC-3.N.2 If sampling without replacement, the standard deviation of the difference in sample proportions is smaller than what is given by the formula above. If the sample sizes are less than 10% of the population sizes, the difference is negligible.

Learning Objective UNC-3.O: Determine whether a sampling distribution for a difference of sample proportions can be described as approximately normal. [Skill 3.C]

• UNC-3.O.1 The sampling distribution of the difference in sample proportions $\hat{p}_1 - \hat{p}_2$ will have an approximate normal distribution provided the sample sizes are large enough: $n_1 p_1 \geq 10, n_1(1-p_1) \geq 10, n_2 p_2 \geq 10, n_2(1-p_2) \geq 10$.

Learning Objective UNC-3.P: Interpret probabilities and parameters for a sampling distribution for a difference in proportions. [Skill 4.B]

• UNC-3.P.1 Parameters for a sampling distribution for a difference of proportions should be interpreted using appropriate units and within the context of a specific populations.

Enduring Understanding (UNC-3): Probabilistic reasoning allows us to anticipate patterns in data.

Learning Objective UNC-3.Q: Determine parameters for a sampling distribution for sample means. [Skill 3.B]

• UNC-3.Q.1 For a numerical variable, when random sampling with replacement from a population with mean $\mu$ and standard deviation, $\sigma$, the sampling distribution of the sample mean has mean $\mu_{\bar{x}} = \mu$ and standard deviation $\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}}$.
• UNC-3.Q.2 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 is negligible.

Learning Objective UNC-3.R: Determine whether a sampling distribution of a sample mean can be described as approximately normal. [Skill 3.C]

• UNC-3.R.1 For a numerical variable, if the population distribution can be modeled with a normal distribution, the sampling distribution of the sample mean, $\bar{x}$, can be modeled with a normal distribution.
• UNC-3.R.2 For a numerical variable, if the population distribution cannot be modeled with a normal distribution, the sampling distribution of the sample mean, $\bar{x}$, can be modeled approximately by a normal distribution, provided the sample size is large enough, e.g., greater than or equal to 30.

Learning Objective UNC-3.S: Interpret probabilities and parameters for a sampling distribution for a sample mean. [Skill 4.B]

• UNC-3.S.1 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.

Enduring Understanding (UNC-3): Probabilistic reasoning allows us to anticipate patterns in data.

Learning Objective UNC-3.T: Determine parameters of a sampling distribution for a difference in sample means. [Skill 3.B]

• UNC-3.T.1 For a numerical variable, when randomly sampling with replacement from two independent populations with population means $\mu_1$ and $\mu_2$ and population standard deviations $\sigma_1$ and $\sigma_2$, the sampling distribution of the difference in sample means $\bar{x}_1 - \bar{x}_2$ has mean $\mu_{(\bar{x}_1 - \bar{x}_2)} = \mu_1 - \mu_2$ and standard deviation, $\sigma_{(\bar{x}_1 - \bar{x}_2)} = \sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}$.
• UNC-3.T.2 If sampling without replacement, the standard deviation of the difference in sample means is smaller than what is given by the formula above. If the sample sizes are less than 10% of the population sizes, the difference is negligible.

Learning Objective UNC-3.U: Determine whether a sampling distribution of a difference in sample means can be described as approximately normal. [Skill 3.C]

• UNC-3.U.1 The sampling distribution of the difference in sample means $\bar{x}_1 - \bar{x}_2$ can be modeled with a normal distribution if the two population distributions can be modeled with a normal distribution.
• UNC-3.U.2 The sampling distribution of the difference in sample means $\bar{x}_1 - \bar{x}_2$ can be modeled approximately by a normal distribution if the two population distributions cannot be modeled with a normal distribution but both sample sizes are greater than or equal to 30.

Learning Objective UNC-3.V: Interpret probabilities and parameters for a sampling distribution for a difference in sample means. [Skill 4.B]

• UNC-3.V.1 Probabilities and parameters for a sampling distribution for a difference of sample means should be interpreted using appropriate units and within the context of a specific populations.

Enduring Understanding (VAR-1): Given that variation may be random or not, conclusions are uncertain.

Learning Objective VAR-1.H: Identify questions suggested by variation in the shapes of distributions of samples taken from the same population. [Skill 1.A]

• VAR-1.H.1 Variation in shapes of data distributions may be random or not.

Enduring Understanding (UNC-4): An interval of values should be used to estimate parameters, in order to account for uncertainty.

Learning Objective UNC-4.A: Identify an appropriate confidence interval procedure for a population proportion. [Skill 1.D]

• UNC-4.A.1 The appropriate confidence interval procedure for a one-sample proportion for one categorical variable is a one sample $z$-interval for a proportion.

Learning Objective UNC-4.B: Verify the conditions for calculating confidence intervals for a population proportion. [Skill 4.C]

• UNC-4.B.1 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.
• UNC-4.B.2 In order to calculate a confidence interval to estimate a population proportion, $p$, we must check for independence and that the sampling distribution is approximately normal.
  • a. To check for independence:
    • i. Data should be collected using a random sample or a randomized experiment.
    • ii. When sampling without replacement, check that $n \leq 10\%N$, where $N$ is the size of the population.
  • b. To check that the sampling distribution of $\hat{p}$ is approximately normal (shape):
    • i. For categorical variables, check that both the number of successes, $n\hat{p}$, and the number of failures, $n(1-\hat{p})$ are at least 10 so that the sample size is large enough to support an assumption of normality.

Learning Objective 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. [Skill 3.D]

• UNC-4.C.1 Based on sample data, the standard error of a statistic is an estimate for the standard deviation for the statistic. The standard error of $\hat{p}$ is $SE_{\hat{p}} = \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$.
• UNC-4.C.2 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.
• UNC-4.C.3 For categorical variables, the margin of error is the critical value ($z^*$) times the standard error (SE) of the relevant statistic, which equals $z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$ for a one sample proportion.
• UNC-4.C.4 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. For this purpose, use a guess for $\hat{p}$ or use $\hat{p} = 0.5$ in order to find an upper bound for the sample size that will result in a given margin of error.

Learning Objective UNC-4.D: Calculate an appropriate confidence interval for a population proportion. [Skill 3.D]

• UNC-4.D.1 In general, an interval estimate can be constructed as point estimate ± (margin of error). For a one-sample proportion, the interval estimate is $\hat{p} \pm z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$.
  • Clarifying statement: Formulas for interval estimates do not appear explicitly on the AP Statistics Formula Sheet provided with the AP Statistics Exam. However, these formulas do not need to be memorized, as they can be constructed based on the general test statistic formula and the relevant standard error formulas that are provided on the formula sheet.
• UNC-4.D.2 Critical values represent the boundaries encompassing the middle C% of the standard normal distribution, where C% is an approximate confidence level for a proportion.

Learning Objective UNC-4.E: Calculate an interval estimate based on a confidence interval for a population proportion. [Skill 3.D]

• UNC-4.E.1 Confidence intervals for population proportions can be used to calculate interval estimates with specified units.

Enduring Understanding (VAR-6): The normal distribution may be used to model variation.

Learning Objective VAR-6.D: Identify the null and alternative hypotheses for a population proportion. [Skill 1.F]

• VAR-6.D.1 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.
• VAR-6.D.2 For hypotheses about parameters, the null hypothesis contains an equality reference (=, ≥, or ≤), while the alternative hypothesis contains a strict inequality (<, >, or ≠). The type of inequality in the alternative hypothesis is based on the question of interest. Alternative hypotheses with < or > are called one-sided, and alternative hypotheses with ≠ are called two-sided. Although the null hypothesis for a one-sided test may include an inequality symbol, it is still tested at the boundary of equality.
• VAR-6.D.3 The null hypothesis for a population proportion is: $H_0 : p = p_0$, where $p_0$ is the null hypothesized value for the population proportion.
• VAR-6.D.4 A one-sided alternative hypothesis for a proportion is either $H_a : p < p_0$ or $H_a : p > p_0$. A two-sided alternate hypothesis is $H_a : p_1 \neq p_2$.
• VAR-6.D.5 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.

Learning Objective VAR-6.E: Identify an appropriate testing method for a population proportion. [Skill 1.E]

• VAR-6.E.1 For a single categorical variable, the appropriate testing method for a population proportion is a one-sample $z$-test for a population proportion.

Learning Objective VAR-6.F: Verify the conditions for making statistical inferences when testing a population proportion. [Skill 4.C]

• VAR-6.F.1 In order to make statistical inferences when testing a population proportion, we must check for independence and that the sampling distribution is approximately normal:
  • a. To check for independence:
    • i. Data should be collected using a random sample or a randomized experiment.
    • ii. When sampling without replacement, check that $n \leq 10\%N$.
  • b. To check that the sampling distribution of $\hat{p}$ is approximately normal (shape):
    • i. Assuming that $H_0$ is true $(p = p_0)$, verify that both the number of successes, $np_0$, and the number of failures, $n(1-p_0)$ are at least 10 so that that the sample size is large enough to support an assumption of normality.

Enduring Understanding (DAT-3): Significance testing allows us to make decisions about hypotheses within a particular context.

Learning Objective DAT-3.B: Justify a claim about the population based on the results of a significance test for a population proportion. [Skill 4.E]

• DAT-3.B.1 The significance level, $\alpha$, is the predetermined probability of rejecting the null hypothesis given that it is true.
• DAT-3.B.2 A formal decision explicitly compares the $p$-value to the significance level, $\alpha$. If the $p$-value $\leq \alpha$, reject the null hypothesis. If the $p$-value $> \alpha$, fail to reject the null hypothesis.
• DAT-3.B.3 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.
• DAT-3.B.4 The conclusion about the alternative hypothesis must be stated in context.
• DAT-3.B.5 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. Lack of statistical evidence for the alternative hypothesis is not the same as evidence for the null hypothesis.
• DAT-3.B.6 Small $p$-values indicate that the observed value of the test statistic would be unusual if the null hypothesis and probability model were true, and so provide evidence for the alternative. The lower the $p$-value, the more convincing the statistical evidence for the alternative hypothesis.
• DAT-3.B.7 $p$-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, so do not provide convincing statistical evidence for the alternative hypothesis nor do they provide evidence that the null hypothesis is true.
• DAT-3.B.8 A formal decision explicitly compares the $p$-value to the significance $\alpha$. If the $p$-value $\leq \alpha$, then reject the null hypothesis, $H_0 : p = p_0$. If the $p$-value $> \alpha$, then fail to reject the null hypothesis.
• DAT-3.B.9 The results of a significance test for a population proportion can serve as the statistical reasoning to support the answer to a research question about the population that was sampled.