AP Statistics course outline

1.1.A.1 · A statistical study is a study in which data are collected from a sample to answer an investigative question about a larger population.

1.1.A.2 · Statistical studies are necessary when the population is too large or it is too difficult to collect data from every item or individual in the population.

1.1.A.3 · A datum (singular form of data) is a piece of information about an item or individual. A collection of data is called a data set.

1.1.A.4 · A population consists of all items or individuals of interest. The population size is represented by the symbol N.

1.1.A.5 · A sample selected for study is a subset of the population from which data are obtained. The number of items in the sample, called the sample size, is represented by the symbol n.

1.1.A.6 · Each component of a statistical study and the resulting calculations can be related to an aspect of the corresponding real-world context from which the components were derived. This identification of a statistical result with the corresponding contextual component is what is meant by “in context.”

1.1.B.1 · An investigative question for a specific study should have a defined purpose and should not be changed based on the data analysis or results.

1.1.B.2 · An investigative question should be posed so that the required data can be collected and analyzed.

1.2.A.1 · An observational unit is an item or individual from which a datum is collected.

1.2.A.2 · A variable is a characteristic that may change from one observational unit to another.

1.2.A.3 · Data collected on numerical and categorical variables measured on observational units, including photographs, sounds, videos, and text, can convey meaningful information.

1.2.A.4 · A parameter is a numerical attribute or summary of the variable of interest for a population.

1.2.A.5 · A statistic is a numerical attribute or summary of the variable of interest for a sample. The value of a statistic from a certain sample is often not equal to the unknown value of the population parameter but may provide the basis for making inferences about the population parameter.

1.2.B.1 · A categorical variable, also called a qualitative variable, takes on values that are category names or group labels.

1.2.B.2 · A quantitative variable, also called a numerical variable, takes on numerical values for a measured or counted quantity and generally has units of measure.

1.2.C.1 · A discrete quantitative variable can take on a countable number of values. The number of values may be finite or countably infinite, as with the whole numbers.

1.2.C.2 · A continuous quantitative variable can take on an infinite number of possible values within a given interval. The number of values the variable can take on is measurable but not countable. This variable can take on all possible values between any pair of values.

1.3.A.1 · A frequency table shows the number of observational units in each category of a categorical variable.

1.3.A.2 · A relative frequency table shows the proportion of observational units in each category of a categorical variable.

1.3.B.1 · Percentages, relative frequencies, and ratios all provide the same information as proportions.

1.3.B.2 · Counts and relative frequencies of categorical variables reveal information that can be used to justify claims about the variables in context.

1.4.A.1 · Bar charts, also called bar graphs, display frequencies (counts) or relative frequencies (proportions) for the categories of a single categorical variable. Each bar on a bar chart represents a category of the categorical variable of interest. The height or length of each bar corresponds to the frequency or relative frequency of the observational units in each category.

1.4.A.2 · Pie charts are used to display frequencies (counts) or relative frequencies (proportions) for categorical data. Each slice on a pie chart represents a category of the categorical variable of interest. The area of each slice, as a fraction of the total area, corresponds to the relative frequency of observational units falling within each category. The sum of the slices’ areas together will equal 1, or 100% of the total area.

1.4.B.1 · Graphical representations of a categorical variable reveal information that can be used to justify claims about the variable in context.

1.4.C.1 · Frequency and relative frequency tables, bar charts, and pie charts can be used to compare two or more data sets in terms of the same categorical variable.

1.5.A.1 · Histograms, stem-and-leaf plots, and dotplots provide a visual representation of the distribution of the values of a quantitative variable. These graphs show the frequency or relative frequency of the quantitative variable values or intervals of values and maintain the natural ordering, smallest to largest, of the quantitative variable.

1.5.A.2 · A histogram places the observed values of the quantitative variable into ordered intervals, or bins, along the horizontal axis. Each bar represents an interval or bin, and the height of each bar shows the frequency or relative frequency of the observations within that interval. Altering the interval widths, or bin widths, can change the appearance of the histogram. Alternatively, a histogram can be constructed with bins on the vertical axis with bars appearing horizontally.

1.5.A.3 · A stem-and-leaf plot splits each value of the quantitative variable into two parts: a “stem” (the first digit or digits) and a “leaf” (usually the single digit after the stem digit or digits). Both stems and leaves are ordered from smallest to largest.

1.5.A.4 · A dotplot represents each value of the quantitative variable by a dot. Each dot is placed above the horizontal or beside the vertical axis corresponding to the value of that observation, with nearly identical values stacked on top of each other.

1.6.A.1 · Descriptions of the distribution of one quantitative variable include shape, center, and variability (spread) as well as any unusual features such as outliers, gaps, or clusters in context.

1.6.A.2 · The shape of the distribution of one quantitative variable is skewed to the right (positively skewed) if the right tail (toward larger values) is longer than the left. The shape of the distribution is skewed to the left (negatively skewed) if the left tail (toward smaller values) is longer than the right. The shape of the distribution is approximately symmetric if the left half is approximately the mirror image of the right half.

1.6.A.3 · Distributions of one quantitative variable with one main peak are called unimodal. Distributions with two prominent peaks are called bimodal. A distribution in which each frequency or each relative frequency is approximately the same with no prominent peaks is approximately uniform.

1.6.A.4 · Outliers for one quantitative variable are data points that are unusually small or large relative to the rest of the data.

1.6.A.5 · A gap is a region in a distribution between two values in which there are no observed data.

1.6.A.6 · Clusters are concentrations of values usually separated by gaps.

1.6.B.1 · Graphical representations of a quantitative variable may reveal information that can be used to justify claims about the variable in context.

1.7.A.1 · Two commonly used measures of center in the distribution of a quantitative variable are the mean and median.

1.7.A.2 · The mean is the sum of all the values divided by the number of values and can be found with and without using technology. For a sample, the mean is denoted by x-bar: $\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i$, where $x_i$ represents the ith data point in the sample and $n$ represents the number of data values in the sample.

1.7.A.3 · The median is the middle value when the data set is ordered from smallest to largest and can be found with and without using technology. One common method for determining the median of a data set with an even number of values is to use the mean of the two middle values. A common method for determining the median of a data set with an odd number of values is to use the value in the middle of all the values.

1.7.A.4 · In an ordered data set, the smallest value is the minimum value, and the largest value is the maximum value.

1.7.A.5 · The first quartile, denoted by Q1, is the median value of the lower half of the ordered data set from the minimum value to the position of the median. Approximately 25% of the values in the data set are less than or equal to Q1. The third quartile, denoted by Q3, is the median value of the upper half of the ordered data set from the position of the median to the maximum value. Approximately 75% of the values in the data set are less than or equal to Q3. The second quartile, Q2, is also the median of the data set. Q1 and Q3 form the boundaries for the middle 50% of values in an ordered data set.

1.7.A.6 · The pth percentile is the value that has p% of the data less than or equal to it when the data set is ordered from smallest to largest. The first and third quartiles are the 25th and 75th percentiles, respectively.

1.7.B.1 · Three commonly used measures of variability (or spread) in the distribution of a quantitative variable are the range, interquartile range, and standard deviation.

1.7.B.2 · The range is the difference between the maximum data value and the minimum data value.

1.7.B.3 · The interquartile range (IQR) is the difference between the third and first quartiles: $\mathrm{Q3}-\mathrm{Q1}$.

1.7.B.4 · The standard deviation is a typical deviation of the data values from their mean and can be found with and without using technology. The sample standard deviation is denoted by $s$ and calculated by $s=\sqrt{\frac{1}{n-1}\sum(x_i-\bar{x})^2}$, where $x_i$ is the data value, $\bar{x}$ is the mean, and $n$ is the number of data values in the sample. The square of the sample standard deviation, $s^2$, is called the sample variance.

1.7.C.1 · Changing units of measurement affects the values of the calculated statistics.

1.7.D.1 · There are many methods for determining potential outliers. Two methods frequently used are as follows:

1.7.D.1.i · An outlier is a value located more than $1.5\times\mathrm{IQR}$ above the third quartile or more than $1.5\times\mathrm{IQR}$ below the first quartile.

1.7.D.1.ii · An outlier is a value located more than 2 standard deviations above, or below, the mean.

1.7.E.1 · Summary statistics can be used to compare features of two or more independent samples, including center, variability, shape, and outliers.

1.7.F.1 · The median and IQR are considered a resistant (or robust) measure of center and measure of variability, respectively, because outliers do not greatly (if at all) affect their values. Because outliers can affect their values greatly, the mean is considered a nonresistant (or non-robust) measure of center, and the range and standard deviation are considered nonresistant (or non-robust) measures of variability.

1.7.F.2 · Summary statistics of a quantitative variable may reveal information that can be used to justify claims about the variable in context.

1.8.A.1 · A five-number summary is made up of the minimum data value, the first quartile (Q1), the median, the third quartile (Q3), and the maximum data value.

1.8.A.2 · A boxplot is a graphical representation of the five-number summary (minimum, first quartile, median, third quartile, maximum). The box represents the middle 50% of data, with a line at the median and the ends of the box corresponding to the quartiles. Lines (“whiskers”) that represent 25% of the data extend from the first quartile to the minimum and from the third quartile to the maximum. If there are outliers in the data, the whiskers extend to the most extreme data values that are not outliers, and outliers are usually denoted with an asterisk or other symbol.

1.8.B.1 · If a distribution is relatively symmetric, then the values of the mean and median are relatively close to each other. If a distribution is skewed right, then the value of the mean is usually larger than the median. If the distribution is skewed left, then the value of the mean is usually smaller than the median.

1.9.A.1 · Graphical representations of a quantitative variable can be used to compare important features between two or more distributions of the same quantitative variable. Histograms, back-to-back stem-and-leaf plots, and dotplots may be used to compare center, variability, shape, outliers, clusters, or gaps in two or more distributions. Boxplots may be used to compare center, variability, outliers, and skewness (or symmetry).

1.9.B.1 · A comparison of graphical representations for two or more distributions can include any of the numerical summaries (e.g., mean, standard deviation, etc.).

1.9.C.1 · Multiple quantitative one-variable graphical representations may reveal information that can be used to justify claims about the variable in context.

1.9.D.1 · A standardized score measures the number of standard deviations a data value falls above or below the mean.

1.9.D.2 · A z-score is calculated as $\frac{x_i-\mu}{\sigma}$, where $x_i$ is the data value, $\mu$ is the population mean, and $\sigma$ is the population standard deviation. A z-score measures how many standard deviations a data value is above (positive z-score) or below (negative z-score) the mean. When the population mean and standard deviation are unknown, the sample mean and standard deviation may be used to determine a z-score.

1.9.E.1 · z-scores may be used to compare relative positions of individual values within a distribution or between distributions.

1.10.A.1 · The first component of an investigative question should guide the data collection process and should be phrased in terms of the variable(s) of interest in the study.

1.10.A.2 · The second component of an investigative question should guide the data analysis choice.

1.10.A.2.i · In the case of a hypothesis test, the investigative question should make clear the parameter and the direction of the alternative hypothesis (i.e., not equal, greater than, less than, association, not independent).

1.10.A.2.ii · In the case of a confidence interval, the investigative question should make clear the parameter and the goal of estimation of that parameter within a range of potential values.

1.10.A.3 · The third component of an investigative question should indicate the type(s) of conclusion(s) applicable from the study. The investigative question should provide the population to which the conclusions will be applicable and, in the case of an experiment that uses random assignment, a cause-and-effect conclusion.

1.10.B.1 · A census consists of recording information from all items or individuals in a population.

1.10.C.1 · An experiment is a study in which a researcher assigns conditions, or treatments, to experimental units to explore an investigative question of interest about the population.

1.10.C.2 · The experimental unit is the observational unit to which the treatment is assigned. When experimental units consist of people, they are sometimes referred to as subjects or participants.

1.10.C.3 · An explanatory variable, or factor, is a variable whose different categories, or levels, are imposed on the experimental units. The different categories, or levels, of the explanatory variable are called treatments. When there is more than one explanatory variable, the combinations of the categories, or levels, of the explanatory variables are called treatments.

1.10.C.4 · A response variable is an outcome measured on each experimental unit after the treatment has been administered.

1.10.D.1 · An observational study is a study where treatments are not imposed. The researcher records the values of the variables of interest in order to explore an investigative question of interest.

1.10.D.2 · A prospective study is one in which the observational units of study are selected at a point in time, and data are gathered both at that time and into the future.

1.10.D.3 · A retrospective study is one in which the observational units of study are selected at a point in time and data from the past are gathered.

1.10.D.4 · A survey is an observational study in which the data are collected from humans using a standard set of questions.

1.10.D.5 · A confounding variable in an observational study provides an alternative explanation for the observed relationship between the explanatory and response variables determined in the study, thereby reducing the possibility of concluding a causal relationship between the explanatory and response variables of interest. To be a confounding variable, a variable must be associated with both the explanatory variable and the response variable.

1.10.E.1 · A sample is considered random when all observational units in the sample are selected from the population using some type of random mechanism, such as a random number generator.

1.10.E.2 · When observational units, or experimental units, in a sample are randomly selected from a population, it is appropriate to make generalizations about the entire population of individuals from which the sample was selected.

1.10.E.3 · A sample is not randomly selected when observational units are deliberately chosen or volunteer themselves to be in the sample.

1.10.E.4 · When observational units, or experimental units, in a sample are not randomly selected from a population, it is appropriate to make generalizations only about a population of individuals that are similar to those used in the study.

1.11.A.1 · Sampling without replacement is a sampling strategy in which an observational unit from a population can be selected only once. The observational unit is not returned to the population before subsequent selections of observational units are made, so there is no chance that the observational unit can be selected again.

1.11.A.2 · Sampling with replacement is a sampling strategy in which an observational unit from the population can be selected more than once. The observational unit is returned to the population before subsequent selections of observational units are made, so it is possible that the observational unit could be selected again.

1.11.A.3 · In a simple random sample (SRS) of size n, every sample of the size n has the same chance of being selected. This method is the basis for many types of sampling mechanisms. There are several procedures to obtain a simple random sample; for example, using a random number generator or randomly selecting numbered slips of paper.

1.11.A.4 · A stratified random sample involves the division of all individuals in a population into non-overlapping groups, called strata, based on one or more shared attributes or characteristics (homogeneous grouping). Within each stratum a simple random sample is selected, and the selected individuals are combined to form one sample.

1.11.A.5 · A cluster random sample involves the division of a population into smaller groups, called clusters. Ideally, each cluster mirrors the heterogeneity of the population, with clusters similar to one another. A simple random sample of clusters is selected from the population to form the sample of clusters. Data are collected from all observational units in each of the selected clusters.

1.11.A.6 · 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 between successive sampling units.

1.11.B.1 · Each random sampling method has different characteristics that make it more appropriate for sampling populations depending on the question being investigated.

1.12.A.1 · Bias in a sampling method is a systematic error in the sampling procedure that results in a statistic being consistently larger or consistently smaller than the parameter the statistic is used to estimate.

1.12.A.2 · Voluntary response bias is a bias that may occur when a sample consists entirely of volunteers.

1.12.A.3 · Undercoverage bias may occur when the sampling method fails to include part of the population or a part of the population is less likely to be selected based on the sampling method.

1.12.A.4 · Nonresponse bias may occur because of a failure to obtain responses from some individuals chosen to be sampled. The respondents and nonrespondents could differ significantly in ways that are important for the study.

1.12.A.5 · Response bias may occur when responses to a survey or measurements of observational units tend to differ from the “true” value in one direction. Examples include questions that are confusing or leading (question wording bias) or self-reported responses.

1.12.A.6 · Nonrandom sampling methods (e.g., samples chosen by convenience or voluntary response) introduce potential bias because they do not use random chance to select the individuals.

1.13.A.1 · A well-designed experiment should include the following:

1.13.A.1.i · Comparisons of at least two treatment groups, one of which could be a control group

1.13.A.1.ii · Random assignment of treatments to experimental units

1.13.A.1.iii · Replication

1.13.A.1.iv · Direct control of potential extraneous sources of variation in the response

1.13.A.2 · A control group is a collection of experimental units that are created for comparison. A control group may be given a treatment different from the treatment of interest to determine if the treatment of interest has an effect (e.g., a treatment with an inactive substance, a placebo, may be given).

1.13.A.3 · The placebo effect is the difference between the average response to a placebo and the average response to no treatment.

1.13.A.4 · In a single-blind, also called single-masked, experiment, participants do not know which treatment they are receiving, but members of the research team who interact with them know which treatment each participant is receiving, or vice versa.

1.13.A.5 · In a double-blind, also called double-masked, experiment, neither the participants nor the members of the research team who interact with them know which treatment each participant is receiving.

1.13.A.6 · An extraneous source of variation, also referred to as an extraneous variable, is a variable that is known (or believed) to affect the response but is not an explanatory variable being studied.

1.13.A.7 · The purpose of random assignment is to create treatment groups that are as similar as possible with respect to extraneous sources of variation. If random assignment is successful, the respective distributions of each extraneous variable will be approximately the same for all the treatment groups.

1.13.A.8 · A confounding variable in an experiment is a variable that is related to the explanatory variable in such a way that it is difficult to determine which variable, explanatory or confounding, is influencing the change in the response variable. However, in a well-designed experiment, the potential for confounding variables is reduced.

1.13.A.9 · Replication within an experiment means more than one experimental unit is assigned to each treatment.

1.13.A.10 · Direct control in an experiment means keeping the settings of certain potential extraneous sources of variation in the response variable the same from experimental unit to experimental unit.

1.13.B.1 · In a completely randomized design, treatments are assigned to experimental units completely at random. Often the number of experimental units assigned to each treatment will be the same, but the sample sizes in each treatment do not have to be the same.

1.13.B.2 · A blocking variable is a source of extraneous variation in the response variable. In a randomized block design, the experimental units are first grouped according to similar values of a blocking variable. These groups are called blocks. Units within the same block are homogeneous with respect to the blocking variable. After the blocks are formed, the treatments are randomly assigned to experimental units within each block so that all treatments occur within every block.

1.13.B.3 · The purpose of blocking is to separate the variation in the response caused by the blocking variable from the rest of the extraneous variation in the response. Blocking allows for more precise comparisons of the response across the treatments. Within a block, the treatments can be compared without having to worry about variation in the response caused by changes in the blocking variable.

1.13.B.4 · A matched pairs design is a randomized block design with only two treatments. Experimental units are arranged in pairs by matching on one or more extraneous sources of variation in the response variable. Each pair receives both treatments by randomly assigning one treatment to one member of the pair and the other treatment to the second member of the pair. Alternatively, each experimental unit may get both treatments while the order of the treatments is randomized.

1.13.C.1 · One experimental design may be more appropriate than another experimental design based on the goals of the investigative study, the characteristics of the population, and the sample and variables involved.

1.13.D.1 · Using random assignment of treatments to experimental units allows for cause-and-effect conclusions between the explanatory and the response variables because the potential for confounding variables is reduced.

1.13.D.2 · Depending on the experimental unit, it may be unethical or difficult to randomly select experimental units to participate in an experiment. In that case, the study’s experimental units are obtained from volunteers and will represent the population of experimental units similar to those who participated in the study.

2.1.A.1 · A two-way table, also called a contingency table, can be used to summarize and compare data for two categorical variables. The entries in the cells of the table can be frequencies (i.e., counts) or relative frequencies (i.e., proportions).

2.1.A.2 · Side-by-side bar charts, segmented bar charts, and mosaic plots are examples of graphs used to display the relationship between two categorical variables. In these graphs, the frequency or relative frequency of each category, or level, of one of the categorical variables is displayed for each category of the other categorical variable.

2.1.A.3 · Graphical representations of two categorical variables can be used to compare the relationship of one categorical variable across the levels of the other categorical variable and determine whether the two variables are associated.

2.1.B.1 · Tabular and graphical representations for the distributions of two categorical variables may reveal information that can be used to justify claims about the variable in context.

2.2.A.1 · A joint relative frequency in a two-way table is a cell frequency divided by the total for the entire table.

2.2.A.2 · A marginal relative frequency in a two-way table is a row total divided by the total for the entire table or a column total divided by the total for the entire table.

2.2.A.3 · A conditional relative frequency is a relative frequency computed by restricting to a particular level, or category of interest. A conditional relative frequency can be a cell frequency in a row divided by the total for that row or it can be a cell frequency in a column divided by the total for that column.

2.2.B.1 · Summary statistics for two categorical variables can be used to compare distributions for evidence of association between the two variables.

2.2.C.1 · Summary statistics for two categorical variables may reveal information that can be used to justify claims about the variables in context.

2.3.A.1 · A random process generates results that are determined by chance.

2.3.A.2 · An outcome is the result of one trial of a random process.

2.3.A.3 · An event is a collection of outcomes.

2.3.A.4 · Simulation is a way to model random events such that simulated outcomes closely match real-world outcomes. All possible outcomes are associated with a value to be determined by chance. Record the counts of simulated outcomes and the count total.

2.3.A.5 · The probability of an outcome or event is its long-run relative frequency—that is, its relative frequency over a large number of trials.

2.3.A.6 · The relative frequency of an outcome or event determined from empirical data can be used to estimate the actual, or true, probability of that outcome or event.

2.3.A.7 · The law of large numbers states that for independent trials, as the number of trials increases, the long-run relative frequency of the outcome or event gets closer and closer to a single value.

2.4.A.1 · The sample space of a random process is the set of all possible nonoverlapping outcomes. The probability of the sample space is 1.

2.4.A.2 · If all outcomes in the sample space are equally likely, then the theoretical probability an event $E$ will occur is $\frac{\text{number of outcomes in event }E}{\text{total number of outcomes in the sample space}}$. The probability of event $E$ occurring is written as $P(E)$.

2.4.A.3 · The probability of an event is a number between 0 and 1, inclusive.

2.4.A.4 · The probability of the complement of an event $E$, which can be written as $E^{\prime}$, $\overline{E}$, or $E^C$ (i.e., the probability of “not $E$”) is equal to $1-P(E)$.

2.5.A.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 $B$. Joint probability is defined as $P(A\text{ intersect }B)$ or $P(A\cap B)$.

2.5.A.2 · Two events are mutually exclusive, or disjoint, if they cannot occur at the same time. This means that if two events are mutually exclusive, then $P(A\cap B)=0$.

2.6.A.1 · The probability that event $A$ will occur given that event $B$ has occurred is called a conditional probability and is written as $P(A\mid B)$. Conditional probability is defined as $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$.

2.6.A.2 · The general 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 conditional probability that event $B$ will occur given that event $A$ has occurred. The multiplication rule is defined as $P(A\cap B)=P(A)\cdot P(B\mid A)$.

2.7.A.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. When events $A$ and $B$ are independent, then $P(A\mid B)=P(A)$, $P(B\mid A)=P(B)$, and $P(A\cap B)=P(A)\cdot P(B)$.

2.7.A.2 · The probability that event $A$ or event $B$ (or both) will occur is the probability of $A$ union $B$. The probability of the union is defined as $P(A\cup B)$.

2.7.A.3 · $P(A\text{ union }B)=P(A)+P(B)-P(A\text{ intersect }B)$, or $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.

2.8.A.1 · A random variable is a variable whose values have numerical outcomes that result from a random phenomenon.

2.8.A.2 · A probability distribution for a discrete random variable shows the probability associated with every possible value of the random variable. The sum of the probabilities over all possible values of a discrete random variable is 1.

2.8.A.3 · A discrete probability distribution can be determined using the rules of probability or estimated with a simulation.

2.8.A.4 · A discrete probability distribution can be represented as a graph, table, or function showing the probabilities associated with values of a random variable.

2.8.A.5 · A cumulative probability distribution can be represented as a table or function and shows the probability of being less than or equal to each value of the discrete random variable.

2.9.A.1 · A numerical value measuring a characteristic of a probability distribution of a random variable, or a population, is a parameter. The value of a parameter is a single, fixed value.

2.9.A.2 · The expected value (or mean) of a probability distribution is a parameter and is denoted by $E(X)$ or $\mu_X$. For a discrete random variable $X$, the expected value is calculated as $\mu_X=\sum x_i\cdot P(x_i)$, where $x_i$ is the possible value of the random variable and $P(x_i)$ is the probability of the possible value of the random variable. The expected value can be interpreted as the long-run average outcome of the random variable. The discrete random variable can only take on values that are countable or finite.

2.9.A.3 · The standard deviation of a probability distribution is a parameter represented by $SD(X)$ or $\sigma_X$. For a discrete random variable $X$, the standard deviation is calculated as $\sigma_X=\sqrt{\sum (x_i-\mu_X)^2\cdot P(x_i)}$, where $x_i$ is the possible value of the random variable, $\mu_X$ is the mean, and $P(x_i)$ is the probability of the possible value of the random variable. The standard deviation can be interpreted as the typical deviation of the values of the random variable from the mean value (or expected value) of the random variable over the long run. The square of the standard deviation of a random variable is called the variance of the random variable and is denoted as $V(X)$ or $\sigma_X^2$.

2.9.B.1 · The mean and standard deviation for the probability distribution of a discrete random variable should be interpreted in the context of a specific population.

2.10.A.1 · A binomial random variable, $X$, is a discrete random variable that counts the number of successes in repeated independent trials, $n$, that have only two possible outcomes (success or failure), with the probability of success $p$ and the probability of failure $1-p$.

2.10.B.1 · If a random variable is binomial, its mean, $\mu_X$, is $np$ and its standard deviation, $\sigma_X$, is $\sqrt{np(1-p)}$.

2.10.C.1 · The mean, standard deviation, and probabilities for a binomial distribution should be interpreted in context.

2.10.D.1 · A probability distribution can be constructed using the rules of probability or estimated with a simulation.

2.10.E.1 · The probability that a binomial random variable, $X$, has exactly $x$ successes for $n$ independent trials, when the probability of success is $p$, is calculated as $P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$, $x=0,1,2,\ldots,n$. This is called the binomial probability function.

2.11.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.

2.11.A.2 · Many continuous random variables are well-modeled by a normal distribution.

2.11.A.3 · A normal distribution can be described as a continuous, unimodal, bell-shaped, and symmetric curve.

2.11.A.4 · A normal curve can be used to model a distribution of data and a continuous random variable.

2.11.A.5 · The normal distribution, or the normal curve, is identified by two parameters, the mean, $\mu$, and the standard deviation, $\sigma$. The smaller the standard deviation, the taller and more concentrated the normal curve is around its mean. The larger the standard deviation, the shorter and less concentrated the normal curve is around its mean.

2.11.B.1 · A standard normal distribution is a normal distribution with mean $\mu=0$ and standard deviation $\sigma=1$.

2.11.C.1 · The empirical rule can be used to estimate the area of a region under the graph of the normal distribution curve. For a normal distribution, approximately 68% of 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, or the 68–95–99.7 rule.

2.11.D.1 · If the distribution of a random variable is approximately normal, the probability that the random variable takes on values within a particular interval of the random variable is determined by the area under the normal curve within that interval. The total probability or area under the normal curve is 1.

2.11.E.1 · The boundaries of an interval associated with a given area in a normal distribution can be determined using technology or using z-scores and a standard normal table.

2.11.E.2 · Intervals associated with a given area in a normal distribution can be determined by assigning appropriate inequalities to the boundaries of the intervals. To determine the intervals, $p$ is defined as a number between 0 and 100, $x_a$ is the lower bound, and $x_b$ is the upper bound on a normal distribution.

2.11.E.2.i · $P(X<x_a)=\frac{p}{100}$ means that the lowest $p$% of the values lie to the left of $x_a$.

2.11.E.2.ii · $P(x_a<X<x_b)=\frac{p}{100}$ means that $p$% of the values lie between $x_a$ and $x_b$.

2.11.E.2.iii · $P(X>x_b)=\frac{p}{100}$ means that the highest $p$% of the values lie to the right of $x_b$.

2.11.E.2.iv · To determine the most extreme $p$% of values on both sides requires dividing the area associated with $p$% into two equal areas on either extreme of the distribution: $P(X<x_a)=\frac{1}{2}\left(\frac{p}{100}\right)$ and $P(X>x_b)=\frac{1}{2}\left(\frac{p}{100}\right)$ mean 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 $x_b$.

2.11.F.1 · Percentiles and proportions may be used to compare relative positions of individual values within a normal distribution or between normal distributions.

2.12.A.1 · A sampling distribution of a statistic is the distribution of values of the statistic for all possible samples of a given size from a given population.

2.12.A.2 · The sampling distribution of a statistic can be simulated by repeatedly generating a large number of random samples from the population assuming known value(s) for the parameter(s). The value of the statistic is determined and recorded for each sample. The resulting distribution of the sample statistic values approximates the sampling distribution of the statistic.

2.12.A.3 · A randomization distribution is the distribution of a statistic generated by simulation from repeatedly randomly reallocating, or reassigning, the response values to treatment groups. The value of the statistic is determined and recorded for each reallocation, or reassignment. The resulting distribution of the statistic values approximates the sampling distribution of the statistic.

2.12.A.4 · The central limit theorem (CLT) states that the sampling distribution of a mean of a random sample has a shape that can be approximated by a normal distribution. The larger the sample is, the better the approximation will be.

3.1.A.1 · When estimating a population parameter, an estimator is unbiased if, on average, the value of the estimator does not underestimate or overestimate the population parameter.

3.1.B.1 · A sample statistic is a point estimator of the corresponding population parameter and can be thought of as the estimate of the population parameter. For example, the sample proportion $\hat p$ is a point estimator for the population proportion $p$.

3.2.A.1 · For a population with population proportion $p$, when the sampled values are independent, the sampling distribution of a sample proportion $\hat p$ has a mean $\mu_{\hat p}=p$ and a standard deviation $\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n}}$.

3.2.B.1 · Sampling without replacement requires that two conditions be met:

3.2.B.1.i · The randomization condition—the data should be collected using a random sample.

3.2.B.1.ii · The 10% condition—the population size must be at least 10 times larger than the sample size $(n\leq 10\%N)$, where $N$ is the size of the population and $n$ is the sample size.

3.2.B.2 · The sampling distribution of the sample proportion $\hat p$ is approximately normal provided the sample size is large enough. To ensure the sample size is large enough, the following condition must be met: $np\geq 10$ and $n(1-p)\geq 10$, where $np$ is the expected number of successes and $n(1-p)$ is the expected number of failures.

3.2.C.1 · The mean, standard deviation, and probabilities for a sampling distribution of a sample proportion should be interpreted in the context of a specific population.

3.3.A.1 · A confidence interval is an interval estimate for a population parameter. Based on the sample proportion, a confidence interval can be calculated to estimate the value of a single population proportion. The appropriate confidence interval procedure is a one-sample z-interval for a population proportion.

3.3.A.2 · The parameter for a confidence interval for a population proportion should reference the proportion, the response variable, and the population in context.

3.3.B.1 · A one-sample z-interval for a population proportion requires that three conditions be met:

3.3.B.1.i · The randomization condition—the data should be collected using a random sample.

3.3.B.1.ii · The 10% condition—when sampling without replacement, the population size must be at least 10 times larger than the sample size $(n\le10\%N)$, where $N$ is the size of the population and $n$ is the sample size.

3.3.B.1.iii · The normality condition—the observed number of successes, $n\hat p$, and the observed number of failures, $n(1-\hat p)$, should be at least 10.

3.3.C.1 · $z^*$ denotes a critical value, such that $-z^*$ and $+z^*$ represent the boundaries enclosing the middle C% of the standard normal distribution, in which C% is an approximate confidence level with which the population proportion is estimated.

3.3.C.2 · An interval estimate can be constructed as $\text{point estimate}\pm(\text{margin of error})$. For a population proportion, the one-sample z-interval estimate is $\hat p\pm z^*\sqrt{\frac{\hat p(1-\hat p)}{n}}$.

3.3.D.1 · The standard error (SE) of a statistic is an estimate of the standard deviation of the sampling distribution of the statistic. The standard error of the sample proportion $\hat p$ is $SE_{\hat p}=\sqrt{\frac{\hat p(1-\hat p)}{n}}$.

3.3.D.2 · The standard error quantifies the typical amount that a statistic will vary from the value of the corresponding population parameter.

3.3.D.3 · The margin of error of $\hat p$ is half the width of the confidence interval and is calculated as the critical value $(z^*)$ times the standard error (SE) of $\hat p$, which equals $z^*\sqrt{\frac{\hat p(1-\hat p)}{n}}$.

3.3.D.4 · The formula for the margin of error (MOE) can be rearranged to solve for $n$, $n=\frac{(z^*)^2(\hat p)(1-\hat p)}{(MOE)^2}$, the minimum sample needed to achieve a given margin of error. For this purpose, if $\hat p$ is not defined or unable to be calculated, use $\hat p=0.5$ in order to find the upper bound for the sample size that will result in a given margin of error.

3.4.A.1 · Because the confidence interval for a population proportion is calculated based on a sample from a population, the computed interval may or may not contain the value of the population proportion.

3.4.A.2 · The interpretation of the confidence level is that in repeated random sampling with the same sample size, approximately C% of confidence intervals calculated will capture the population proportion, with C representing the numerical value of the confidence level used.

3.4.A.3 · When interpreting a C% confidence interval for a population proportion, we say we are C% confident that the interval $(a,b)$ contains the true value of the parameter for the population, where $a$ represents the lower limit and $b$ represents the upper limit. An interpretation of a confidence interval for a population proportion includes a reference to the parameter with details about the population it represents in the context of the study.

3.4.B.1 · A confidence interval for a population proportion provides a range of plausible values that may serve as convincing evidence to support a particular claim about the population proportion.

3.4.C.1 · For a given sample, increasing the confidence level will result in the following:

3.4.C.1.i · The critical value will increase.

3.4.C.1.ii · The margin of error will increase.

3.4.C.1.iii · The width of the confidence interval will increase.

3.4.C.2 · Increasing the sample size decreases the standard error. Thus, 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 confidence interval for a population proportion with a given confidence level, the width of the interval is approximately proportional to $\frac{1}{\sqrt n}$.

3.5.A.1 · A hypothesis test is a statistical inference procedure that is used to make a decision about the value of a population parameter. The appropriate hypothesis testing procedure is a one-sample z-test for a population proportion.

3.5.A.2 · The parameter for a hypothesis test for a population proportion should reference the population parameter, the response variable, and the population in context.

3.5.B.1 · In the hypothesis testing procedure, the null hypothesis, $H_0$, is the statement about a parameter that is assumed to be correct unless there is convincing statistical evidence suggesting otherwise. It is the status quo condition. The alternative hypothesis, $H_a$, is the claim or belief about a parameter for which evidence is being collected. A researcher’s claim or belief about the population parameter is represented by the alternative hypothesis.

3.5.B.2 · The null hypothesis contains an equality reference $(=,\ge,\text{ or }\le)$. Although the null hypothesis for a one-sided test may include an inequality symbol, in AP Statistics it is tested at the boundary of equality. The alternative hypothesis with $<$ or $>$ is called one-sided, and the alternative hypothesis with $\ne$ is called two-sided.

3.5.B.3 · The null hypothesis for a one-sample z-test for a population proportion is as follows: $H_0:p=p_0$, where $p_0$ is the null hypothesized value for the population proportion. A one-sided alternative hypothesis for a one-sample z-test for a population proportion is either $H_a:p<p_0$ or $H_a:p>p_0$. A two-sided alternative hypothesis is $H_a:p\ne p_0$.

3.5.C.1 · A one-sample z-test for a population proportion requires that three conditions be met:

3.5.C.1.i · The randomization condition—the data should be collected using a random sample.

3.5.C.1.ii · The 10% condition—when sampling without replacement, the population size must be at least 10 times larger than the sample size $(n\le10\%N)$, where $N$ is the size of the population and $n$ is the sample size.

3.5.C.1.iii · The normality condition—the expected number of successes, $np_0$, and the expected number of failures, $n(1-p_0)$, should be at least 10.

3.6.A.1 · Given the null hypothesis is true, there is a probability distribution of the test statistic called the null distribution. Using the null distribution, the p-value is the probability of obtaining a test statistic as extreme or more extreme (i.e., in the direction of the alternative hypothesis) than the test statistic that is observed given that the null hypothesis is true. That is, when $x$ is the test statistic, the p-value is determined by finding the following:

3.6.A.1.i · The probability at or above the observed value of the test statistic $(P(z\ge x))$, if the alternative is $>$

3.6.A.1.ii · The probability at or below the observed value of the test statistic $(P(z\le x))$, if the alternative is $<$

3.6.A.1.iii · The probability less than or equal to the negative of the absolute value of the test statistic plus the probability greater than or equal to the absolute value of the test statistic, $(P(z\le-|x|))+(P(z\ge|x|))$, if the alternative is $\ne$

3.6.A.2 · If the distribution of the test statistic has been simulated, the p-value is the proportion of values in the null distribution that are as extreme or more extreme than the observed value of the test statistic. This is as follows:

3.6.A.2.i · The proportion at or above the observed value of the test statistic, if the alternative is $>$

3.6.A.2.ii · The proportion at or below the observed value of the test statistic, if the alternative is $<$

3.6.A.2.iii · The proportion less than or equal to the negative of the absolute value of the test statistic plus the proportion greater than or equal to the absolute value of the test statistic, if the alternative is $\ne$

3.6.A.3 · An interpretation of the p-value of a hypothesis test for a population proportion should include a statement that the p-value is computed by assuming the null hypothesis is true (i.e., by assuming the true population proportion is equal to the particular value stated in the null hypothesis in context).

3.6.A.4 · Small p-values indicate that the observed value of the test statistic would be unusual if the null hypothesis were true and therefore provide evidence for the alternative hypothesis. The lower the p-value, the more convincing the statistical evidence for the alternative hypothesis.

3.6.A.5 · p-values that are not small indicate that the observed value of the test statistic would not be unusual if the null hypothesis were true and therefore do not provide convincing statistical evidence for the alternative hypothesis, nor do they provide evidence that the null hypothesis is true.

3.7.A.1 · The test statistic for testing a population proportion is $z=\frac{\hat p-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$. The z-statistic has a standard normal distribution when the null hypothesis is true.

3.7.A.2 · The distribution of the test statistic assuming the null hypothesis is true (null distribution) can be approximated by a probability model (e.g., a theoretical distribution such as the standard normal distribution).

3.7.A.3 · The p-value of a one-sample z-test for a population proportion is found from the standard normal distribution using a table or technology.

3.7.B.1 · The significance level of a hypothesis test, denoted by $\alpha$, is the predetermined probability of rejecting the null hypothesis given that it is true. The significance level may be given or determined by the researcher. The relationship between a p-value and the significance level of a hypothesis test determines whether a result is statistically significant.

3.7.B.2 · A formal decision in a hypothesis test explicitly compares the p-value to the significance level, $\alpha$. If the p-value $\le\alpha$, then reject the null hypothesis, $H_0:p=p_0$. If the p-value $>\alpha$, then fail to reject the null hypothesis.

3.7.B.3 · Rejecting the null hypothesis means there is convincing statistical evidence to support the alternative hypothesis. Failing to reject the null hypothesis means there is not convincing statistical evidence to support the alternative hypothesis.

3.7.B.4 · A hypothesis 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.

3.7.B.5 · The results of a hypothesis test for a population proportion can serve as the statistical reasoning to support the answer to an investigative question about the population that was sampled.

3.7.B.6 · A conclusion for the hypothesis test for a population proportion is stated in context consistent with, and in terms of, the alternative hypothesis using non-definitive language. The conclusion should contain a reference to the parameter and the population.

3.8.A.1 · A Type I error occurs when there is convincing statistical evidence that the alternative hypothesis is true (due to the small p-value), but it is not.

3.8.A.2 · A Type II error occurs when there is not convincing statistical evidence that the alternative hypothesis is true (due to the large p-value), but it is.

3.8.A.3 · The power of a hypothesis test is the probability that a hypothesis test will correctly reject the false null hypothesis.

3.8.B.1 · The probability of making a Type I error is defined as the significance level, $\alpha$. For a given study and hypothesis test, the probability of making a Type I error is typically set to a small value (e.g., 0.01, 0.05, 0.10) prior to collecting the data.

3.8.B.2 · The probability of making a Type II error is $1-\text{power}$.

3.8.C.1 · For a given study and hypothesis test, the probability of a Type II error should ideally be small, and thus, the power will be large (e.g., $P(\text{Type II error})=0.20$ and power = 0.80). The probability of a Type II error decreases and the power increases when any one of the following occurs, provided the others do not change:

3.8.C.1.i · Sample size(s) increases.

3.8.C.1.ii · Standard error decreases.

3.8.C.1.iii · True parameter value is farther from the null hypothesis.

3.8.C.1.iv · Significance level $(\alpha)$ of a test increases.

3.8.D.1 · In some studies, making a Type I error may have more serious consequences than making a Type II error. In other studies, making a Type II error may have more serious consequences than making a Type I error. The consequences of each error should be considered prior to conducting the study.

3.8.D.2 · Because the significance level, $\alpha$, is the probability of making a Type I error, the consequences of a Type I error influence decisions about a significance level.

3.8.D.3 · Because sample size influences the probability of making a Type II error, the consequences of a Type II error influence decisions about how large the sample size should be.

3.9.A.1 · For two independent populations, with population proportions $p_1$ and $p_2$, when the sampled values are independent, the sampling distribution for the difference in sample proportions, $\hat p_1-\hat p_2$, has a mean, $\mu_{\hat p_1-\hat p_2}=p_1-p_2$, and standard deviation, $\sigma_{\hat p_1-\hat p_2}=\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}$.

3.9.B.1 · When sampling without replacement, two conditions must be met:

3.9.B.1.i · The randomization condition—the data should be collected using two independent random samples.

3.9.B.1.ii · The 10% condition—the size of each sample should be less than or equal to 10% of the respective population size: $n_1\leq 10\%N_1$ and $n_2\leq 10\%N_2$, where $N_1$ is the size of population 1 and $N_2$ is the size of population 2. The sample sizes are represented as $n_1$ and $n_2$.

3.9.B.2 · If the data come from an experiment, the data only need to meet the randomization condition. The treatments must be randomly assigned to the experimental units to meet the randomization condition.

3.9.B.3 · The sampling distribution for the difference between sample proportions, $\hat p_1-\hat p_2$, will have an approximately normal distribution provided both sample sizes are large enough. To ensure that both samples are large enough, the data must meet the following conditions: $n_1p_1\geq 10$, $n_1(1-p_1)\geq 10$, $n_2p_2\geq 10$, and $n_2(1-p_2)\geq 10$, where $n_1p_1$ and $n_2p_2$ are the expected number of successes and $n_1(1-p_1)$ and $n_2(1-p_2)$ are the expected number of failures.

3.9.C.1 · The mean, standard deviation, and probabilities for the sampling distribution for the difference between two sample proportions should be interpreted within the context of two specific populations.

3.10.A.1 · Based on the sample data, a confidence interval can be calculated to estimate the difference between two population proportions. The appropriate confidence interval procedure is a two-sample z-interval for a difference between population proportions.

3.10.A.2 · The parameters of a confidence interval for the difference between two population proportions should refer to the difference in the proportions, the response variable, and the populations in context.

3.10.B.1 · A two-sample z-interval for a difference between two population proportions requires that three conditions be met:

3.10.B.1.i · The randomization condition—the data should be collected using two independent random samples or a randomized experiment.

3.10.B.1.ii · The 10% condition—when sampling without replacement, the size of each sample should be less than or equal to 10% of the respective population size: $n_1\le10\%N_1$ and $n_2\le10\%N_2$, where $N_1$ is the size of population 1 and $N_2$ is the size of population 2. The sample sizes are represented as $n_1$ and $n_2$. (Note: This condition is unnecessary when the data are from a randomized experiment.)

3.10.B.1.iii · The normality condition—the number of observed successes, $n_1\hat p_1$ and $n_2\hat p_2$, and number of observed failures, $n_1(1-\hat p_1)$ and $n_2(1-\hat p_2)$, for both samples are all at least 10.

3.10.C.1 · The point estimate for the difference between two population proportions is $(\hat p_1-\hat p_2)$.

3.10.C.2 · For the difference between two population proportions, the interval estimate can be constructed as point estimate $\pm$ (margin of error). The interval estimate for the difference between two population proportions is $(\hat p_1-\hat p_2)\pm z^*\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2}}$.

3.10.D.1 · The standard error $(SE)$ for the difference between two population proportions is $SE_{\hat p_1-\hat p_2}=\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2}}$.

3.10.D.2 · For the difference between two population proportions, the margin of error is the critical value $(z^*)$ times the standard error $(SE)$ of the difference between the two proportions, which equals $z^*\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2}}$.

3.11.A.1 · Because the confidence interval for the difference between two population proportions is calculated based on samples from two populations, the computed interval may or may not contain the value for the difference between those two population proportions.

3.11.A.2 · The interpretation of the confidence level is as follows: In repeated random sampling with the same sample sizes from the same populations, approximately C% of confidence intervals created will capture the difference between the two population proportions, where C represents the numerical value of the confidence level used.

3.11.A.3 · When interpreting a C% confidence interval for the difference between two population proportions, we say we are C% confident that the interval $(a,b)$ contains the parameter for the difference between the populations, where $a$ represents the lower limit and $b$ represents the upper limit. An interpretation of a confidence interval for the difference between two population proportions includes a reference to the parameter with the details about the populations it represents in the context of the study.

3.11.B.1 · A confidence interval for the difference between two population proportions provides an interval of values that may provide convincing evidence to support a particular claim about the difference between the two population proportions. For example, if the interval contains 0, then there is insufficient evidence to conclude there is a difference between the two population proportions. If the interval does not contain 0, there is sufficient evidence to conclude there is a difference between the two population proportions.

3.12.A.1 · The appropriate testing method for the difference between two population proportions is a two-sample z-test for the difference between two population proportions.

3.12.A.2 · The parameters for a hypothesis test for the difference between two population proportions should reference the population parameters, the response variables, and the populations in context.

3.12.B.1 · For a two-sample z-test for the difference between two population proportions, the null hypothesis indicates no difference. The null hypothesis for the difference between two population proportions can be written as either $H_0:p_1=p_2$ or $H_0:p_1-p_2=0$. A one-sided alternative hypothesis for the difference between two population proportions can be written as either $H_a:p_1<p_2$ or equivalently $H_a:p_1-p_2<0$, or $H_a:p_1>p_2$ or equivalently $H_a:p_1-p_2>0$. A two-sided alternative hypothesis for the difference between two population proportions can be written as either $H_a:p_1\ne p_2$ or equivalently $H_a:p_1-p_2\ne0$.

3.12.C.1 · A two-sample z-test for a difference between two population proportions requires that three conditions be met:

3.12.C.1.i · The randomization condition—the data should be collected using two independent random samples or a randomized experiment.

3.12.C.1.ii · The 10% condition—when sampling without replacement, the size of each sample should be less than or equal to 10% of the respective population size: $n_1\le10\%N_1$ and $n_2\le10\%N_2$, where $N_1$ is the size of population 1 and $N_2$ is the size of population 2. The sample sizes are represented as $n_1$ and $n_2$. (Note: This condition is unnecessary when the data are from a randomized experiment.)

3.12.C.1.iii · The normality condition—$n_1\hat p_c$, $n_1(1-\hat p_c)$, $n_2\hat p_c$, and $n_2(1-\hat p_c)$ must all be at least 10, with $\hat p_c=\frac{n_1\hat p_1+n_2\hat p_2}{n_1+n_2}$ being the combined (or pooled) proportion assuming that $H_0$ is true $(H_0:p_1=p_2\text{ or }H_0:p_1-p_2=0)$.

3.13.A.1 · The test statistic for the difference between two population proportions is $z=\frac{(\hat p_1-\hat p_2)-0}{\sqrt{\hat p_c(1-\hat p_c)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$, where $\hat p_c=\frac{n_1\hat p_1+n_2\hat p_2}{n_1+n_2}$ is the proportion of successes for the two groups combined. The z-statistic has a standard normal distribution when the null hypothesis is true.

3.13.A.2 · The p-value for a two-sample z-test for the difference between two population proportions can be found from the standard normal distribution using a table or technology.

3.13.B.1 · The p-value is the probability of obtaining a test statistic as extreme or more extreme than the test statistic that was observed (i.e., in the direction of the alternative hypothesis) given that the null hypothesis is true. An interpretation of the p-value of a hypothesis test for a difference between two population proportions should include a statement that the p-value is computed assuming the null hypothesis is true (i.e., by assuming that the true population proportions are equal to each other in context).

3.13.C.1 · A formal decision in a hypothesis test for the difference between two population proportions explicitly compares the p-value to the significance level, $\alpha$. If the p-value $\le\alpha$, then reject the null hypothesis, $H_0:p_1=p_2$ or $H_0:p_1-p_2=0$. If the p-value $>\alpha$, then fail to reject the null hypothesis.

3.13.C.2 · The results of a hypothesis test for the difference between two population proportions can serve as the statistical reasoning to support the answer to an investigative question about the two populations that were sampled.

3.13.C.3 · A conclusion for the hypothesis test for the difference between two population proportions is stated in context consistent with, and in terms of, the alternative hypothesis using non-definitive language. The conclusion should contain a reference to the parameters and the populations.

3.14.A.1 · The chi-square statistic measures the distance between observed and expected counts relative to expected counts.

3.14.A.2 · Chi-square distributions have positive values and are skewed right. Within this family of density curves, the skew becomes less pronounced with increasing degrees of freedom.