AP Statistics Comprehensive Syllabus

Unit 1: Exploring One-Variable Data

Subtopic Number	Subtopic	Key Points
1.1	Introducing statistics: What can we learn from data?	Statistics is the science of collecting, analyzing, and interpreting data in order to make informed decisions and predictions. Statistics can help us identify patterns and relationships in data, as well as test hypotheses and make predictions about future outcomes. By studying statistics, we can gain a deeper understanding of the world around us and make more informed decisions based on the data available to us.
1.2	The Language of Variation: Variables	Variables are characteristics or attributes that can vary among individuals or objects in a population. In statistics, we use variables to represent the different factors we are studying, such as age, gender, or height. Variables can be classified as categorical or quantitative, and understanding their nature and properties is essential to the process of data analysis and interpretation.
1.3	Representing a Categorical Variable with Tables	Categorical variables are qualitative variables that can be divided into distinct categories, while quantitative variables are numerical variables that can be measured and analyzed. Tables can be used to organize and display categorical data, making it easier to see patterns and relationships in the data. Frequency tables show the number of times each category occurs in a dataset, while relative frequency tables show the proportion or percentage of the total dataset represented by each category. Contingency tables, also known as two-way tables, allow us to examine the relationship between two categorical variables by displaying the frequencies or relative frequencies of each combination of categories.
1.4	Representing a Categorical Variable with Graphs	Graphs can be used to visually represent categorical data, allowing us to see patterns and relationships in the data more easily. Bar graphs are often used to represent categorical data, with the height of each bar corresponding to the frequency or relative frequency of each category. Pie charts are another way to represent categorical data, with each slice of the pie representing a different category and the size of the slice corresponding to the frequency or relative frequency of that category.
1.5	Representing a Quantitative Variable with Graphs	Graphs can be used to visually represent quantitative data, allowing us to see patterns and relationships in the data more easily. Histograms are often used to represent quantitative data, with the bars of the histogram representing the frequency or relative frequency of each interval of values. Box plots, also known as box-and-whisker plots, are another way to represent quantitative data, showing the distribution of the data in terms of its median, quartiles, and range.
1.6	Describing the Distribution of a Quantitative Variable	Describing the distribution of a quantitative variable involves identifying its center, spread, and shape. Measures of center include the mean, median, and mode, while measures of spread include the range, interquartile range, and standard deviation. The shape of the distribution can be described in terms of its symmetry, skewness, and the presence of any outliers or gaps.
1.7	Summary Statistics for a Quantitative Variable	Summary statistics are numerical values that provide an overall summary of a quantitative variable. Common summary statistics include measures of center (such as the mean and median) and measures of spread (such as the standard deviation and range). Summary statistics can be used to compare different datasets or to identify patterns and trends in the data.
1.8	Graphical Representations of Summary Statistics	Graphical representations of summary statistics can help us to better understand the distribution of a quantitative variable. Box plots, histograms, and stem-and-leaf plots are common graphical representations of summary statistics. These graphical representations can help us to identify outliers, see the overall shape of the distribution, and compare different datasets.
1.9	Comparing Distributions of a Quantitative Variable	Comparing distributions of a quantitative variable involves examining how the values of the variable differ between different groups or datasets. Common methods for comparing distributions include using box plots, histograms, and summary statistics such as means and standard deviations. Comparing distributions can help us to identify similarities and differences between groups or datasets, and to draw conclusions about the factors that may be influencing the variable.
1.10	The Normal Distribution	A normal distribution is a symmetric bell-shaped curve that is often used to model many naturally occurring phenomena. Normal distributions are characterized by their mean and standard deviation, which determine the location and spread of the curve. The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1.

Unit 2: Exploring Two-Variable Data

Subtopic Number	Subtopic	Key Points
2.1	Introducing Statistics: Are Variables Related?	This subtopic explores the relationships between different variables. This unit covers topics such as correlation, regression, and the principles of experimental design. The goal of this unit is to provide students with a foundational understanding of statistical concepts and methods that will be used throughout the rest of the course. By the end of this unit, students should be able to design and conduct experiments, analyze data using appropriate statistical methods, and communicate their findings effectively.
2.2	Representing 2 categorical variables	Representing 2 categorical variables involves examining the relationship between two different categorical variables. Common methods for representing two categorical variables include contingency tables, segmented bar graphs, and mosaic plots. These methods can help to visually represent the distribution of the variables and to identify patterns or trends in the data. Understanding the relationship between two categorical variables is important for making inferences about the population and for developing predictive models.
2.3	Statistics for 2 categorical variables	Statistics for 2 categorical variables involves using statistical methods to analyze the relationship between two different categorical variables. Common statistical methods for analyzing two categorical variables include chi-square tests of independence, odds ratios, and relative risk. These methods can help to determine whether there is a significant association between the two variables and to quantify the strength of the association. Understanding the statistical relationship between two categorical variables is important for making informed decisions and for developing effective strategies in fields such as medicine, public health, and social sciences.
2.4	Representing the relationships between two Quantitative variables	A scatter plot can show the direction and form of the relationship between two variables. Correlation measures the strength and direction of the linear relationship between two variables.
2.5	Correlation	Correlation is a measure of the strength and direction of the relationship between two quantitative variables. Correlation coefficients range from -1 to 1, with values closer to -1 indicating a strong negative correlation, values closer to 1 indicating a strong positive correlation, and values close to 0 indicating no correlation. Correlation does not imply causation, as a strong correlation between two variables may be due to a third variable that is affecting both of them. Spearman’s rank correlation coefficient is a nonparametric measure of correlation that is based on the ranks of the data, rather than their actual values.
2.6	Linear regression models	Linear regression models aim to find the line of best fit for a set of bivariate quantitative data. The strength of the linear relationship between the variables can be measured using the coefficient of determination (R-squared).
2.7	Residuals	Residuals are the differences between the actual and predicted values in a regression model. Residual plots help identify patterns in the residuals, such as nonlinearity or heteroscedasticity.
2.8	Least Squares Regression	Least squares regression is a method for fitting a linear equation to a set of data points. The goal of least squares regression is to minimize the sum of the squared differences between the predicted values and the actual values of the response variable. The slope and intercept of the least squares regression line can be used to predict the value of the response variable for a given value of the explanatory variable. The coefficient of determination, also known as R-squared, is a measure of the proportion of the variation in the response variable that can be explained by the explanatory variable.
2.9	Analyzing Departures from linearity	Nonlinear relationships can occur between variables, which cannot be modeled by a linear regression model. Residual plots can help detect nonlinearity in the data, and transformations can be applied to variables to better approximate a linear relationship.

Unit 3: Collecting Data

Subtopic Number	Subtopic	Key Points
3.1	Introducing Statistics: Do the Data we Collected Tell the truth?	This unit covers topics such as sampling methods, experimental design, bias, and the principles of inference. The goal of this unit is to help students develop critical thinking skills and to understand the importance of using appropriate methods to collect and analyze data. By the end of this unit, students should be able to design and conduct experiments, interpret and communicate the results of statistical analyses, and evaluate the validity of statistical claims.
3.2	Introduction to Planning a study	Define the research question and the population of interest. Select an appropriate sampling method and determine the sample size. Choose a data collection method and develop a data analysis plan.
3.3	Random Sampling and Data Collection	Simple random sampling: Each member of the population has an equal chance of being selected. Stratified random sampling: Dividing the population into homogeneous subgroups and selecting a random sample from each group.
3.4	Potential Problems with Sampling	Selection bias can occur if the sampling method is not random or representative of the population. Measurement bias can occur if the method of collecting data is inaccurate or flawed.
3.5	Introduction to Experimental Design	This unit includes topics such as identifying research questions, selecting appropriate study designs, and controlling for confounding variables. The goal of this unit is to help students understand how to design and conduct experiments that produce reliable and valid results. By the end of this unit, students should be able to identify appropriate study designs for different research questions, develop experimental protocols, and analyze and interpret experimental results using statistical methods.
3.6	Selecting an Experimental Design	Clearly define the research question and identify the explanatory and response variables. Randomize treatment assignments to avoid bias and control for confounding variables. Replicate the experiment to increase the precision of the estimates.
3.7	Inference and Experiments	Compare treatment groups to control group Use statistical tests to determine if differences are significant This unit covers topics such as hypothesis testing, confidence intervals, and the principles of experimental design. Consider practical significance and potential confounding variables.

Unit 4: Probability, Random Variables and Probability Distributions

Subtopic Number	Subtopic	Key Points
4.1	Introducing Statistics: Random and Non-Random Patterns?	Statistics is the science of patterns, specifically random and non-random patterns. Random patterns are those that occur by chance, while non-random patterns are those that have some underlying cause or explanation. Randomness is a fundamental concept in statistics, as it helps us understand the uncertainty and variability in our data. By analyzing and interpreting data, we can uncover patterns that help us make informed decisions and predictions about future outcomes.
4.2	Estimating probabilities using Simulation	Simulation involves using a model to generate data that represents a real-world scenario. Probability can be estimated by repeating the simulation many times and analyzing the resulting data.
4.3	Introduction to Probability	Probability is the measure of the likelihood of an event occurring. Probability is expressed as a number between 0 and 1, where 0 represents impossibility and 1 represents certainty. The addition rule of probability states that the probability of either of two mutually exclusive events occurring is the sum of their individual probabilities. The multiplication rule of probability states that the probability of two independent events occurring together is the product of their individual probabilities.
4.4	Mutually Exclusive Events	Two events are mutually exclusive if they cannot occur at the same time. If two events are mutually exclusive, the probability of either one occurring is the sum of their individual probabilities. Mutually exclusive events are also known as disjoint events. The intersection of mutually exclusive events is the empty set.
4.5	Conditional Probability	Conditional probability is the probability of an event given that another event has occurred. The conditional probability of event A given event B is calculated by dividing the probability of the intersection of A and B by the probability of event B. Bayes’ Theorem is a formula for calculating conditional probabilities. Conditional probabilities can help us make predictions and decisions based on additional information that becomes available.
4.6	Independent events and Union of events	Two events are independent if the occurrence of one event does not affect the probability of the other event. The multiplication rule of probability can be used to calculate the probability of two independent events occurring together. The union of two events is the set of outcomes that are in either event or both events. The addition rule of probability can be used to calculate the probability of the union of two events.
4.7	Introduction to Random variables and probability distributions	Random variables are numerical outcomes of random events or experiments. Probability distributions describe the probabilities of each possible value of a random variable.
4.8	Mean and Standard Deviation of Random Variables	The mean of a random variable is the average value of the outcomes, weighted by their probabilities.” The variance of a random variable is a measure of how spread out its values are around the mean. The standard deviation of a random variable is the square root of its variance. The mean and standard deviation of a random variable are important summary statistics that help us understand its properties and behavior.
4.9	Combining Random Variables	The sum of two random variables is another random variable whose mean is the sum of the individual means. The variance of the sum of two independent random variables is the sum of their individual variances. The difference of two random variables is another random variable whose mean is the difference of the individual means. The variance of the difference of two independent random variables is the sum of their individual variances.
4.10	Introduction to Binomial Distribution	The binomial distribution models the number of successes in a fixed number of independent trials. The mean of the binomial distribution is the product of the number of trials and the probability of success, and the variance is the product of the number of trials, the probability of success, and the probability of failure.
4.11	Parameters for a binomial distribution	The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. The binomial distribution is characterized by two parameters: the probability of success in each trial (p) and the number of trials (n). The probability of success must be the same in each trial, and the trials must be independent.
4.12	The geometric distribution	The geometric distribution models the probability of the first success in a sequence of independent trials. The geometric distribution is a probability distribution that models the number of independent Bernoulli trials needed to obtain the first success, where each trial has only two possible outcomes (success or failure) and the probability of success is constant and denoted by p.

Unit 5: Sampling Distributions

Subtopic Number	Subtopic	Key Points
5.1	Introducing Statistics: Why Is My Sample Not Like Yours?	Samples from the same population have variability in their statistics due to random sampling. The standard deviation of the sample mean decreases with increasing sample size.
4.2	The Normal Distribution, Revisited	The normal distribution is a continuous probability distribution that is symmetric and bell-shaped. The standard normal distribution has a mean of 0 and a standard deviation of 1, and it is often used as a reference distribution for hypothesis testing and confidence intervals. The empirical rule states that for a normal distribution, about 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations
5.3	The central limit theorem	The Central Limit Theorem states that the sampling distribution of the mean of any independent, random variables will be approximately normal, regardless of the underlying distribution of the population.
5.4	Biased and unbiased point estimates	The use of unbiased point estimates is generally preferred in statistical inference, as biased estimates can lead to incorrect conclusions.
5.5	Sampling distributions for sample proportions	A sample proportion is an unbiased estimator of the population proportion. The sampling distribution of a sample proportion is approximately normal if the sample size is sufficiently large.
5.6	Sampling distributions for differences in sample proportions	The standard error of the difference in sample proportions is a measure of the variability of the differences between two independent proportions, and it decreases as the sample size increases. The sampling distribution of the difference in sample proportions is approximately normal if the sample sizes are large enough and the population proportions are not too close to 0 or 1.
5.7	Sampling distributions for sample means	The sampling distribution of sample means approaches a normal distribution as sample size increases, regardless of the shape of the population distribution. The standard error of the mean, which represents the variability of sample means, decreases as sample size increases.
5.8	Sampling distributions for differences in sample means	A sampling distribution is the distribution of sample statistics, such as sample means, based on repeated random samples from a population. The sampling distribution of the difference in sample means measures the variability in the differences between two independent samples. The standard error of the difference in sample means is a measure of the variability of the differences between two independent samples, and it decreases as the sample size increases.

Unit 6: Inference for Categorical Data: Proportions

Subtopic Number	Subtopic	Key Points
6.1	Introducing Statistics: Why Be Normal?	Even when the population is not normal, the distribution of sample means tends to become approximately normal as the sample size increases, due to the central limit theorem.
6.2	Constructing a Confidence Interval for a Population Proportion	A confidence interval is a range of values that is likely to contain the true population proportion with a specified level of confidence.
6.3	Justifying a Claim Based on a Confidence Interval for a Population Proportion	A confidence interval is a range of plausible values for a population parameter, such as a proportion, based on a sample statistic and a level of confidence. To use a confidence interval to justify a claim, we must ensure that the interval excludes values that are inconsistent with the claim and includes values that are consistent with the claim. The level of confidence and the margin of error of a confidence interval are inversely related, meaning that increasing the level of confidence increases the width of the interval and vice versa.
6.4	Setting up a test for a population proportion	Null and alternative hypotheses are used to set up the test for a population proportion. A test statistic is calculated from the sample data and compared to a critical value or p-value to determine if the null hypothesis should be rejected.
6.5	Interpreting  p-values	A p-value is the probability of observing a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true.
6.6	Concluding a test for Population Proportion	To conclude a test for a population proportion, we compare the calculated test statistic to the critical value or p-value based on the chosen level of significance. If the test statistic falls within the rejection region or the p-value is less than the chosen level of significance, we reject the null hypothesis in favor of the alternative hypothesis. The conclusion of a test for a population proportion should be interpreted in the context of the research question and the potential consequences of making a type I or type II error.
6.7	Potential Errors when Performing tests	Type I error occurs when we reject a true null hypothesis, and the probability of making a type I error is equal to the level of significance. Type II error occurs when we fail to reject a false null hypothesis, and the probability of making a type II error is affected by the sample size, the level of significance, and the effect size. The power of a test is the probability of rejecting a false null hypothesis, and it increases with the sample size, the effect size, and the level of significance.
6.8	Confidence Intervals for the Difference of Two Proportions	Confidence intervals for the difference of two proportions are used to estimate the difference between two population proportions based on sample data. The formula for the standard error of the difference of two proportions takes into account the sample sizes and sample proportions of both groups. The confidence level and margin of error of a confidence interval for the difference of two proportions depend on the sample sizes and proportions, as well as the level of confidence chosen
6.9	Justifying a Claim Based on a Confidence Interval for a Difference of Population Proportions	To justify a claim based on a confidence interval for the difference of population proportions, we must ensure that the interval excludes values that are inconsistent with the claim and includes values that are consistent with the claim. If the confidence interval for the difference of population proportions does not include zero, we can conclude that the difference between the two population proportions is statistically significant. The interpretation of a confidence interval for the difference of population proportions should be based on the context of the research question and the potential consequences of making a type I or type II error
6.10	Setting Up a Test for the Difference of Two Population Proportions	To set up a test for the difference of two population proportions, we first formulate the null and alternative hypotheses based on the research question. The test statistic for the difference of two population proportions follows an approximate normal distribution if certain conditions are met, such as independent and random samples with large enough sample sizes. The level of significance chosen for the test determines the critical value or p-value that is compared to the calculated test statistic to determine the conclusion of the test.
6.11	Carrying Out a Test for the Difference of Two Population Proportions	To carry out a test for the difference of two population proportions, we calculate the test statistic using the sample proportions and sample sizes of both groups. If the calculated test statistic falls within the rejection region or the p-value is less than the level of significance, we reject the null hypothesis in favor of the alternative hypothesis.

Unit 7: Inference for Quantitative Data: Means

Subtopic Number	Subtopic	Key Points
7.1	Introducing Statistics: Should I Worry About Error?	“Introducing Statistics: Should I Worry About Error?” discusses the concept of sampling variability and how it affects the accuracy and precision of statistical estimates. The video emphasizes the importance of understanding the sources of error in statistical analyses, such as sampling error, measurement error, and bias, and how to minimize them. By acknowledging and accounting for sources of error, statisticians can improve the validity and reliability of their conclusions and make more informed decisions based on data.
7.2	Constructing a Confidence Interval for a Population Mean	Constructing a Confidence Interval for a Population Mean is a statistical method that provides an estimated range of values for a population mean based on a sample mean and sample size. It explains how to calculate the standard error of the mean, select the appropriate confidence level, and determine the margin of error to construct a confidence interval. By using this method, researchers can estimate the true value of a population mean with a certain degree of confidence, providing valuable information for decision-making and further analysis.
7.3	Justifying a Claim About a Population Mean Based on a Confidence Interval	This method involves calculating a confidence interval for the population mean based on a sample mean and sample size, and then comparing the claim to the interval to determine if it falls within the range of plausible values. By using this method, researchers can determine if a claim about a population mean is supported by the data and make informed decisions based on the results.
7.4	Setting Up a Test for a Population Mean	Specify the null and alternative hypotheses for the test of a population mean. Check the conditions for conducting the test, including independence and normality. Choose an appropriate test statistic and calculate the p-value or critical value for the test.
7.5	Carrying Out a Test for a Population Mean	Use the test statistic and p-value or critical value to make a decision about rejecting or failing to reject the null hypothesis. Interpret the results of the test in the context of the problem. Calculate and report the confidence interval for the population mean, if applicable.
7.6	Confidence Intervals for the Difference of Two Means	A confidence interval for the difference of two means is calculated by subtracting the lower confidence limit of one sample mean from the upper confidence limit of the other sample mean. The formula for calculating the margin of error for a confidence interval for the difference of two means involves the standard errors of the two means, which are combined using a formula that takes into account the sample sizes of the two populations. In order to construct a confidence interval for the difference of two means, we need to assume that the two populations have the same variance, or that the sample sizes are large enough to allow us to use the pooled standard deviation.
7.7	Justifying a Claim About the Difference of Two Means Based on a Confidence Interval	We are 95% confident that the true difference between the means falls within the range of our confidence interval, suggesting that there is a significant difference between the two population means. When the confidence interval for the difference between the means does not include zero, we can conclude that there is a statistically significant difference between the two populations. The fact that the confidence interval for the difference between the means is narrow indicates that our sample size was large enough to produce a reliable estimate of the true difference between the two population means
7.8	Setting Up a Test for the Difference of Two Population Means	To test the difference of two population means, we can use a two-sample t-test, assuming that the samples are independent and the populations have equal variances. We can set up a null hypothesis that there is no difference between the means of the two populations and an alternative hypothesis that there is a significant difference. To conduct the test, we can calculate the t-statistic using the sample means, sample standard deviations, and sample sizes, and compare it to the critical t-value from the t-distribution with degrees of freedom based on the sample sizes and assuming equal variances
7.9	Carrying Out a Test for the Difference of Two Population Means	After setting up the null and alternative hypotheses and determining the significance level, we can calculate the t-statistic and degrees of freedom using the sample data. We can use a t-table or statistical software to find the critical t-value for the given degrees of freedom and level of significance. If the calculated t-statistic falls within the rejection region determined by the critical t-value and level of significance, we reject the null hypothesis and conclude that there is a significant difference between the population means.
7.10	Skills Focus: Selecting, Implementing, and Communicating Inference Procedures	When selecting an inference procedure, consider the type of data, the research question, and any assumptions that need to be made. To implement the procedure, collect relevant data, check assumptions, calculate test statistics, and interpret results. Communicate the inference procedure and results clearly, including appropriate visual displays, measures of center and spread, and confidence or significance levels

Unit 8: Inference for Categorical Data: Chi-Square

Subtopic Number	Subtopic	Key Points
8.1	Introducing Statistics: Are My Results Unexpected?	Statistical inference allows us to make conclusions about a population based on a sample, and to assess the degree of uncertainty associated with those conclusions. The null hypothesis represents the status quo or default assumption, and the alternative hypothesis represents the research question or claim that we want to test. The p-value measures the strength of evidence against the null hypothesis and is compared to the level of significance to determine whether we reject or fail to reject the null hypothesis
8.2	Setting up a chi-square test for goodness of fit	The chi-square test for goodness of fit is used to compare observed frequencies in a single categorical variable to expected frequencies based on a specified distribution or proportion. To set up a chi-square test for goodness of fit, we must identify the null and alternative hypotheses, choose a level of significance, calculate the expected frequencies, and calculate the test statistic. The degrees of freedom for a chi-square test for goodness of fit depend on the number of categories minus one, and the critical value can be found using a chi-square distribution table or calculator.
8.3	Carrying Out a Chi-Square Test for Goodness of Fit	To carry out a chi-square test for goodness of fit, we calculate the test statistic by summing the squared differences between the observed and expected frequencies divided by the expected frequencies. The p-value for a chi-square test for goodness of fit is the probability of observing a test statistic as extreme or more extreme than the one calculated, assuming the null hypothesis is true. If the p-value is less than the level of significance, we reject the null hypothesis and conclude that there is evidence of a difference between the observed and expected frequencies in the categorical variable.
8.4	Expected Counts in Two-Way Tables	Expected counts in two-way tables are calculated based on the assumption of independence between two categorical variables. To calculate expected counts, we multiply the row total, column total, and grand total and divide by the total number of observations in the table. Expected counts can be used to test the independence between two categorical variables using the chi-square test for independence
8.5	Setting Up a Chi-Square Test for Homogeneity or Independence	A chi-square test for homogeneity or independence is used to determine whether there is evidence of a relationship between two or more categorical variables. To set up a chi-square test for homogeneity or independence, we first state the null and alternative hypotheses, choose a level of significance, calculate the expected counts, and compute the test statistic. The degrees of freedom for a chi-square test for homogeneity or independence depend on the number of categories and the number of variables, and the critical value can be found using a chi-square distribution table or calculator
8.6	Carrying Out a Chi-Square Test for Homogeneity or Independence	To carry out a chi-square test for homogeneity or independence, we first calculate the test statistic and find its associated p-value using a chi-square distribution table or calculator. If the p-value is less than the level of significance, we reject the null hypothesis and conclude that there is evidence of a relationship between the categorical variables. The conclusions from a chi-square test for homogeneity or independence should be supported by appropriate visual aids, such as contingency tables, bar charts, or mosaic plots.
8.7	Skills Focus: Selecting an Appropriate Inference Procedure for Categorical Data	If comparing proportions from two independent groups, use a two-sample z-test or two-sample chi-square test.

Unit 9: Inference for Quantitative Data: Slopes

Subtopic Number	Subtopic	Key Points
9.1	Introducing Statistics: Do Those Points Align?	Scatterplots are used to visualize the relationship between two quantitative variables, and can show whether there is a linear relationship between the two variables. The correlation coefficient measures the strength and direction of the linear relationship between two variables, and ranges from -1 to 1, with values close to -1 or 1 indicating a strong linear relationship. The coefficient of determination (R-squared) measures the proportion of variability in the response variable that can be explained by the linear relationship with the explanatory variable.
9.2	Confidence intervals for the slope of a regression model	Confidence intervals for the slope estimate in regression help to determine the range of values that the true slope is likely to fall within.
9.3	Justifying a Claim About the Slope of a Regression Model Based on a Confidence Interval	To justify a claim about the slope of a regression model based on a confidence interval, we must ensure that the interval excludes values that are inconsistent with the claim and includes values that are consistent with the claim. If the confidence interval for the slope of a regression model does not include zero, we can conclude that there is a statistically significant linear relationship between the two variables.
9.4	Setting up a test for the slope of a regression model	The null hypothesis states that there is no linear relationship between the two variables, while the alternative hypothesis suggests there is a significant linear relationship.
9.5	Carrying out a test for the slope of a regression model	The null hypothesis states that there is no linear relationship between the two variables, while the alternative hypothesis suggests there is a significant linear relationship.
9.6	Skills Focus: Selecting an appropriate inference procedure	The choice of inference procedure depends on the type of data, the research question, and the sample size. For categorical data, we use chi-square tests for independence, homogeneity, or goodness of fit, while for quantitative data, we use t-tests, ANOVA, or regression analysis.

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