Exploring Two-Variable Data · ⁨استكشاف بيانات متغيرين⁩

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

Learning Objective VAR-1.D: Identify questions to be answered about possible relationships in data. [Skill 1.A]

• VAR-1.D.1 Apparent patterns and associations in data may be random or not.

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

Learning Objective UNC-1.P: Compare numerical and graphical representations for two categorical variables. [Skill 2.D]

• UNC-1.P.1 Side-by-side bar graphs, segmented bar graphs, and mosaic plots are examples of bar graphs for one categorical variable, broken down by categories of another categorical variable.
• UNC-1.P.2 Graphical representations of two categorical variables can be used to compare distributions and/or determine if variables are associated.
• UNC-1.P.3 A two-way table, also called a contingency table, is used to summarize two categorical variables. The entries in the cells can be frequency counts or relative frequencies.
• UNC-1.P.4 A joint relative frequency is a cell frequency divided by the total for the entire table.

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

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

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

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

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

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

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

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

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

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

• DAT-1.A.1 A description of a scatter plot includes form, direction, strength, and unusual features.
• DAT-1.A.2 The direction of the association shown in a scatterplot, if any, can be described as positive or negative.
• DAT-1.A.3 A positive association means that as values of one variable increase, the values of the other variable tend to increase. A negative association means that as values of one variable increase, values of the other variable tend to decrease.
• DAT-1.A.4 The form of the association shown in a scatterplot, if any, can be described as linear or non-linear to varying degrees.
• DAT-1.A.5 The strength of the association is how closely the individual points follow a specific pattern, e.g., linear, and can be shown in a scatterplot. Strength can be described as strong, moderate, or weak.
• DAT-1.A.6 Unusual features of a scatter plot include clusters of points or points with relatively large discrepancies between the value of the response variable and a predicted value for the response variable.

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

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

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

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

• DAT-1.C.1 The correlation, $r$, is unit-free, and always between $-1$ and 1, inclusive. A value of $r = 0$ indicates that there is no linear association. A value of $r = 1$ or $r = -1$ indicates that there is a perfect linear association.
• DAT-1.C.2 A perceived or real relationship between two variables does not mean that changes in one variable cause changes in the other. That is, correlation does not necessarily imply causation.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• DAT-1.I.1 An outlier in regression is a point that does not follow the general trend shown in the rest of the data and has a large residual when the Least Squares Regression Line (LSRL) is calculated.
• DAT-1.I.2 A high-leverage point in regression has a substantially larger or smaller $x$-value than the other observations have.
• DAT-1.I.3 An influential point in regression is any point that, if removed, changes the relationship substantially. Examples include much different slope, $y$-intercept, and/or correlation. Outliers and high leverage points are often influential.

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

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