Unit 5 · Regression Analysis

Bivariate quantitative data ● (core concept) — A data set of ordered pairs from two quantitative variables measured on the same individuals, used to construct a scatterplot.

Scatterplot ● (core concept) — A graph showing the relationship between two quantitative variables, with the explanatory variable on the x-axis and the response variable on the y-axis.

Explanatory variable ● (core concept) — The variable placed on the x-axis whose values are used to explain or predict the corresponding values of the response variable.

Response variable ● (core concept) — The variable placed on the y-axis whose values are explained or predicted by the explanatory variable.

Form of association ● (core concept) — The shape of the pattern in a scatterplot, described as linear or nonlinear.

Direction of association ● (core concept) — Positive if response values tend to increase as explanatory values increase; negative if they tend to decrease.

Strength of association ● (core concept) — How closely the points in a scatterplot follow the general pattern, described as strong, moderate, or weak.

Unusual features in a scatterplot ● (core concept) — Clusters of points or individual points that do not fit the general pattern of association between the two variables.

Correlation coefficient (r) ● (core concept) — Summarizes the strength and direction of the linear association between two quantitative variables. It is unit-free and always between -1 and 1, inclusive.

Interpreting the correlation coefficient ● (core concept) — The closer r is to -1 or 1, the stronger the linear association; r = 0 means no linear association and r = ±1 means a perfect linear association. A value near ±1 does not guarantee a linear model is appropriate.

Correlation does not imply causation ● (core concept) — A perceived or real relationship between two variables does not mean that changes in one variable cause changes in the other.

Linear regression model ● (core concept) — A linear equation that approximates the relationship between x and y when the form appears linear, using an explanatory variable x to predict the response variable y.

Predicted response value (y-hat) ● (core concept) — The value predicted by the regression model for a given x, calculated as y-hat = a + bx, where a is the y-intercept and b is the slope.

Slope of the regression line ● (core concept) — The value b in y-hat = a + bx; the predicted change in the response variable for a one-unit increase in the explanatory variable.

y-intercept of the regression line ● (core concept) — The value a in y-hat = a + bx; the predicted value of the response variable when the explanatory variable equals 0.

Extrapolation ● (core concept) — Predicting a response value using an x-value beyond the interval of x-values used to determine the regression line; the prediction is less reliable the further it is extrapolated.

Interpolation ● (core concept) — Predicting a response value using an x-value within the interval of x-values used to determine the regression line.

Residual ● (core concept) — The difference between the observed and predicted response for a given x: residual = observed y - predicted y-hat. A positive residual means the model underpredicts; a negative residual means it overpredicts.

Residual plot ● (core concept) — A scatterplot of the residuals versus the predicted response values (or the explanatory variable values), used to investigate whether a linear regression model is appropriate.

Assessing linearity with residual plots ● (core concept) — Apparent randomness in a residual plot confirms a linear form and indicates the linear model is appropriate; curvature in the plot suggests the linear model is not the most appropriate model.

Least-squares regression line (LSRL) ● (core concept) — The regression line fit by minimizing the sum of the squares of the residuals, calculated using technology; it passes through the point (x-bar, y-bar). Also called the line of best fit.

Coefficient of determination (r-squared) ● (core concept) — The square of the correlation coefficient; the proportion of variation in the response variable that is explained by the linear relationship with the explanatory variable.

Interpreting the slope in context ● (core concept) — The slope is interpreted as the predicted increase or decrease in the response variable for a one-unit increase in the explanatory variable, stated in the context of the data.

Interpreting the y-intercept in context ● (core concept) — The predicted value of the response variable when x = 0, stated in context. It may have no reasonable interpretation if x = 0 is extrapolation or gives an impossible value (e.g., a negative height).