A study collects data on average combined SAT scores (Math +...

A study collects data on average combined SAT scores (Math + Evidence-Based Reading and Writing) and percentage of students who took the exam at 100 randomly selected high schools. The following is part of the computer printout for regression: Which of the following is a correct conclusion? (A) "SAT" in the variable column indicates that SAT score is the dependent (response) variable. (B) The correlation is 0.875 . (C) The $y$ -intercept indicates the mean combined SAT score if percent of students taking the exam has no effect on combined SAT scores. (D) The $r^{2}$ value indicates that the residual plot does not show a strong pattern. (E) Schools with lower percentages of students taking the exam tend to have higher average combined SAT scores.

A study collects data on average combined SAT scores (Math + Evidence-Based Reading and Writing) and percentage of students who took the exam at 100 randomly selected high schools. The following is part of the computer printout for regression:
Which of the following is a correct conclusion?
(A) "SAT" in the variable column indicates that SAT score is
the dependent (response) variable.
(B) The correlation is 0.875 .
(C) The $y$ -intercept indicates the mean combined SAT score if percent of students taking the exam has no effect on combined SAT scores.
(D) The $r^{2}$ value indicates that the residual plot does not show a strong pattern.
(E) Schools with lower percentages of students taking the exam tend to have higher average combined SAT scores.

Key Concepts

Dependent Variable

In regression analysis, the dependent variable (also known as the response variable) is the outcome that the model is intended to predict or explain. It is distinguished from the independent variable(s) or predictor(s), which are used to explain the variation in the dependent variable.

Correlation Coefficient

The correlation coefficient quantifies the strength and direction of the linear relationship between two variables. It is a standardized measure that ranges from -1 to 1, where values near 1 or -1 indicate a strong linear relationship and values near 0 indicate a weak linear relationship.

Regression Intercept

The regression intercept is the predicted value of the dependent variable when all independent variables are equal to zero. It serves as the baseline level of the outcome but may not always have a practical interpretation if a value of zero for the predictors is not meaningful.

Coefficient of Determination

The coefficient of determination, denoted as r², represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s). It provides an indication of the overall fit of the model, but it does not directly comment on the residual patterns in the data.

Residual Analysis

Residual analysis involves examining the differences between observed and predicted values from a regression model. A residual plot is used to detect non-linearity, heteroscedasticity, or outliers, and helps validate the assumptions of the regression model.

Direction of Association

The direction of association in regression is determined by the sign of the slope coefficient. A positive slope suggests that as the independent variable increases, the dependent variable tends to increase, whereas a negative slope indicates that as the independent variable increases, the dependent variable tends to decrease.

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