As part of a study at a large university, data were collected on n = 224 freshmen computer science (CS)majors in a particular year. The researchers were interested in modeling y, a studentʹs grade
Point average (GPA)after three semesters, as a function of the following independent variables
(recorded at the time the students enrolled in the university): $$\begin{array} { l }
x _ { 1 } = \text { average high school grade in mathematics (HSM) } \
x _ { 2 } = \text { average high school grade in science (HSS) } \
x _ { 3 } = \text { average high school grade in English (HSE) } \
x _ { 4 } = \text { SAT mathematics score (SATM) } \
x _ { 5 } = \text { SAT verbal score (SATV) }
\end{array}$$
A first-order model was fit to data with $R ^ { 2 } = 0.211$.
What is the correct interpretation of $R ^ { 2 }$, the coefficient of determination for the model?
A) Approximately $79 \%$ of the sample variation in GPAs can be explained by the first-order model.
B) We are $79 \%$ confident that the model is useful for predicting $y$.
C) We expect to predict GPA to within approximately .21 of its true value.
D) Approximately $21 \%$ of the sample variation in GPAs can be explained by the first-order model.

Question

As part of a study at a large university, data were collected on n = 224 freshmen computer science (CS)majors in a particular year. The researchers were interested in modeling y, a studentʹs grade
Point average (GPA)after three semesters, as a function of the following independent variables
(recorded at the time the students enrolled in the university): $$\begin{array} { l } 
x _ { 1 } = \text { average high school grade in mathematics (HSM) } \
x _ { 2 } = \text { average high school grade in science (HSS) } \
x _ { 3 } = \text { average high school grade in English (HSE) } \
x _ { 4 } = \text { SAT mathematics score (SATM) } \
x _ { 5 } = \text { SAT verbal score (SATV) }
\end{array}$$
A first-order model was fit to data with $R ^ { 2 } = 0.211$.
What is the correct interpretation of $R ^ { 2 }$, the coefficient of determination for the model?
A) Approximately $79 \%$ of the sample variation in GPAs can be explained by the first-order model.
B) We are $79 \%$ confident that the model is useful for predicting $y$.
C) We expect to predict GPA to within approximately .21 of its true value.
D) Approximately $21 \%$ of the sample variation in GPAs can be explained by the first-order model.

Quizplus · Accepted Answer

The coefficient of determination, $R^2$, indicates the proportion of the variance in the dependent variable that is predictable from the independent variables. In this case, $R^2 = 0.211$ means that approximately 21% of the variation in students' GPAs can be explained by the model that includes their high school grades and SAT scores.

As Part of a Study at a Large University, Data