As Part of a Study at a Large University, Data $x _ { 1 } =$

Short Answer

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 student's 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): $x _ { 1 } =$ average high school grade in mathematics (HSM)
$x _ { 2 } =$ average high school grade in science (HSS)
$x _ { 3 } =$ average high school grade in English (HSE)
$x _ { 4 } =$ SAT mathematics score (SATM)
$x _ { 5 } =$ SAT verbal score (SATV)
A first-order model was fit to data with the following results:
$\begin{array}{lrrrrr}\hline \text { SOURCE } & \text { DF } & \text { SS } & \text { MS } & \text { FVALUE } & \text { PROB }>\text { F } \\\text { MODEL } & 5 & 28.64 & 5.73 & 11.69 & .0001 \\\text { ERROR } & 218 & 106.82 & 0.49 & & \\\text { TOTAL } & 223 & 135.46 & & &\end{array}$

$\begin{array}{llll}\text { ROOT MSE } & 0.700 & \text { R-SOUARE } & 0.211\end{array}$
$\begin{array}{llll}\text { DEP MEAN } & 4.635 & \text { ADJ R-5Q } & 0.193\end{array}$

$\begin{array}{lrrrr}&\text { PARAMETER }&\text {STANDARD } & \text { T FOR O: }\\ \text { VARIABLE } & \text { ESTIMATE } & \text { ERROR } & \text { PARAMETER }=0 & \text { PROB }>|T|\\\text { INTERCEPT } & 2.327 & 0.039 & 5.817 & 0.0001 \\\text { X1 (HSM) } & 0.146 & 0.037 & 3.718 & 0.0003 \\\text { X2 (HSS) } & 0.036 & 0.038 & 0.950 & 0.3432 \\\text { X3 (HSE) } & 0.055 & 0.040 & 1.397 & 0.1637 \\\text { X4 (SATM) } & 0.00094 & 0.00068 & 1.376 & 0.1702 \\\text { X5 (SATV) } & -0.00041 & 0.00059 & -0.689 & 0.4915 \\\hline\end{array}$

Interpret the value under the column heading $\mathrm { PROB } > \mathrm { F }$ .
A) There is sufficient evidence (at $\alpha = .01$ ) to conclude that the first-order model is statistically useful for predicting GPA.
B) There is insufficient evidence (at $\alpha = .01$ ) to conclude that the first-order model is statistically useful for predicting GPA.
C) Over $99 \%$ of the variation in GPAs can be explained by the model.
D) Accept $H _ { 0 }$ (at $\alpha = .01$ ); at least one of the $\beta$ -coefficients in the first-order model is equal to 0 .