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

question 104

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 .

• 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 .

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Definitions:

Growth Hormone

A hormone secreted by the pituitary gland that stimulates growth, cell reproduction, and regeneration in humans and other animals.

Anterior Pituitary

The front part of the pituitary gland, which produces hormones that regulate various physiological processes, including growth, reproduction, and metabolism.

Adrenocorticotropic Hormone

A hormone produced by the pituitary gland that stimulates the adrenal glands to release cortisol and other hormones.

Pituitary Gland

A small, pea-sized gland located at the base of the brain that produces hormones regulating important bodily functions.