A Statistics Professor Investigated Some of the Factors That Affect $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$

question 83

Essay

A statistics professor investigated some of the factors that affect an individual student's final grade in his or her course. He proposed the multiple regression model: $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ .
Where:
y = final mark (out of 100). $x _ { 1 }$ = number of lectures skipped. $x _ { 2 }$ = number of late assignments. $X _ { 3 }$ = mid-term test mark (out of 100).
The professor recorded the data for 50 randomly selected students. The computer output is shown below. THE REGRESSION EQUATION IS
$= 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$

$\begin{array}{|c|ccc|}\hline \text { Predictor } & \text { Coef } & \text { StDev } & \text { T } \\\hline \text { Constant } & 41.6 & 17.8 & 2.337 \\x_{1} & -3.18 & 1.66 & -1.916 \\x_{2} & -1.17 & 1.13 & -1.035 \\x_{3} & 0.63 & 0.13 & 4.846 \\\hline\end{array}$
$$\mathrm { S } = 13.74 \quad \mathrm { R } - \mathrm { Sq } = 30.0 \%$$
ANALYSIS OF VARIANCE
$\begin{array}{|l|cccc|}\hline \text { Source of Variation } & \text { df } & \mathrm{SS} & \mathrm{MS} & \mathrm{F} \\\hline \text { Regression } & 3 & 3716 & 1238.667 & 6.558 \\\text { Error } & 46 & 8688 & 188.870 & \\\hline \text { Total } & 49 & 12404 & & \\\hline\end{array}$
Do these data provide enough evidence to conclude at the 5% significance level that the model is useful in predicting the final mark?