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 23

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
ŷ = $$41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$$ $$\begin{array} { | c | c c c | } \hline \text { Predictor } & \text { Coef } & \text { StDev } & \mathrm { T } \\\hline \text { Constant } & 41.6 & 17.8 & 2.337 \\x _ { 1 } & - 3.18 & 1.66 & - 1.916 \\x _ { 2 } & - 1.17 & 1.13 & - 1.035 \\x _ { 3 } & 0.63 & 0.13 & 4.846 \\\hline\end{array}$$ se = 13.74, R2 = 30.0%. $$\begin{array}{l}\text { ANALYSIS OF VARIANCE }\\\begin{array} { | l | c c c c | } \hline \text { Source of Variation } & \mathrm { df } & \text { SS } & \text { MS } & \text { F } \\\hline \text { Regression } & 3 & 3716 & 1238.667 & 6.558 \\\text { Error } & 46 & 8688 & 188.870 & \\\hline \text { Total } & 49 & 12404 & & \\\hline\end{array}\end{array}$$ Do these data provide enough evidence at the 1% significance level to conclude that the final mark and the mid-term mark are positively linearly related?

---

Definitions:

John Maynard Keynes

A British economist whose theories on the influences of macroeconomic factors on economic output and government interventions shaped modern economics.

Savings

The portion of disposable income not spent on current consumption but set aside for future use, often in a deposit account or as investments.

Investment

The action or process of allocating resources, usually money, with the expectation of generating an income or profit.

Aggregate Demand

The total demand for goods and services within an economy at a given overall price level and in a given time period.