An Avid Football Fan Was in the Process of Examining $y = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 } + \varepsilon$

question 14

Essay

An avid football fan was in the process of examining the factors that determine the success or failure of football teams. He noticed that teams with many rookies and teams with many veterans seem to do quite poorly. To further analyse his beliefs, he took a random sample of 20 teams and proposed a second-order model with one independent variable. The selected model is: $y = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 } + \varepsilon$ .
where
y = winning team's percentage.
x = average years of professional experience.
The computer output is shown below:
THE REGRESSION EQUATION IS: $y =$ $32.6 + 5.96 x - 0.48 x ^ { 2 }$ $$\begin{array} { | c | r c r | } \hline \text { Predictor } & \text { Coef } & S 2 D e v & T \\\hline \text { Constant } & 32.6 & 19.3 & 1.689 \\x & 5.96 & 2.41 & 2.473 \\x ^ { 2 } & - 0.48 & 0.22 & - 2.182 \\\hline\end{array}$$ S = 16.1 R-Sq = 43.9%.
ANALYSIS OF VARIANCE $$\begin{array} { | l | c c c c | } \hline \text { Source of Variation } & d f & \text { SS } & \text { MS } & F \\\hline \text { Regression } & 2 & 3452 & 1726 & 6.663 \\\text { Error } & 17 & 4404 & 259.059 & \\\hline \text { Total } & 19 & 7856 & & \\\hline\end{array}$$ Test to determine at the 10% significance level whether the $x ^ { 2 }$ term should be retained.

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Definitions:

Clarifying Additive

A substance added to liquids to remove or to reduce the appearance of cloudiness or turbidity.

Polypropylene

A thermoplastic polymer used in a wide variety of applications, known for its toughness, flexibility, and resistance to various chemical solvents, acids, and bases.

Millad NX8000

A highly efficient clarifier used in the production of transparent polypropylene plastics, enhancing clarity and physical properties.

Acetonide

An acetonide is a ketal formed from the diol (a molecule with two hydroxyl groups) and acetone, often used as a protective group in organic synthesis.