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

question 83

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 $$\begin{array} { | l | c c c c | } \hline \text { Source of Variation } & d f & \text { SS } & \text { MS } & F \\\hline \text { Regression } & 2 & 3452 & 1726 & 6.663 \\\text { Error } & 17 & 4404 & 259.059 & \\\hline \text { Total } & 19 & 7856 & & \\\hline\end{array}$$ Do these results allow us to conclude at the 5% significance level that the model is useful in predicting the team's winning percentage?

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Definitions:

Y Chromosome

One of the two sex chromosomes of mammals and some other organisms; the Y chromosome determines male gender in humans; normal females are XX, normal males are XY. Compare with X chromosome.

Cystic Fibrosis

A hereditary disorder affecting the exocrine glands, causing the production of abnormally thick mucus, leading to blockage of the pancreatic ducts, intestines, and bronchi and resulting in respiratory and digestive problems.

Heterozygous

A genetic condition where an individual has two different alleles at a particular gene locus, one from each parent.

Mendel's Principle

The foundational principles of genetics discovered by Gregor Mendel, including the laws of segregation and independent assortment.