A Traffic Consultant Has Analysed the Factors That Affect the Number

question 14

Essay

A traffic consultant has analysed the factors that affect the number of traffic fatalities. She has come to the conclusion that two important variables are the number of cars and the number of tractor-trailer trucks. She proposed the second-order model with interaction: $y =$ $\beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } ^ { 2 } + \beta _ { 4 } x _ { 2 } ^ { 2 } + \beta _ { 5 } x _ { 1 } x _ { 2 } + \varepsilon$ .
Where:
y = number of annual fatalities per shire. $x _ { 1 }$ = number of cars registered in the shire (in units of 10 000). $x _ { 2 }$ = number of trucks registered in the shire (in units of 1000).
The computer output (based on a random sample of 35 shires) is shown below.
THE REGRESSION EQUATION IS $y =$ $69.7 + 11.3 x _ { 1 } + 7.61 x _ { 2 } - 1.15 x _ { 1 } ^ { 2 } - 0.51 x _ { 2 } ^ { 2 } - 0.13 x _ { 1 } x _ { 2 }$ . $$\begin{array} { | c | r r c | } \hline \text { Predictor } & \text { Coef } & \text { SiDev } & T \\\hline \text { Constant } & 69.7 & 41.3 & 1.688 \\x _ { 1 } & 11.3 & 5.1 & 2.216 \\x _ { 2 } & 7.61 & 2.55 & 2.984 \\x _ { 1 } ^ { 2 } & - 1.15 & 0.64 & - 1.797 \\x _ { 2 } ^ { 2 } & - 0.51 & 0.20 & - 2.55 \\x _ { 1 } x _ { 2 } & - 0.13 & 0.10 & - 1.30 \\\hline\end{array}$$ S = 15.2 R-Sq = 47.2%.
ANALYSIS OF VARIANCE $$\begin{array} { | l | c c c c | } \hline \begin{array} { l } \text { Source of } \\\text { Variation }\end{array} & d f & \text { SS } & M S & F \\\hline \text { Regression } & 5 & 5959 & 1191.800 & 5.181 \\\text { Error } & 29 & 6671 & 230.034 & \\\hline \text { Total } & 34 & 12630 & & \\\hline\end{array}$$ Test at the 1% significance level to determine whether the $x _ { 1 } ^ { 2 }$ term should be retained in the model.

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Definitions:

Chronic Bronchitis

Inflammation of the bronchi in the lungs that is usually caused by smoking or exposure to environmental contaminants.

Mucus

A slippery secretion produced by mucous membranes and glands for protection and lubrication in various parts of the body.

Bronchioles

The smallest air passages in the lungs, extending from the bronchi and leading to the alveolar sacs.

Cartilage

A flexible and semi-rigid supportive connective tissue found throughout the body, including the joints, ear, nose, and trachea, providing structure and allowing for movement.