Motorcycle Fatalities
a Traffic Consultant Has Analyzed the Factors That

question 116

Essay

Motorcycle Fatalities
A traffic consultant has analyzed the factors that affect the number of motorcycle fatalities.She has come to the conclusion that two important variables are the number of motorcycle and the number of cars.She proposed the model $$\tilde { y } = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } ^ { 2 } + \beta _ { 4 } x _ { 2 } ^ { 2 } + \beta _ { 5 } x _ { 1 } x _ { 2 } + \varepsilon$$ (the second-order model with interaction),where y = number of annual fatalities per county,x₁ = number of motorcycles registered in the county (in 10,000),and x₂ = number of cars registered in the county (in 1000).The computer output (based on a random sample of 35 counties)is shown below:
THE REGRESSION EQUATION IS $$y = 69.7 + 11.3 x _ { 1 } + 7.61 x _ { 2 } - 1.15 x _ { 1 } ^ { 2 } - 0.51 x _ { 2 } ^ { 2 } - 0.13 x _ { 1 } x _ { 2 }$$ $\begin{array}{ccc}\text { Predictor}&\text { Coef } & \text { StDev } & T \\\hline\text { Constant}& 69.7 & 41.3 & 1.688 \\x_{1} &11.3 & 5.1 & 2.216 \\ x_{2}&7.61 & 2.55 & 2.984 \\x_1^2&-1.15 & .64 & -1.797 \\& & \\x_2^2&-.51 & .20 & -2.55 \\& & \\& & \\ x_{1} x_{2}&-.13 & .10 & -1.30\end{array}$ $$S= 15.2 \quad \mathrm { R } - \mathrm { Sq } = 47.2 \%$$ ANALYSIS OF VARIANCE
$$\begin{array} { | l | c c c c | } \hline \text { Source of Variation } & \boldsymbol { df } & \mathbf { S S } & \boldsymbol { M S } & \boldsymbol { F } \\\hline \text { Repressian } & 5 & 5959 & 1191.800 & 5.181 \\\text { Error } & 29 & 6671 & 230.034 & \\\hline \text { Total } & 34 & 12630 & & \\\hline\end{array}$$
-{Motorcycle Fatalities Narrative} What does the coefficient of $x _ { 1 } ^ { 2 }$ tell you about the model?

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Definitions:

Integer Models

Mathematical models that involve decision variables restricted to integer values, commonly used in optimization for solving problems in areas such as scheduling and resource allocation.

Solver

A software tool that finds the optimal solution to a mathematical problem by varying multiple inputs within certain constraints.

Discontinuous

Refers to a function or process that exhibits breaks, jumps, or gaps in its domain or range.

Integer Linear Optimization Models

Mathematical models used in operations research that seek to maximize or minimize a linear objective function subject to a set of linear constraints and integer variables.