A Traffic Consultant Has Analysed the Factors That Affect the Number

question 48

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 $y =$ $69.7 + 11.3 x _ { 1 } + 7.61 x _ { 2 } - 1.15 x _ { 1 } ^ { 2 } - 0.51 x _ { 2 } ^ { 2 } - 0.13 x _ { 1 } x _ { 2 }$ . $$\begin{array} { | c | r r r | } \hline \text { Predictor } & \text { Coef } & \text { SDev } & \multicolumn{1}{c|} { \text { T } } \\\hline \text { Constant } & 69.7 & 41.3 & 1.688 \\x _ { 1 } & 11.3 & 5.1 & 2.216 \\x _ { 2 } & 7.61 & 2.55 & 2.984 \\x _ { 1 } ^ { 2 } & - 1.15 & 0.64 & - 1.797 \\x _ { 2 } ^ { 2 } & - 0.51 & 0.20 & - 2.55 \\x _ { 1 } x _ { 2 } & - 0.13 & 0.10 & - 1.30 \\\hline\end{array}$$ S = 15.2 R-Sq = 47.2%.
ANALYSIS OF VARIANCE $$\begin{array} { | l | c c c c | } \hline \text { Source of } & d f & \text { SS } & M S & F \\\text { Variation } & & & & \\\hline \text { Regression } & 5 & 5959 & 1191.800 & 5.181 \\\text { Error } & 29 & 6671 & 230.034 & \\\hline \text { Total } & 34 & 12630 & & \\\hline\end{array}$$ What is the multiple coefficient of determination? What does this statistic tell you about the model?

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Definitions:

REM Deprivation

The condition of being deprived of Rapid Eye Movement sleep, a crucial stage of the sleep cycle known for dreaming and important for memory consolidation.

REM Rebound

The phenomenon where a person experiences increased amounts of REM sleep after being deprived of REM sleep on previous nights.

Sleeping Medications

Drugs or substances used to induce sleep or improve the quality of sleep in individuals with sleep disorders.

Sleep Deprivation

The condition of not having enough sleep, which can affect mental, physical, and emotional functioning.