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Highway crash data analysis. Researchers at Montana State
Chapter 12, Problem 13E(choose chapter or problem)
Highway crash data analysis. Researchers at Montana State University have written a tutorial on an empirical method for analyzing before and after highway crash data (Montana Department of Transportation, Research Report, May 2004). The initial step in the methodology is to develop a Safety Performance Function (SPF)—a mathematical model that estimates crash occurrence for a given roadway segment. Using data collected for over 100 roadway segments, the researchers fit the model, \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}\), where y = number of crashes per 3 years, \(x_{1}\) = roadway length (miles), and \(x_{2}\) = AADT = average annual daily traffic (number of vehicles). The results are shown in the following tables.
\(\begin{array}{l}
\text { Interstate Highways }\\
\begin{array}{lccc}
\hline \text { Variable } & \begin{array}{c}
\text { Parameter } \\
\text { Estimate }
\end{array} & \begin{array}{c}
\text { Standard } \\
\text { Error }
\end{array} & t \text {-value } \\
\hline \text { Intercept } & 1.81231 & .50568 & 3.58 \\
\text { Length }\left(x_{1}\right) & .10875 & .03166 & 3.44 \\
\text { AADT }\left(x_{2}\right) & .00017 & .00003 & 5.19 \\
\hline
\end{array}
\end{array}\)
\(\begin{array}{l}
\text { Noninterstate Highways }\\
\begin{array}{lccc}
\hline \text { Variable } & \begin{array}{c}
\text { Parameter } \\
\text { Estimate }
\end{array} & \begin{array}{c}
\text { Standard } \\
\text { Error }
\end{array} & t \text {-value } \\
\hline \text { Intercept } & 1.20785 & .28075 & 4.30 \\
\text { Length }\left(x_{1}\right) & .06343 & .01809 & 3.51 \\
\text { AADT }\left(x_{2}\right) & .00056 & .00012 & 4.86 \\
\hline
\end{array}
\end{array}\)
a. Give the least squares prediction equation for the interstate highway model.
b. Give practical interpretations of the \(\beta\) estimates, part a.
c. Refer to part a. Find a 99% confidence interval for \(\beta_{1}\) and interpret the result.
d. Refer to part a. Find a 99% confidence interval for \(\beta_{2}\) and interpret the result.
e. Repeat parts a–d for the noninterstate highway model.
Questions & Answers
QUESTION:
Highway crash data analysis. Researchers at Montana State University have written a tutorial on an empirical method for analyzing before and after highway crash data (Montana Department of Transportation, Research Report, May 2004). The initial step in the methodology is to develop a Safety Performance Function (SPF)—a mathematical model that estimates crash occurrence for a given roadway segment. Using data collected for over 100 roadway segments, the researchers fit the model, \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}\), where y = number of crashes per 3 years, \(x_{1}\) = roadway length (miles), and \(x_{2}\) = AADT = average annual daily traffic (number of vehicles). The results are shown in the following tables.
\(\begin{array}{l}
\text { Interstate Highways }\\
\begin{array}{lccc}
\hline \text { Variable } & \begin{array}{c}
\text { Parameter } \\
\text { Estimate }
\end{array} & \begin{array}{c}
\text { Standard } \\
\text { Error }
\end{array} & t \text {-value } \\
\hline \text { Intercept } & 1.81231 & .50568 & 3.58 \\
\text { Length }\left(x_{1}\right) & .10875 & .03166 & 3.44 \\
\text { AADT }\left(x_{2}\right) & .00017 & .00003 & 5.19 \\
\hline
\end{array}
\end{array}\)
\(\begin{array}{l}
\text { Noninterstate Highways }\\
\begin{array}{lccc}
\hline \text { Variable } & \begin{array}{c}
\text { Parameter } \\
\text { Estimate }
\end{array} & \begin{array}{c}
\text { Standard } \\
\text { Error }
\end{array} & t \text {-value } \\
\hline \text { Intercept } & 1.20785 & .28075 & 4.30 \\
\text { Length }\left(x_{1}\right) & .06343 & .01809 & 3.51 \\
\text { AADT }\left(x_{2}\right) & .00056 & .00012 & 4.86 \\
\hline
\end{array}
\end{array}\)
a. Give the least squares prediction equation for the interstate highway model.
b. Give practical interpretations of the \(\beta\) estimates, part a.
c. Refer to part a. Find a 99% confidence interval for \(\beta_{1}\) and interpret the result.
d. Refer to part a. Find a 99% confidence interval for \(\beta_{2}\) and interpret the result.
e. Repeat parts a–d for the noninterstate highway model.
ANSWER:Step 1 of 5
a)
The sample estimates of \({\beta _0},{\beta _1},{\beta _2}\) are from the interstate highway model:
\({\beta _0} = 1.81231\)
\({\beta _1} = 0.10875\)
\({\beta _2} = 0.00017\)
The least squares prediction equation for the interstate highway model is:
\(\widehat y = {\beta _0} + {\beta _1}{x_1} + {\beta _2}{x_2}\)
\(\widehat y = 1.81231 + 0.10875{x_1} + 0.00017{x_2}\)