Question

Question: Highway crash data analysis. Researchers at Montana State University have written a tutorial on an empiricalmethod for analyzing before and after highway crashdata (Montana Department of Transportation, ResearchReport, May 2004). The initial step in the methodologyis to develop a Safety Performance Function (SPF)—amathematical model that estimates crash occurrence fora given roadway segment. Using data collected for over100 roadway segments, the researchers fit the model, , where y = number ofcrashes per 3 years, roadway length (miles), and (number of vehicles). The results are shown in the following tables.

a. Give the least squares prediction equation for the interstate highway model.

b. Give practical interpretations of the${\mathbf{\beta }}_{\mathbf{1}}$ estimates, part a.

c. Refer to part a. Find a 99% confidence interval for${\mathbf{\beta }}_{\mathbf{1}}$ and interpret the result.

d. Refer to part a. Find a 99% confidence interval for${\mathbf{\beta }}_{\mathbf{2}}$ and interpret the result.

e. Repeat parts a–d for the non-interstate highway model.

f. Write a first-order model for E(y) as a function ofx₁and x₂ that allows the slopes to differ dependingon whether the roadway segment is Interstate ornon-interstate. [Hint: Create a dummy variable forInterstate/non-interstate.]

Short answer

$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3$Answer

a. The least-square prediction equation for the interstate highway model is$E\left(y\right)=1.81231+0.10875x_1+0.00017x_2$.

b. The intercept${\beta }_0$value in the least square prediction equation for the interstate model is 1.81231.They-intercept or the value of E(y) when x₁ and x₂ are zero.is 0.10875 which is positive indicating a positive relationship between y andand since the value is less than 1 the slope of the line would be flatter.value is 0.00017 which is a positive indicating positive relation between y andand since the value is near zero the line will be flatter.

c. The 99% confidence interval for${\beta }_1$is (-8.89373, 9.11123).

d. The 99% confidence interval for ${\beta }_2$is (-13.58206, 13.5824).

e. For the non-interstate highway, the least square prediction equation for the interstate highway model would be $E\left(y\right)=1.20785+0.06343x_1+0.00056x_2$.The interpretation of value in the least square prediction equation for the interstate model ${\beta }_1$is 1.20785. This represents the y-intercept or the value of E(y) when x₁ and are zero. ${\beta }_2$value is 0.06343 which is positive indicating a positive relationship between y and and since the value is less than 1 the slope of the line would be flatter. value is 0.00056 which is a positive indicating positive relation between y and and since the value is near zero the line will be flatter. Confidence intervals - A 99% confidence interval for ${\beta }_1$is (-9.12224, 9.2491) and a 99% confidence interval for ${\beta }_2$ is (-12.71806, 12.71918).

f. A first-order model for E(y) as a function of and which also represents whether the roadway segment is interstate or non-interstate can be written as $E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3$.

Step-by-step solution

Given Information

Let y is the number of crashes per 3 years, x₁= roadway length (in miles), and x₂= AADT (average annual daily traffic).

Least square prediction equation

The least-square prediction equation for the interstate highway model is.

$E\left(y\right)=1.81231+0.10875x_1+0.00017x_2$

 Step 3: Interpretation for β

Beta values helps in explaining the effect of independent variable, x₁ and x₂ on the dependent variable, y. ${\beta }_0$ value in the least square prediction equation for the interstate model is 1.81231. The y-intercept or the value of E(y) when and are zero. A ${\beta }_1$value is 0.10875 which is positive indicating a positive relationship between y and and since the value is less than 1 the slope of the line would be flatter.${\beta }_2$ value is 0.00017 which is positive indicating positive relation between y and and since the value is near to zero the line will be flatter.

99%  confidence interval for β1 

A 99%confidence interval forcan be found as$\left({\hat{\beta }}_1\pm t_{0.005}\times t-value\right)$

That is,$\left(0.10875\pm 2.617\times 3.44\right)$.

The 99% confidence interval for${\beta }_1$ is (-8.89373, 9.11123).

 Step 5: A  99% confidence interval for  β2

A 99%confidence interval forcan be found as$\left({\hat{\beta }}_2\pm t_{0.005}\times t-value\right)$

That is,$\left(0.00017\pm 2.617\times 5.19\right)$.

A 99% confidence interval for${\beta }_2$ is (-13.58206, 13.5824).

For non-interstate highway

Least square prediction equation:$E\left(y\right)=1.20785+0.06343x_1+0.00056x_2$

The least-square prediction equation for the interstate highway model would be.

Interpretation for β:

The ${\beta }_0$value in the least square prediction equation for the interstate model is 1.20785. This represents the y-intercept or the value of E(y) when and are zero. The ${\beta }_1$ value is 0.06343 which is positive indicating a positive relationship between y and x₁ , since the value is less than 1 the slope of the line would be flatter. The ${\beta }_2$value is 0.00056 which is a positive indicating positive relation between y and x₂ , since the value is near zero the line will be flatter.

A 99% confidence interval for ${\beta }_1$:

A 99%confidence interval for${\beta }_1$can be found as

That is,

A 99% confidence interval for${\beta }_1$is (-9.12224, 9.2491).

A 99% confidence interval for${\beta }_2$:

A 99%confidence interval forcan be found as

That is,

A 99% confidence interval for is (-12.71806, 12.71918).

 Step 7: First-order model for E(y)

A first-order model for E(y) as a function of and which also represents whether the roadway segment is interstate or non-interstate can be written as

$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3$

Where,

$$\begin{gathered} x_1=roadwaylength \left(miles\right) \\ {x}_2=averageannualdailytraffic\left(numberofvehicles\right) \\ {x}_3=1,iftheroadwaysegmentisinterstate \\ =0,iftheroadwaysegmentisnon-interstate \end{gathered}$$