91Ó°ÊÓ

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

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

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

Beta values helps in explaining the effect of independent variable, x₁ and x₂ on the dependent variable, y. ${\mathrm{β}}_0$ 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 ${\mathrm{β}}_1$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.${\mathrm{β}}_2$ 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.

A 99%confidence interval forcan be found as$\left({\widehat{\mathrm{β}}}_1Â\pm t_{0.005}×t-value\right)$

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

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

A 99%confidence interval forcan be found as$\left({\widehat{\mathrm{β}}}_2Â\pm t_{0.005}×t-value\right)$

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

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

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 ${\mathrm{β}}_0$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 ${\mathrm{β}}_1$ 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 ${\mathrm{β}}_2$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 ${\mathrm{β}}_1$:

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

That is,

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

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

A 99%confidence interval forcan be found as

That is,

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

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

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

Where,

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