Problem $1$: The Negative Binomial Model was built for fatal crashes and injury crashes for the segment of the roadway as shown in the table. The segment has the following characteristics: $6-$Lanes, $\backslash (2,000\backslash$mathrm$\{$vpd$\},11\backslash )-$ ft lane width, $6-$ft shoulder width, $18-$ ft median width, urban $($coded as $1),$ rolling terrain $($coded as $1),$ curved section $($coded as $1,$ while straight is coded $0),$ and speed limit above $45$ mph $($coded $1).$ The variables are considered significant at $\backslash (95\backslash \%\backslash )$ confidence level.

$\backslash$begin$\{$tabular$\}\{|$l$|$l$|$l$|$l$|$l$|\}$

$\backslash$hline $\backslash$multirow$\{2\}\{*\}\{\}$ & $\backslash$multicolumn$\{2\}\{|$c$|\}\{$ Fatal Crashes $\}$ & $\backslash$multicolumn$\{2\}\{$c$|\}\{$ Injury Crashes $\}\backslash \backslash$

$\backslash$cline $\{2-5\}$ & Coefficient & Z$-$Value & Coefficient & Z$-$Value $\backslash \backslash$

$\backslash$hline Number of Lanes $($N$)$ & $3\mathrm{.}00$ & $2\mathrm{.}55$ & $2\mathrm{.}00$ & $3\mathrm{.}23\backslash \backslash$

$\backslash$hline AADT $($A$)$ & $-0\mathrm{.}020$ & $3\mathrm{.}72$ & $-0\mathrm{.}017$ & $1\mathrm{.}35\backslash \backslash$

$\backslash$hline Lane Width $($L$)$ & $0\mathrm{.}23$ & $1\mathrm{.}69$ & $0\mathrm{.}20$ & $1\mathrm{.}36\backslash \backslash$

$\backslash$hline Shoulder Width $($SW$)$ & $-0\mathrm{.}40$ & $1\mathrm{.}62$ & $0\mathrm{.}60$ & $5\mathrm{.}68\backslash \backslash$

$\backslash$hline Median Width $($M$)$ & $-0\mathrm{.}30$ & $1\mathrm{.}66$ & $1\mathrm{.}00$ & $0\mathrm{.}77\backslash \backslash$

$\backslash$hline Urban Area $($U$)$ & & & $-0\mathrm{.}10$ & $4\mathrm{.}15\backslash \backslash$

$\backslash$hline Rolling Terrain $($R$)$ & & & $1\mathrm{.}50$ & $1\mathrm{.}22\backslash \backslash$

$\backslash$hline Curved Section $($C$)$ & $-2\mathrm{.}00$ & $1\mathrm{.}11$ & $-0\mathrm{.}10$ & $7\mathrm{.}09\backslash \backslash$

$\backslash$hline Speed Limit above $45$ mph $($SL$)$ & $12\mathrm{.}00$ & $2\mathrm{.}41$ & $-0\mathrm{.}40$ & $-3\mathrm{.}21\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

i$.$ Identify and discuss significant variables in each model, justify your variable pick

ii$.$ Formulate the Prediction equation for Fatal Crashes

iii. Formulate the Prediction equation for Injury Crashes

iv$.$ Predict the number of fatal crashes for this segment

$\backslash [$

$\backslash$mu$=$e$^\{\backslash$sum $\backslash$beta$\_\{$i$\}$ X$\_\{$i$\}\}$

$\backslash ]$

v$.$ Predict the number of injury crashes for this segment

vi$.$ What percentage will Injury Crashes increase or decrease if the section is changed from curved to straight

vii. At what percentage will the Fatal crashes increase or decrease if shoulder is increased from $8$ to $10$ ft $?$

viii. What will be the percentage increase or decrease in the number of fatal crashes if the AADT is reduced from $\backslash (2,000\backslash$mathrm$\{$vpd$\}\backslash )$ to $\backslash (1,975\backslash$mathrm$\{$vpd$\}\backslash )$