Linear Regression Application, Interpolation and Extrapolation
Use the data and story to answer the following questions
The table below shows the number of state-registered automatic weapons and the murder rate for several Northwestern states.
\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
\hline$x$ & 11.5 & 8.2 & 6.7 & 3.4 & 2.3 & 2.4 & 2.7 & 0.5 \\
\hline$y$ & 13.9 & 11.3 & 9.8 & 7 & 5.8 & 5.9 & 6.3 & 4.6 \\
\hline
\end{tabular}
$x=$ thousands of automatic weapons
$y=$ murders per 100,000 residents
Use your calculator to determine the equation of the regression line. (Round to 2 decimal places)
Determine the regression equation in $y=a x+b$ form and write it below.

Step 1 ：Given the data of state-registered automatic weapons (in thousands) and the murder rate per 100,000 residents for several Northwestern states, we are asked to determine the equation of the regression line.

Step 2 ：The regression line is a line that best fits the data points on a scatter plot. It is used to predict the value of one variable given the value of another variable.

Step 3 ：The equation of the regression line is given by \(y = ax + b\), where \(a\) is the slope of the line and \(b\) is the y-intercept.

Step 4 ：The slope \(a\) can be calculated as the covariance of \(x\) and \(y\) divided by the variance of \(x\), and the y-intercept \(b\) can be calculated as the mean of \(y\) minus the slope times the mean of \(x\).

Step 5 ：Using the given data, we calculate the slope \(a\) to be approximately 0.995 and the y-intercept \(b\) to be approximately 3.39.

Step 6 ：Thus, the equation of the regression line is \(y = 0.995x + 3.39\).

Step 7 ：\(\boxed{y = 0.995x + 3.39}\) is the final answer.