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$x$ & 11.6 & 8.2 & 6.7 & 3.6 & 2.9 & 2.8 & 2.2 & 0.9 \\ \hline$y$ & 13.6 & 10.7 & 9.8 & 6.8 & 6.3 & 6.5 & 5.8 & 5 \\ \hline \end{tabular} $x=$ thousands of automatic weapons $y=$ murders per 100,000 residents This data can be modeled by the equation $y=0.82 x+4.08$. Use this equation to answer the following; Special Note: I suggest you verify this equation by performing linear regression on your calculator. A) How many murders per 100,000 residents can be expected in a state with 1.2 thousand automatic weapons? Answer $=$ Round to 3 decimal places. B) How many murders per 100,000 residents can be expected in a state with 4.8 thousand automatic weapons? Answer $=$ Round to 3 decimal places.
Solution
Step 1 ：Given the equation \(y=0.82 x+4.08\), where \(x\) is the number of automatic weapons in thousands and \(y\) is the number of murders per 100,000 residents.
Step 2 ：Substitute \(x = 1.2\) into the equation to find the expected number of murders per 100,000 residents in a state with 1.2 thousand automatic weapons. The calculation is \(y = 0.82 * 1.2 + 4.08\).
Step 3 ：The result is \(y = 5.064\). So, for a state with 1.2 thousand automatic weapons, the expected number of murders per 100,000 residents is \(\boxed{5.064}\).
Step 4 ：Substitute \(x = 4.8\) into the equation to find the expected number of murders per 100,000 residents in a state with 4.8 thousand automatic weapons. The calculation is \(y = 0.82 * 4.8 + 4.08\).
Step 5 ：The result is \(y = 8.016\). So, for a state with 4.8 thousand automatic weapons, the expected number of murders per 100,000 residents is \(\boxed{8.016}\).
