12.3 Linear Regression Equation 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$x$ & 11.8 & 8.6 & 7.1 & 3.9 & 2.8 & 2.4 & 2.7 & 0.9 \\ \hline$y$ & 14.1 & 11 & 10 & 7.4 & 6.1 & 6.4 & 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 1 decimal place) Determine the regression equation in $\hat{y}=m x+b$ form and write it below. \[ \hat{y}= \] A) How many murders per 100,000 residents can be expected in a state with 6 thousand automatic weapons? Answer = Round to the nearest whole number. B) How many murders per 100,000 residents can be expected in a state with 2.9 thousand automatic weapons? Answer $=$ Round to the nearest whole number.
Solution
Step 1 ：Calculate the sums: \(\Sigma x = 11.8 + 8.6 + 7.1 + 3.9 + 2.8 + 2.4 + 2.7 + 0.9 = 40.2\), \(\Sigma y = 14.1 + 11 + 10 + 7.4 + 6.1 + 6.4 + 6.3 + 4.6 = 65.9\), \(\Sigma xy = 11.8*14.1 + 8.6*11 + 7.1*10 + 3.9*7.4 + 2.8*6.1 + 2.4*6.4 + 2.7*6.3 + 0.9*4.6 = 334.12\), \(\Sigma x^2 = 11.8^2 + 8.6^2 + 7.1^2 + 3.9^2 + 2.8^2 + 2.4^2 + 2.7^2 + 0.9^2 = 204.86\)
Step 2 ：Substitute these values into the formulas for m and b: \(m = (8*334.12 - 40.2*65.9) / (8*204.86 - 40.2^2) = 1.3\) (rounded to 1 decimal place), \(b = (65.9 - 1.3*40.2) / 8 = 2.1\) (rounded to 1 decimal place)
Step 3 ：The equation of the regression line is: \(\hat{y} = 1.3x + 2.1\)
Step 4 ：For a state with 6 thousand automatic weapons, the expected number of murders per 100,000 residents is: \(\hat{y} = 1.3*6 + 2.1 = 9.9\). Rounded to the nearest whole number, the answer is \(\boxed{10}\)
Step 5 ：For a state with 2.9 thousand automatic weapons, the expected number of murders per 100,000 residents is: \(\hat{y} = 1.3*2.9 + 2.1 = 5.87\). Rounded to the nearest whole number, the answer is \(\boxed{6}\)
