question: the number of newly reported crime cases in a county in new york state is shown in the…

Explanation:

Step1: Calculate the means of x and y

Let (x = [0,1,2,3]) and (y=[911,907,915,981]). The mean of (x), (\bar{x}=\frac{0 + 1+2 + 3}{4}=\frac{6}{4}=1.5) The mean of (y), (\bar{y}=\frac{911+907+915+981}{4}=\frac{3714}{4}=928.5)

Step2: Calculate the slope (m)

[ \begin{align*} m&=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}\ (x_1-\bar{x})(y_1 - \bar{y})&=(0 - 1.5)(911-928.5)=(- 1.5)\times(-17.5)=26.25\ (x_2-\bar{x})(y_2 - \bar{y})&=(1 - 1.5)(907-928.5)=(-0.5)\times(-21.5) = 10.75\ (x_3-\bar{x})(y_3 - \bar{y})&=(2 - 1.5)(915-928.5)=0.5\times(-13.5)=-6.75\ (x_4-\bar{x})(y_4 - \bar{y})&=(3 - 1.5)(981-928.5)=1.5\times52.5 = 78.75\ \sum_{i = 1}^{4}(x_i-\bar{x})(y_i - \bar{y})&=26.25+10.75-6.75 + 78.75=109\ (x_1-\bar{x})^2&=(0 - 1.5)^2=2.25\ (x_2-\bar{x})^2&=(1 - 1.5)^2=0.25\ (x_3-\bar{x})^2&=(2 - 1.5)^2=0.25\ (x_4-\bar{x})^2&=(3 - 1.5)^2=2.25\ \sum_{i=1}^{4}(x_i-\bar{x})^2&=2.25 + 0.25+0.25+2.25 = 5\ m&=\frac{109}{5}=21.8 \end{align*} ]

Step3: Calculate the y - intercept (b)

[b=\bar{y}-m\bar{x}=928.5-21.8\times1.5=928.5 - 32.7=895.8] The linear - regression equation is (y = 21.8x+895.8)

Step4: Find the value of x when y = 1230

[ \begin{align*} 1230&=21.8x+895.8\ 21.8x&=1230 - 895.8\ 21.8x&=334.2\ x&=\frac{334.2}{21.8}\approx15.33 \end{align*} ] Since (x) represents the number of years since 2011, the year is (2011 + 15.33\approx2026)

Answer:

The linear - regression equation is (y = 21.8x+895.8), and the year when the number of new cases reaches 1230 is approximately 2026.