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

Explanation:

Step1: Calculate the means of x and y

Let $n = 6$. $\bar{x}=\frac{0 + 1+2+3+4+5}{6}=\frac{15}{6}=2.5$ $\bar{y}=\frac{816 + 802+809+780+754+712}{6}=\frac{4673}{6}\approx778.8$

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}\ \sum_{i = 1}^{6}(x_i-\bar{x})(y_i - \bar{y})&=(0 - 2.5)(816-778.8)+(1 - 2.5)(802 - 778.8)+(2 - 2.5)(809-778.8)+(3 - 2.5)(780 - 778.8)+(4 - 2.5)(754-778.8)+(5 - 2.5)(712-778.8)\ &=(- 2.5)\times37.2+(-1.5)\times23.2+(-0.5)\times30.2 + 0.5\times1.2+1.5\times(-24.8)+2.5\times(-66.8)\ &=-93-34.8 - 15.1+0.6-37.2 - 167\ &=-346.5 \end{align*} ] [ \begin{align*} \sum_{i=1}^{6}(x_i-\bar{x})^2&=(0 - 2.5)^2+(1 - 2.5)^2+(2 - 2.5)^2+(3 - 2.5)^2+(4 - 2.5)^2+(5 - 2.5)^2\ &=6.25 + 2.25+0.25 + 0.25+2.25+6.25\ &=17.5 \end{align*} ] $m=\frac{-346.5}{17.5}\approx - 19.8$

Step3: Calculate the y - intercept $b$

Using the point - slope form $y=mx + b$ and the point $(\bar{x},\bar{y})$, we substitute $x = 2.5$, $y=778.8$ and $m=-19.8$ $778.8=-19.8\times2.5 + b$ $778.8=-49.5 + b$ $b=778.8 + 49.5=828.3$

The linear regression equation is $y=-19.8x + 828.3$

Step4: Estimate the year when $y = 604$

Set $y = 604$ in the equation $y=-19.8x + 828.3$ $604=-19.8x+828.3$ $19.8x=828.3 - 604$ $19.8x=224.3$ $x=\frac{224.3}{19.8}\approx11.3$

Since $x$ represents the number of years since 2013, the year is $2013+11.3\approx2024$

Answer:

The linear regression equation is $y=-19.8x + 828.3$. The calendar year is approximately 2024.