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

Explanation:

Step1: Calculate sums

Let (n = 4) (number of data - points). (\sum_{i = 1}^{n}x_{i}=0 + 1+2 + 3=6) (\sum_{i = 1}^{n}y_{i}=1172 + 1130+1140 + 1092 = 4534) (\sum_{i = 1}^{n}x_{i}^{2}=0^{2}+1^{2}+2^{2}+3^{2}=0 + 1+4 + 9 = 14) (\sum_{i = 1}^{n}x_{i}y_{i}=0\times1172+1\times1130 + 2\times1140+3\times1092=0 + 1130+2280 + 3276=6686)

Step2: Calculate slope (m)

The formula for the slope (m) of the regression line is (m=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}) Substitute the values: [ \begin{align*} m&=\frac{4\times6686-6\times4534}{4\times14 - 6^{2}}\ &=\frac{26744-27204}{56 - 36}\ &=\frac{-460}{20}\ &=- 23 \end{align*} ]

Step3: Calculate y - intercept (b)

The formula for the y - intercept (b) is (b=\frac{\sum_{i = 1}^{n}y_{i}-m\sum_{i = 1}^{n}x_{i}}{n}) Substitute (m=-23), (\sum_{i = 1}^{n}x_{i}=6) and (\sum_{i = 1}^{n}y_{i}=4534) and (n = 4) [ \begin{align*} b&=\frac{4534-(-23)\times6}{4}\ &=\frac{4534 + 138}{4}\ &=\frac{4672}{4}\ &=1168 \end{align*} ]

The regression equation is (y=-23x + 1168)

Step4: Estimate the year

Set (y = 887) in the equation (y=-23x + 1168) [ \begin{align*} 887&=-23x+1168\ 23x&=1168 - 887\ 23x&=281\ x&=\frac{281}{23}\approx12.22 \end{align*} ] Since (x) represents the number of years since 1995, the year is (1995 + 12.22\approx2007)

Answer:

Regression Equation: (y=-23x + 1168) Final Answer: 2007