Question

the number of newly reported crime cases in a county in new york state is shown in the accompanying table, where x represents the number of years since 1998, and y represents the number of new cases. write the linear regression equation that represents this set of data, rounding all coefficients to the nearest hundredth. using this equation, find the projected number of new cases for 2008, rounded to the nearest whole number.
years since 1998 (x) new cases (y)
0 703
1 675
2 706
3 643
4 660
5 605

Explanation:

Step1: Recall linear - regression formula

The linear - regression equation is of the form $y = ax + b$, where $a=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}$ and $b=\overline{y}-a\overline{x}$, with $n$ being the number of data points, $\overline{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$ and $\overline{y}=\frac{\sum_{i = 1}^{n}y_i}{n}$.
We have $n = 6$, $\sum_{i=1}^{6}x_i=0 + 1+2 + 3+4 + 5=15$, $\sum_{i=1}^{6}y_i=703+675+706+643+660+605 = 3992$, $\sum_{i = 1}^{6}x_i^{2}=0^{2}+1^{2}+2^{2}+3^{2}+4^{2}+5^{2}=0 + 1+4 + 9+16+25 = 55$, and $\sum_{i = 1}^{6}x_iy_i=0\times703+1\times675+2\times706+3\times643+4\times660+5\times605=0 + 675+1412+1929+2640+3025 = 9681$.

Step2: Calculate $a$

$a=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}=\frac{6\times9681-15\times3992}{6\times55 - 15^{2}}=\frac{58086-59880}{330 - 225}=\frac{-1794}{105}\approx - 17.09$.

Step3: Calculate $\overline{x}$ and $\overline{y}$

$\overline{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{15}{6}=2.5$, $\overline{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{3992}{6}\approx665.33$.

Step4: Calculate $b$

$b=\overline{y}-a\overline{x}=665.33-(-17.09)\times2.5=665.33 + 42.73=708.06$.
The linear - regression equation is $y=-17.09x + 708.06$. Rounding to the nearest hundredth, $a\approx - 17.09$ and $b\approx708.06$.

If we want to find the projected number of new cases for 2008 ($x = 10$):
$y=-17.09\times10+708.06=-170.9+708.06 = 537.16\approx537$.

Answer:

The linear - regression equation is $y=-17.09x + 708.06$. The projected number of new cases for 2008 is 537.