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 1995, and y represents 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, estimate the calendar year in which the number of new cases would reach 887.
years since 1995 (x) new cases (y)
0 1172
1 1130
2 1140
3 1092
answer attempt 1 out of 3
regression equation:
final answer:

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