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 2004, 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, find the projected number of new cases for 2014, rounded to the nearest whole number.
years since 2004 (x) new cases (y)
0 1077
1 1041
2 1076
3 1004
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answer attempt 2 out of 3
regression equation:
final answer:

Explanation:

Step1: Calculate sums

Let \(n = 4\) (number of data - points).
\(\sum_{i = 1}^{n}x_{i}=0 + 1+2 + 3=6\)
\(\sum_{i = 1}^{n}y_{i}=1077+1041 + 1076+1004 = 4198\)
\(\sum_{i = 1}^{n}x_{i}^{2}=0^{2}+1^{2}+2^{2}+3^{2}=0 + 1+4 + 9 = 14\)
\(\sum_{i = 1}^{n}x_{i}y_{i}=0\times1077+1\times1041+2\times1076+3\times1004=0 + 1041+2152+3012 = 6205\)

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 \(n = 4\), \(\sum_{i = 1}^{n}x_{i}=6\), \(\sum_{i = 1}^{n}y_{i}=4198\), \(\sum_{i = 1}^{n}x_{i}^{2}=14\) and \(\sum_{i = 1}^{n}x_{i}y_{i}=6205\) into the formula:
\[

$$\begin{align*} m&=\frac{4\times6205-6\times4198}{4\times14 - 6^{2}}\\ &=\frac{24820-25188}{56 - 36}\\ &=\frac{-368}{20}\\ &=- 18.4 \end{align*}$$

\]

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 \(m=-18.4\), \(n = 4\), \(\sum_{i = 1}^{n}x_{i}=6\) and \(\sum_{i = 1}^{n}y_{i}=4198\) into the formula:
\[

$$\begin{align*} b&=\frac{4198-(-18.4)\times6}{4}\\ &=\frac{4198 + 110.4}{4}\\ &=\frac{4308.4}{4}\\ &=1077.1 \end{align*}$$

\]

The linear regression equation is \(y=-18.4x + 1077.1\)

Step4: Find \(x\) value for 2014

Since \(x\) represents the number of years since 2004, for 2014, \(x = 2014 - 2004=10\)

Step5: Predict number of new cases

Substitute \(x = 10\) into the regression equation \(y=-18.4x + 1077.1\)
\(y=-18.4\times10+1077.1=-184 + 1077.1 = 893.1\approx893\)

Answer:

Regression Equation: \(y=-18.40x + 1077.10\)
Final Answer: 893