Question

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 2000, and y represents number of new cases. write the linear regression equation that represents this set of data, rounding all coefficients to the nearest tenth. using this equation, find the projected number of new cases for 2008, rounded to the nearest whole number.

years since 2000 (x)	new cases (y)
1	1047
2	1106
3	1084

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answer attempt 1 out of 2
regression equation:
final answer:
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Explanation:

Step1: Calculate mean of x and y

First, find the mean of \( x \) values: \( x = [0, 1, 2, 3] \), so \( \bar{x}=\frac{0 + 1+2 + 3}{4}=\frac{6}{4}=1.5 \)
Then, find the mean of \( y \) values: \( y = [998, 1047, 1106, 1084] \), so \( \bar{y}=\frac{998+1047 + 1106+1084}{4}=\frac{4235}{4}=1058.75 \)

Step2: Calculate slope (m)

The formula for slope \( m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2} \)
Calculate \( (x_i-\bar{x})(y_i-\bar{y}) \) for each \( i \):

• For \( i = 1 \): \( (0 - 1.5)(998 - 1058.75)=(-1.5)(-60.75)=91.125 \)
• For \( i = 2 \): \( (1 - 1.5)(1047 - 1058.75)=(-0.5)(-11.75)=5.875 \)
• For \( i = 3 \): \( (2 - 1.5)(1106 - 1058.75)=(0.5)(47.25)=23.625 \)
• For \( i = 4 \): \( (3 - 1.5)(1084 - 1058.75)=(1.5)(25.25)=37.875 \)

Sum of these: \( 91.125+5.875 + 23.625+37.875 = 158.5 \)

Calculate \( (x_i-\bar{x})^2 \) for each \( i \):

• For \( i = 1 \): \( (0 - 1.5)^2 = 2.25 \)
• For \( i = 2 \): \( (1 - 1.5)^2 = 0.25 \)
• For \( i = 3 \): \( (2 - 1.5)^2 = 0.25 \)
• For \( i = 4 \): \( (3 - 1.5)^2 = 2.25 \)

Sum of these: \( 2.25+0.25 + 0.25+2.25 = 5 \)

So, \( m=\frac{158.5}{5}=31.7 \)

Step3: Calculate y-intercept (b)

Using the formula \( \bar{y}=m\bar{x}+b \), we can solve for \( b \):
\( 1058.75 = 31.7\times1.5 + b \)
\( 1058.75 = 47.55 + b \)
\( b = 1058.75 - 47.55 = 1011.2 \)

So the linear regression equation is \( y = 31.7x + 1011.2 \)

Step4: Predict for 2008

2008 is \( 2008 - 2000 = 8 \) years since 2000, so \( x = 8 \)
Substitute \( x = 8 \) into the equation: \( y = 31.7\times8+1011.2 \)
\( y = 253.6+1011.2 = 1264.8 \approx 1265 \)

Answer:

Regression Equation: \( y = 31.7x + 1011.2 \)
Final Answer: \( 1265 \)