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:

Explanation:

Step1: Calculate mean of x and y

First, find the mean of \( x \) values: \( x = [0, 1, 2, 3] \), so \( \bar{x} = \frac{0 + 1 + 2 + 3}{4} = 1.5 \).
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)

Slope formula: \( m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \).
Compute \( (x_i - \bar{x})(y_i - \bar{y}) \) for each \( i \):

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

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

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

• \( (0 - 1.5)^2 = 2.25 \)
• \( (1 - 1.5)^2 = 0.25 \)
• \( (2 - 1.5)^2 = 0.25 \)
• \( (3 - 1.5)^2 = 2.25 \)

Sum: \( 2.25 + 0.25 + 0.25 + 2.25 = 5 \).

Thus, \( m = \frac{158.5}{5} = 31.7 \) (rounded to nearest tenth).

Step3: Calculate y-intercept (b)

Use \( \bar{y} = m\bar{x} + b \).
\( 1058.75 = 31.7(1.5) + b \).
\( 1058.75 = 47.55 + b \).
\( b = 1058.75 - 47.55 = 1011.2 \) (rounded to nearest tenth).

Step4: Linear regression equation

Equation: \( y = 31.7x + 1011.2 \).

Step5: Predict for 2008 (x=8)

Substitute \( x = 8 \) into the equation:
\( y = 31.7(8) + 1011.2 = 253.6 + 1011.2 = 1264.8 \approx 1265 \).

Answer:

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