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 2006, 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 2014, rounded to the nearest whole number.

years since 2006 (x)	new cases (y)
1	825
2	855
3	806

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Explanation:

Step1: Calculate mean of x and y

First, find the mean of \( x \) values (\( \bar{x} \)) and mean of \( y \) values (\( \bar{y} \)).
\( x \) values: \( 0, 1, 2, 3 \)
\( \bar{x} = \frac{0 + 1 + 2 + 3}{4} = \frac{6}{4} = 1.5 \)
\( y \) values: \( 843, 825, 855, 806 \)
\( \bar{y} = \frac{843 + 825 + 855 + 806}{4} = \frac{3329}{4} = 832.25 \)

Step2: Calculate slope (m)

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

• For \( x = 0 \), \( y = 843 \): \( (0 - 1.5)(843 - 832.25) = (-1.5)(10.75) = -16.125 \)
• For \( x = 1 \), \( y = 825 \): \( (1 - 1.5)(825 - 832.25) = (-0.5)(-7.25) = 3.625 \)
• For \( x = 2 \), \( y = 855 \): \( (2 - 1.5)(855 - 832.25) = (0.5)(22.75) = 11.375 \)
• For \( x = 3 \), \( y = 806 \): \( (3 - 1.5)(806 - 832.25) = (1.5)(-26.25) = -39.375 \)

Sum of numerator: \( -16.125 + 3.625 + 11.375 - 39.375 = -40.5 \)
Calculate \( (x_i - \bar{x})^2 \) for each \( i \):

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

Sum of denominator: \( 2.25 + 0.25 + 0.25 + 2.25 = 5 \)
So, \( m = \frac{-40.5}{5} = -8.1 \)

Step3: Calculate y-intercept (b)

Use the formula \( \bar{y} = m\bar{x} + b \)
\( 832.25 = -8.1(1.5) + b \)
\( 832.25 = -12.15 + b \)
\( b = 832.25 + 12.15 = 844.4 \) (rounded to nearest tenth)

Step4: Linear regression equation

The equation is \( y = -8.1x + 844.4 \)

Step5: Project for 2014

2014 is \( 2014 - 2006 = 8 \) years since 2006, so \( x = 8 \)
Substitute \( x = 8 \) into the equation:
\( y = -8.1(8) + 844.4 \)
\( y = -64.8 + 844.4 \)
\( y = 779.6 \approx 780 \) (rounded to nearest whole number)

Answer:

Linear regression equation: \( y = -8.1x + 844.4 \)
Projected number of new cases for 2014: \( 780 \)