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 2012, 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 2024, rounded to the nearest whole number.

years since 2012 (x)	new cases (y)
1	887
2	915
3	995

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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 \( \bar{x}=\frac{0 + 1+ 2 + 3}{4}=\frac{6}{4}=1.5 \).
Mean of \( y \) values: \( y = [912, 887, 915, 995] \), so \( \bar{y}=\frac{912 + 887+ 915 + 995}{4}=\frac{3709}{4}=927.25 \).

Step2: Calculate slope (m)

Use the 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 \( i \):

• \( i=1 \): \( (0 - 1.5)(912 - 927.25)=(-1.5)(-15.25)=22.875 \)
• \( i=2 \): \( (1 - 1.5)(887 - 927.25)=(-0.5)(-40.25)=20.125 \)
• \( i=3 \): \( (2 - 1.5)(915 - 927.25)=(0.5)(-12.25)=-6.125 \)
• \( i=4 \): \( (3 - 1.5)(995 - 927.25)=(1.5)(67.75)=101.625 \)

Sum: \( 22.875 + 20.125 - 6.125 + 101.625 = 138.5 \).

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

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

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

Thus, \( m=\frac{138.5}{5}=27.7 \).

Step3: Calculate y-intercept (b)

Use \( \bar{y}=m\bar{x}+b \).
Substitute \( \bar{y}=927.25 \), \( m = 27.7 \), \( \bar{x}=1.5 \):
\( 927.25 = 27.7(1.5)+b \)
\( 927.25 = 41.55 + b \)
\( b = 927.25 - 41.55 = 885.7 \).

So the linear regression equation is \( y = 27.7x + 885.7 \).

Step4: Project for 2024

2024 is \( 2024 - 2012 = 12 \) years since 2012, so \( x = 12 \).
Substitute into the equation: \( y = 27.7(12)+885.7 \)
\( y = 332.4 + 885.7 = 1218.1 \), rounded to nearest whole number is 1218.

Answer:

Regression Equation: \( y = 27.7x + 885.7 \)
Final Answer: 1218