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 2014, 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 2026, rounded to the nearest whole number.

years since 2014 (x)	new cases (y)
1	1020
2	989
3	943
4	964
5	880

Explanation:

Step1: Calculate mean of x and y

First, we find the mean of \( x \) values (\( \bar{x} \)) and mean of \( y \) values (\( \bar{y} \)).
The \( x \) values are \( 0, 1, 2, 3, 4, 5 \).
\( \bar{x}=\frac{0 + 1+2 + 3+4 + 5}{6}=\frac{15}{6}=2.5 \)
The \( y \) values are \( 1047, 1020, 989, 943, 964, 880 \).
\( \bar{y}=\frac{1047+1020 + 989+943+964+880}{6}=\frac{5843}{6}\approx973.83 \)

Step2: Calculate slope (m)

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

• For \( x = 0,y = 1047 \): \( (0 - 2.5)(1047 - 973.83)=(- 2.5)(73.17)=-182.925 \)
• For \( x = 1,y = 1020 \): \( (1 - 2.5)(1020 - 973.83)=(-1.5)(46.17)=-69.255 \)
• For \( x = 2,y = 989 \): \( (2 - 2.5)(989 - 973.83)=(-0.5)(15.17)=-7.585 \)
• For \( x = 3,y = 943 \): \( (3 - 2.5)(943 - 973.83)=(0.5)(-30.83)=-15.415 \)
• For \( x = 4,y = 964 \): \( (4 - 2.5)(964 - 973.83)=(1.5)(-9.83)=-14.745 \)
• For \( x = 5,y = 880 \): \( (5 - 2.5)(880 - 973.83)=(2.5)(-93.83)=-234.575 \)

Sum of these products: \( -182.925-69.255 - 7.585-15.415-14.745-234.575=-524.5 \)

Now, calculate \( (x_i-\bar{x})^2 \) for each \( i \):

• For \( x = 0 \): \( (0 - 2.5)^2 = 6.25 \)
• For \( x = 1 \): \( (1 - 2.5)^2 = 2.25 \)
• For \( x = 2 \): \( (2 - 2.5)^2 = 0.25 \)
• For \( x = 3 \): \( (3 - 2.5)^2 = 0.25 \)
• For \( x = 4 \): \( (4 - 2.5)^2 = 2.25 \)
• For \( x = 5 \): \( (5 - 2.5)^2 = 6.25 \)

Sum of these squares: \( 6.25+2.25 + 0.25+0.25+2.25+6.25 = 17.5 \)

So, \( m=\frac{-524.5}{17.5}\approx - 29.97\approx - 30.0 \) (rounded to nearest tenth)

Step3: Calculate y-intercept (b)

Using the formula \( \bar{y}=m\bar{x}+b \)
\( 973.83=-30.0\times2.5 + b \)
\( 973.83=-75 + b \)
\( b=973.83 + 75=1048.83\approx1048.8 \) (rounded to nearest tenth)

So, the linear regression equation is \( y=-30.0x + 1048.8 \)

Step4: Find x for 2026

2026 - 2014 = 12, so \( x = 12 \)

Step5: Predict y for x = 12

Substitute \( x = 12 \) into the equation \( y=-30.0\times12 + 1048.8 \)
\( y=-360+1048.8 = 688.8\approx689 \) (rounded to nearest whole number)

Answer:

The linear regression equation is \( y = - 30.0x+1048.8 \) and the projected number of new cases for 2026 is \( 689 \)