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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Explanation:

Step1: Calculate necessary sums

We have data points \((x_1,y_1)=(0,912)\), \((x_2,y_2)=(1,887)\), \((x_3,y_3)=(2,915)\), \((x_4,y_4)=(3,995)\).

First, calculate \(\sum x\), \(\sum y\), \(\sum xy\), \(\sum x^2\):

\(\sum x = 0 + 1 + 2 + 3 = 6\)

\(\sum y = 912 + 887 + 915 + 995 = 3709\)

\(\sum xy = (0\times912)+(1\times887)+(2\times915)+(3\times995)=0 + 887+1830 + 2985=5702\)

\(\sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 = 0 + 1 + 4 + 9 = 14\)

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the linear regression line \(y = mx + b\) is:

\(m=\frac{n\sum xy-\sum x\sum y}{n\sum x^2 - (\sum x)^2}\)

where \(n = 4\) (number of data points).

Substitute the values:

\(m=\frac{4\times5702 - 6\times3709}{4\times14 - 6^2}=\frac{22808 - 22254}{56 - 36}=\frac{554}{20}=27.7\)

Step3: Calculate y-intercept \(b\)

The formula for \(b\) is:

\(b=\frac{\sum y - m\sum x}{n}\)

Substitute the values:

\(b=\frac{3709 - 27.7\times6}{4}=\frac{3709 - 166.2}{4}=\frac{3542.8}{4}=885.7\)

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

Step4: Find \(x\) for 2024

Since \(x\) is the number of years since 2012, for 2024, \(x = 2024 - 2012 = 12\)

Step5: Predict \(y\) for \(x = 12\)

Substitute \(x = 12\) into the equation \(y = 27.7x + 885.7\):

\(y = 27.7\times12 + 885.7 = 332.4 + 885.7 = 1218.1\approx1218\)

Answer:

The linear regression equation is \(y = 27.7x + 885.7\) and the projected number of new cases for 2024 is \(\boxed{1218}\)