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 1999, and y represents number of new cases. write the linear regression equation that represents this set of data, rounding all coefficients to the nearest hundredth. using this equation, find the projected number of new cases for 2011, rounded to the nearest whole number.

years since 1999 (x)	new cases (y)
1	865
2	823
3	793
4	745

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Explanation:

Step1: Calculate necessary sums

First, we list the data points: \((0, 885)\), \((1, 865)\), \((2, 823)\), \((3, 793)\), \((4, 745)\)

Calculate \(\sum x\), \(\sum y\), \(\sum xy\), and \(\sum x^2\):

\(\sum x = 0 + 1 + 2 + 3 + 4 = 10\)

\(\sum y = 885 + 865 + 823 + 793 + 745 = 4111\)

\(\sum xy = (0\times885) + (1\times865) + (2\times823) + (3\times793) + (4\times745) = 0 + 865 + 1646 + 2379 + 2980 = 7870\)

\(\sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 0 + 1 + 4 + 9 + 16 = 30\)

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 \(n = 5\) (number of data points)

Substitute the values:

\(m=\frac{5\times7870 - 10\times4111}{5\times30 - 10^2}=\frac{39350 - 41110}{150 - 100}=\frac{-1760}{50}=-35.2\)

Step3: Calculate y-intercept \(b\)

The formula for the y-intercept \(b\) is:

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

Substitute the values:

\(b=\frac{4111 - (-35.2)\times10}{5}=\frac{4111 + 352}{5}=\frac{4463}{5}=892.6\)

So the linear regression equation is \(y = -35.2x + 892.6\)

Step4: Find \(x\) for 2011

2011 is \(2011 - 1999 = 12\) years since 1999, so \(x = 12\)

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

Substitute \(x = 12\) into the equation:

\(y = -35.2\times12 + 892.6=-422.4 + 892.6 = 470.2\approx470\)

Answer:

The linear regression equation is \(y = -35.20x + 892.60\) (rounded to nearest hundredth), and the projected number of new cases for 2011 is \(\boxed{470}\)