Question

8 the number of newly reported crime cases in a county in new york state is shown in the accompanying table. write the linear regression equation that represents this set of data. (let x = 0 represent 1999.) using this equation, find the projected number of new cases for 2009, rounded to the nearest whole number.

year (x)	new cases (y)
2000	457
2001	369
2002	351

Explanation:

Step1: Calculate sums

Let \(n = 4\) (number of data - points).
\(\sum_{i = 1}^{n}x_{i}=0 + 1+2 + 3=6\)
\(\sum_{i = 1}^{n}y_{i}=440 + 457+369+351 = 1617\)
\(\sum_{i = 1}^{n}x_{i}^{2}=0^{2}+1^{2}+2^{2}+3^{2}=0 + 1+4 + 9 = 14\)
\(\sum_{i = 1}^{n}x_{i}y_{i}=0\times440+1\times457+2\times369+3\times351=457 + 738+1053 = 2248\)

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the linear - regression line \(y=mx + b\) is \(m=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}\)

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Step3: Calculate y - intercept \(b\)

The formula for the y - intercept \(b\) is \(b=\frac{\sum_{i = 1}^{n}y_{i}-m\sum_{i = 1}^{n}x_{i}}{n}\)

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The linear - regression equation is \(y=-35.5x + 457.5\)

Step4: Find \(x\) value for 2009

Since \(x = 0\) represents 1999, for 2009, \(x=2009 - 1999=10\)

Step5: Predict number of new cases

Substitute \(x = 10\) into the equation \(y=-35.5x + 457.5\)

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Answer:

The linear regression equation is \(y=-35.5x + 457.5\), and the projected number of new cases for 2009 is 103.