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 2001, 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 2010, rounded to the nearest whole number.

years since 2001 (x) | new cases (y)
--- | ---
0 | 868
1 | 834
2 | 803
3 | 793
4 | 780
5 | 751

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Explanation:

Step1: Calculate necessary sums

First, we list the data points: \((0, 868)\), \((1, 834)\), \((2, 803)\), \((3, 793)\), \((4, 780)\), \((5, 751)\)

Number of data points \(n = 6\)

Calculate \(\sum x\): \(0 + 1 + 2 + 3 + 4 + 5 = 15\)

Calculate \(\sum y\): \(868 + 834 + 803 + 793 + 780 + 751 = 4829\)

Calculate \(\sum xy\): \((0\times868)+(1\times834)+(2\times803)+(3\times793)+(4\times780)+(5\times751)\)
\(= 0 + 834 + 1606 + 2379 + 3120 + 3755 = 11694\)

Calculate \(\sum x^2\): \(0^2 + 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 0 + 1 + 4 + 9 + 16 + 25 = 55\)

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the linear regression line is:
\[m=\frac{n\sum xy-\sum x\sum y}{n\sum x^2 - (\sum x)^2}\]
Substitute the values:
\[m=\frac{6\times11694 - 15\times4829}{6\times55 - 15^2}\]
First, calculate numerator: \(6\times11694 = 70164\), \(15\times4829 = 72435\), so numerator \(= 70164 - 72435 = -2271\)
Denominator: \(6\times55 = 330\), \(15^2 = 225\), so denominator \(= 330 - 225 = 105\)
Then \(m=\frac{-2271}{105}\approx - 21.63\) (rounded to nearest hundredth)

Step3: Calculate y-intercept \(b\)

The formula for the y-intercept \(b\) is:
\[b=\frac{\sum y - m\sum x}{n}\]
Substitute the values:
\[b=\frac{4829 - (-21.63)\times15}{6}\]
First, calculate \((-21.63)\times15=-324.45\), so \(4829 - (-324.45)=4829 + 324.45 = 5153.45\)
Then \(b=\frac{5153.45}{6}\approx858.91\) (rounded to nearest hundredth)

So the linear regression equation is \(y = -21.63x + 858.91\)

Step4: Find x for 2010

2010 is \(2010 - 2001 = 9\) years since 2001, so \(x = 9\)

Step5: Predict y for \(x = 9\)

Substitute \(x = 9\) into the equation:
\[y=-21.63\times9 + 858.91\]
Calculate \(-21.63\times9=-194.67\)
Then \(y=-194.67 + 858.91 = 664.24\approx664\) (rounded to nearest whole number)

Answer:

The linear regression equation is \(y = -21.63x + 858.91\) and the projected number of new cases for 2010 is \(\boxed{664}\)