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 tenth. 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)
00	715
1    669
2	651
3	615
4	611

To solve this problem, we'll go through the following steps:

1. **Determine the linear regression equation**: The general form of the linear regression equation is:
   \[
   y = mx + b
   \]
   where \( m \) is the slope and \( b \) is the y-intercept.

2. **Calculate the slope (\( m \))**:
   \[
   m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
   \]
   where \( n \) is the number of data points.

3. **Calculate the y-intercept (\( b \))**:
   \[
   b = \frac{(\sum y) - m(\sum x)}{n}
   \]

4. **Substitute the values into the linear equation** and use it to predict the number of new cases in 2011.

### Step 1: Prepare the data

Given dataTo solve this problem, we'll go through the following steps:

1. **Determine the linear regression equation**: The general form of the linear regression equation is:
   \[
   y = mx + b
   \]
   where \( m \) is the slope and \( b \) is the y-intercept.

2. **Calculate the slope (\( m \))**:
   \[
   m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
   \]
   where \( n \) is the number of data points.

3. **Calculate the y-intercept (\( b \))**:
   \[
   b = \frac{(\sum y) - m(\sum x)}{n}
   \]

4. **Substitute the values into the linear equation** and use it to predict the number of new cases in 2011.

### Step 1: Prepare the data

Given 
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 715 \\
1 & 669 \\
2 & 651 \\
3 & 615 \\
4 & 611 \\
\hline
\end{array}
\]
We'll calculate the following sums:
\[
\sum x = 0 + 1 + 2 + 3 + 4 = 10
\]
\[
\sum y = 715 + 669 + 651 + 615 + 611 = 3261
\]
\[
\sum xy = (0 \times 715) + (1 \times 669) + (2 \times 651) + (3 \times 615) + (4 \times 611) = 0 + 669 + 1302 + 1845 + 2444 = 6260
\]
\[
\sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 0 + 1 + 4 + 9 + 16 = 30
\]

### Step 2: Calculate the slope (\( m \))

\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
\[
m = \frac{5(6260) - (10)(3261)}{5(30) - (10)^2}
\]
\[
m = \frac{31300 - 32610}{150 - 100} = \frac{-1310}{50} = -26.2
\]

### Step 3: Calculate the y-intercept (\( b \))

\[
b = \frac{(\sum y) - m(\sum x)}{n}
\]
\[
b = \frac{3261 - (-26.2)(10)}{5} = \frac{3261 + 262}{5} = \frac{3523}{5} = 704.6
\]

### Step 4: Write the linear regression equation

The linear regression equation is:
\[
y = -26.2x + 704.6
\]

### Step 5: Project the number of new cases for 2011

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

\[
y = -26.2(12) + 704.6
\]
\[
y = -314.4 + 704.6 = 390.2
\]

Rounding to the nearest whole number, the projected number of new cases in 2011 is **390**.

Would you like further details on this process or have any questions?

Here are 5 related questions for further exploration:
1. How would the linear regression equation change if an additional year of data were added?
2. What is the significance of the slope in the context of crime cases?
3. How can we assess the accuracy of this linear model in predicting future cases?
4. What other methods, besides linear regression, can be used to predict crime trends?
5. How would the projection change if we used a different starting year for \( x \)?

**Tip**: When interpreting the slope in a regression equation, it indicates the rate of change of the dependent variable (in this case, the number of new cases) per unit increase in the independent variable (years since 1999).

Explore the linear regression equation for crime cases in a New York County since 1999, predicting future trends based on historical data. Learn how to calculate the slope and y-intercept, and project the number of new cases for specific years. Suitable for high school students studying statistics and regression analysis.

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