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 1998, 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 2006, rounded to the nearest whole number.
Years since 1998 (x)	New Cases (y)
0	  1129
1    1126
2	  1149
3 	  1194
4	  1197

To solve this, we need to:

1. Perform **linear regression** on the given dataset to find the equation of the line of best fit.
2. Use that equation to project the number of new cases for the year 2006.

The data points from the table are:

\[
(x, y) = \{(0, 1129), (1, 1126), (2, 1149), (3, 1194), (4, 1197)\}
\]

The linear regression equation takes the form:

\[
y = mx + b
\]

Where:
- \( m \) is the slope of the line,
- \( b \) is the y-intercept.

Let me compute the values of \( m \) and \( b \) and then use the equation to predict the number of cases for 2006, which corresponds to \( x = 2006 - 1998 = 8 \) years since 1998.

I'll compute the linear regression now.The linear regression equation that represents the given data is:

\[
y = 20.40x + 1118.20
\]

Using this equation to predict the number of new cases for 2006 (which corresponds to \(x = 8\)):

\[
y = 20.40(8) + 1118.20 = 1281.4
\]

Rounding this to the nearest whole number, the projected number of new cases for 2006 is **1281**.

Would you like further details or have any questions? Here are 5 related questions you might explore:

1. How would the prediction change if another year of data were added?
2. Can this linear model be used to predict cases beyond 2006?
3. How accurate is a linear model in predicting crime cases over time?
4. What is the correlation coefficient of the given data?
5. How would you interpret the slope in the context of this problem?

**Tip**: In linear regression, the slope represents the rate of change, and the intercept gives the starting value when \(x = 0\).

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 1998, 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 2006, rounded to the nearest whole number.

Explore how to write a linear regression equation based on crime data from a New York county since 1998. Learn how to round coefficients to the nearest hundredth and use the equation to project the number of new cases in 2006. This problem covers linear regression, algebra, and statistics, suitable for Grades 10-12.

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