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To solve this problem, we need to perform linear regression using the given data and determine the linear equation in the form:

\[
y = mx + b
\]

Where:
- \(x\) is the number of years since 2010,
- \(y\) is the number of new cases,
- \(m\) is the slope (rate of change),
- \(b\) is the y-intercept (the value of \(y\) when \(x = 0\)).

### Data Points:
The data from the table is:
\[
(0, 954), (1, 989), (2, 1008), (3, 1021), (4, 1033)
\]

We will:
1. Compute the slope (\(m\)) using the formula:
\[
m = \frac{n \sum xy - (\sum x)(\sum y)}{n \sum x^2 - (\sum x)^2}
\]
2. Calculate the y-intercept (\(b\)) using the equation:
\[
b = \frac{\sum y - m \sum x}{n}
\]

Once the linear equation is determined, we will use it to estimate the number of years (\(x\)) when the number of new cases (\(y\)) reaches 1142.

Let me calculate that for you.The linear regression equation for the given data is:

\[
y = 19.0x + 963.0
\]

Now, to estimate the year when the number of new cases reaches 1142, we need to solve for \(x\) in the equation:

\[
1142 = 19.0x + 963.0
\]

Let's solve this equation.The number of new cases is estimated to reach 1142 approximately 9.42 years after 2010. Since we are asked to estimate the calendar year, this would be:

\[
2010 + 9.42 \approx 2019.42
\]

Thus, the number of new cases is expected to reach 1142 sometime in the year **2019**.

Would you like any further details or explanations on this? Here are five related questions to consider:

1. How is the slope \(m\) interpreted in the context of this problem?
2. How would the regression equation change if additional data points were included?
3. What is the meaning of the y-intercept in this linear model?
4. Could a non-linear regression model provide a better fit for the data?
5. How does rounding affect the accuracy of predictions in this case?

**Tip:** When using linear regression, always check if the data seems to follow a linear trend. If not, other models might give better predictions!

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 2010, and y represents the 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, estimate the calendar year in which the number of new cases would reach 1142.

Using linear regression, learn how to predict the number of crime cases in a New York county. This example uses real data, estimates the linear equation, and calculates when the number of new cases would reach 1142. Suitable for Grades 9-12.

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