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 2013, 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, estimate the calendar year in which the number of new cases would reach 604.
Years since 2013 (x) New Cases (y)
00 816816
11 802802
22 809809
33 780780
44 754754
55 712712
Answer
Attempt 1 out of 2

Regression Equation:

Final Answer:

Answer 1

To find the linear regression equation based on the provided data, we will first denote the years since 2013 as \(x\) and the number of new cases as \(y\). The values of \(x\) and \(y\) are as follows:

\[
\begin{align*}
x: & \quad 0, \quad 1, \quad 2, \quad 3, \quad 4, \quad 5 \\
y: & \quad 816816, \quad 802802, \quad 809809, \quad 780780, \quad 754754, \quad 712712 \\
\end{align*}
\]

### Step 1: Calculate the Linear Regression Equation

We can calculate the linear regression coefficients (slope \(m\) and intercept \(b\)) using the following formulas:

- Slope (\(m\)):
\[
m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2}
\]

- Intercept (\(b\)):
\[
b = \frac{\sum y - m(\sum x)}{N}
\]

Where \(N\) is the number of points, \(\sum\) denotes the summation, and \(xy\) is the product of \(x\) and \(y\).

### Step 2: Compute the Necessary Sums

Let's compute the required values:

1. \(N = 6\)
2. \(\sum x = 0 + 1 + 2 + 3 + 4 + 5 = 15\)
3. \(\sum y = 816816 + 802802 + 809809 + 780780 + 754754 + 712712 = 4670673\)
4. \(\sum xy = 0\cdot816816 + 1\cdot802802 + 2\cdot809809 + 3\cdot780780 + 4\cdot754754 + 5\cdot712712\)

\[
= 0 + 802802 + 1619618 + 2342340 + 3019016 + 3563560 = 10382236
\]

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

### Step 3: Plug Values into Formulas

Now we plug the sums into the formulas for \(m\) and \(b\):

- Calculate \(m\):
\[
m = \frac{6(10382236) - (15)(4670673)}{6(55) - (15)^2}
\]
\[
= \frac{62293416 - 70010095}{330 - 225}
\]
\[
= \frac{-7717985}{105} \approx -73509.4
\]

- Calculate \(b\):
\[
b = \frac{4670673 - (-73509.4)(15)}{6}
\]
\[
= \frac{4670673 + 1102641}{6}
\]
\[
= \frac{5773314}{6} \approx 962218.9
\]

### Step 4: Formulate the Linear Regression Equation

The linear regression equation is:
\[
y = -73509.4x + 962218.9
\]

### Step 5: Estimate the year when \(y = 604\)

We need to set \(y\) to 604 and solve for \(x\):
\[
604 = -73509.4x + 962218.9
\]
Rearranging gives:
\[
73509.4x = 962218.9 - 604
\]
\[
73509.4x = 962218.3
\]
\[
x \approx \frac{962218.3}{73509.4} \approx 13.1
\]

This means that it would take approximately 13.1 years since 2013, which corresponds to the year:
\[
2013 + 13.1 \approx 2026
\]

### Final Answers

- **Regression Equation:** \(y = -73509.4x + 962218.9\)
- **Estimated Year for 604 Cases:** 2026