Sure, let's work through the problem step-by-step.
### Step 1: Organize the Data
First, we organize the given data:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Years since 2011 (x)} & \text{New Cases (y)} \\
\hline
0 & 814 \\
1 & 825 \\
2 & 854 \\
3 & 869 \\
4 & 947 \\
\hline
\end{array}
\][/tex]
### Step 2: Calculate the Linear Regression Equation
To find the linear regression equation [tex]\( y = mx + b \)[/tex], we need to determine the slope [tex]\( m \)[/tex] and the intercept [tex]\( b \)[/tex].
From statistical calculations (using linear regression tools), we find:
- Slope ([tex]\(m\)[/tex]): 31.0 (rounded to the nearest hundredth)
- Intercept ([tex]\(b\)[/tex]): 799.8 (rounded to the nearest hundredth)
Thus, the linear regression equation is:
[tex]\[
y = 31.00x + 799.80
\][/tex]
### Step 3: Estimate the Year When the Number of New Cases Will Reach 1141
We need to estimate the calendar year when the number of new cases [tex]\( y \)[/tex] would reach 1141. Set [tex]\( y = 1141 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[
1141 = 31.00x + 799.80
\][/tex]
Subtract 799.80 from both sides:
[tex]\[
1141 - 799.80 = 31.00x
\][/tex]
[tex]\[
341.20 = 31.00x
\][/tex]
Divide both sides by 31.00:
[tex]\[
x = \frac{341.20}{31.00} \approx 11.01
\][/tex]
### Step 4: Determine the Calendar Year
Since [tex]\( x \)[/tex] represents the number of years since 2011:
[tex]\[
\text{Estimated year} = 2011 + 11.01 \approx 2022.01
\][/tex]
Hence, the number of new cases is projected to reach 1141 around early 2022.
### Final Answer
- The linear regression equation that represents the given set of data is:
[tex]\[
y = 31.00x + 799.80
\][/tex]
- The estimated calendar year when the number of new cases would reach 1141 is approximately:
[tex]\[
\text{early 2022}
\][/tex]
