To determine the equation of the line of best fit for Natalie's data, we need to follow a step-by-step approach to perform linear regression. This will help us find the slope (m) and the y-intercept (b) of the linear equation, which has the general form:
[tex]\[ y = mx + b \][/tex]
### Step 1: List the Data
Here is the data table given:
[tex]\[
\begin{array}{|c|c|}
\hline \text{Length of Catapult Arm} \, ( \text{cm} ) & \text{Horizontal Distance} \, ( \text{cm} ) \\
\hline 25 & 290.8 \\
\hline 35 & 325.4 \\
\hline 30 & 315.2 \\
\hline 60 & 420 \\
\hline 65 & 435.8 \\
\hline 50 & 385.1 \\
\hline 45 & 355 \\
\hline 40 & 362 \\
\hline 50 & 378.3 \\
\hline 40 & 333.9 \\
\hline
\end{array}
\][/tex]
### Step 2: Determine the Slope (m) and Intercept (b)
The best way to determine the line of best fit is to perform linear regression. This involves some statistical calculation to minimize the distance between the data points and the line.
From the calculations, we find:
- The slope [tex]\( m \)[/tex] is approximately 3.6
- The intercept [tex]\( b \)[/tex] is approximately 202.6
These values are rounded to the nearest tenth.
### Step 3: Write the Equation of the Line of Best Fit
Using the values for the slope and intercept, we can now write the equation of the line of best fit:
[tex]\[ y = 3.6x + 202.6 \][/tex]
### Conclusion
The equation of the line of best fit for the data set provided by Natalie is:
[tex]\[ y = 3.6x + 202.6 \][/tex]
Make sure to replace [tex]\( m \)[/tex] and [tex]\( b \)[/tex] with 3.6 and 202.6, respectively, in the linear equation format for the final answer.
