To determine the line of best fit, also known as the linear regression equation, for Natalie's data, we need to follow these steps:
1. Collect the Data Points:
We have the following pairs of data (Length of Catapult Arm in cm, Horizontal Distance in cm):
[tex]\[
\begin{align*}
(25, 290.8), \\
(35, 325.4), \\
(30, 315.2), \\
(60, 420), \\
(65, 435.8), \\
(50, 385.1), \\
(45, 355), \\
(40, 362), \\
(50, 378.3), \\
(40, 333.9).
\end{align*}
\][/tex]
2. Calculate the Means:
- Mean of lengths [tex]\(\bar{x}\)[/tex]:
[tex]\[
\bar{x} = \frac{1}{10}(25 + 35 + 30 + 60 + 65 + 50 + 45 + 40 + 50 + 40) = \frac{440}{10} = 44
\][/tex]
- Mean of distances [tex]\(\bar{y}\)[/tex]:
[tex]\[
\bar{y} = \frac{1}{10}(290.8 + 325.4 + 315.2 + 420 + 435.8 + 385.1 + 355 + 362 + 378.3 + 333.9) = \frac{3601.5}{10} = 360.15
\][/tex]
3. Calculate the Slopes and Intercepts:
We calculate the slope [tex]\(m\)[/tex] using the formula:
[tex]\[
m = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n} (x_i - \bar{x})^2}
\][/tex]
The numerator can be calculated as:
[tex]\[
\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) = (25-44)(290.8-360.15) + (35-44)(325.4-360.15) + \ldots + (40-44)(333.9-360.15)
\][/tex]
The denominator can be calculated as:
[tex]\[
\sum_{i=1}^{n} (x_i - \bar{x})^2 = (25-44)^2 + (35-44)^2 + \ldots + (40-44)^2
\][/tex]
Plugging values:
[tex]\[
\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) = (25-44)(290.8-360.15) + (35-44)(325.4-360.15) + \ldots + (40-44)(333.9-360.15)
\][/tex]
After performing the calculations:
[tex]\[
m \approx 2.7
\][/tex]
Then, to find [tex]\( b \)[/tex], the y-intercept formula is:
[tex]\[
b = \bar{y} - m\bar{x}
\][/tex]
[tex]\[
b = 360.15 - (2.7 \times 44) = 360.15 - 118.8 \approx 241.35
\][/tex]
Thus, the equation of the line of best fit is:
[tex]\[
y = 2.7x + 241.4
\][/tex]
