IDNLearn.com: Your reliable source for finding precise answers. Our experts provide prompt and accurate answers to help you make informed decisions on any topic.
Sagot :
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]
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]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Your search for solutions ends here at IDNLearn.com. Thank you for visiting, and come back soon for more helpful information.