Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Connect with professionals ready to provide precise answers to your questions on our comprehensive Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To find the slope-intercept form of the equation of the line that best fits the given data points, we can use the method of linear regression. Linear regression will give us the best-fit line equation in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Here are the data points provided:
[tex]\[ \begin{array}{c|c} X & Y \\ \hline 0.7 & 5 \\ 0.9 & 6 \\ 0.1 & 1 \\ 0.3 & 1 \\ 0.4 & 2 \\ 0.8 & 5 \\ 0.3 & 1 \\ 0.9 & 6 \\ 0.2 & 2 \\ 1 & 7 \\ \end{array} \][/tex]
Given these data points, we aim to calculate:
1. Slope (m): This measures the steepness of the line.
2. Intercept (b): This is the value of the y-coordinate when x is 0.
Step-by-Step Solution:
### 1. Calculate the Slope (m)
The slope [tex]\( m \)[/tex] can be interpreted as the change in [tex]\( Y \)[/tex] for a unit change in [tex]\( X \)[/tex]. Using the method of least squares, we find that the slope [tex]\( m \)[/tex] is approximately:
[tex]\[ m = 7.01 \][/tex]
### 2. Calculate the Intercept (b)
The intercept [tex]\( b \)[/tex] is the point where the line crosses the y-axis (i.e., when [tex]\( X = 0 \)[/tex]). From our calculations, the intercept [tex]\( b \)[/tex] is approximately:
[tex]\[ b = -0.33 \][/tex]
### 3. Form the Equation of the Line
Substituting the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] into the slope-intercept form, we get the equation of the best-fit line:
[tex]\[ y = 7.01x - 0.33 \][/tex]
In summary, the equation of the line that best fits the given data points is:
[tex]\[ y = 7.01x - 0.33 \][/tex]
[tex]\[ \begin{array}{c|c} X & Y \\ \hline 0.7 & 5 \\ 0.9 & 6 \\ 0.1 & 1 \\ 0.3 & 1 \\ 0.4 & 2 \\ 0.8 & 5 \\ 0.3 & 1 \\ 0.9 & 6 \\ 0.2 & 2 \\ 1 & 7 \\ \end{array} \][/tex]
Given these data points, we aim to calculate:
1. Slope (m): This measures the steepness of the line.
2. Intercept (b): This is the value of the y-coordinate when x is 0.
Step-by-Step Solution:
### 1. Calculate the Slope (m)
The slope [tex]\( m \)[/tex] can be interpreted as the change in [tex]\( Y \)[/tex] for a unit change in [tex]\( X \)[/tex]. Using the method of least squares, we find that the slope [tex]\( m \)[/tex] is approximately:
[tex]\[ m = 7.01 \][/tex]
### 2. Calculate the Intercept (b)
The intercept [tex]\( b \)[/tex] is the point where the line crosses the y-axis (i.e., when [tex]\( X = 0 \)[/tex]). From our calculations, the intercept [tex]\( b \)[/tex] is approximately:
[tex]\[ b = -0.33 \][/tex]
### 3. Form the Equation of the Line
Substituting the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] into the slope-intercept form, we get the equation of the best-fit line:
[tex]\[ y = 7.01x - 0.33 \][/tex]
In summary, the equation of the line that best fits the given data points is:
[tex]\[ y = 7.01x - 0.33 \][/tex]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.