Get personalized and accurate responses to your questions with IDNLearn.com. Ask anything and receive immediate, well-informed answers from our dedicated community of experts.
Sagot :
To find the equation of the line that best fits the data provided in the table, we will employ the method of linear regression. This involves finding the coefficients \( m \) (slope) and \( c \) (intercept) such that the line \( y = mx + c \) minimizes the sum of the squared differences between the observed values (GPA) and the predicted values on the line. Here are the steps to achieve this:
1. Organize the data points:
[tex]\[ \begin{array}{|c|c|} \hline IQ & GPA \\ \hline 117 & 3.9 \\ 93 & 2.7 \\ 102 & 2.9 \\ 110 & 3.1 \\ 88 & 2.4 \\ 75 & 1.9 \\ \hline \end{array} \][/tex]
2. Create the design matrix \( A \):
In linear regression, each row of matrix \( A \) should consist of an \((x, 1)\) pair from your data.
[tex]\[ A = \begin{pmatrix} 117 & 1 \\ 93 & 1 \\ 102 & 1 \\ 110 & 1 \\ 88 & 1 \\ 75 & 1 \\ \end{pmatrix} \][/tex]
3. Construct the \( y \) vector:
The \( y \) vector contains the GPA values corresponding to each IQ.
[tex]\[ y = \begin{pmatrix} 3.9 \\ 2.7 \\ 2.9 \\ 3.1 \\ 2.4 \\ 1.9 \\ \end{pmatrix} \][/tex]
4. Calculate the coefficients \( m \) and \( c \):
Using the least squares method, we solve for the coefficients that best fit the data. The least squares solution seeks to minimize \( \|Ax - y\| \), where \( x \) is a vector containing the coefficients \( m \) and \( c \).
5. Results:
After performing the calculations, the resulting values for the slope \( m \) and the intercept \( c \) are found to be:
[tex]\[ m = 0.0427354069024286 \\ c = -1.350035506320122 \][/tex]
6. Formulate the equation:
Substituting \( m \) and \( c \) into the linear equation \( y = mx + c \), we get:
[tex]\[ y = 0.0427354069024286 \cdot x - 1.350035506320122 \][/tex]
7. Compare with given choices:
We now compare the calculated equation with the given choices:
[tex]\[ \begin{align*} \text{Choice 1:} & \quad y = 1.09x - 2.55 \\ \text{Choice 2:} & \quad y = 0.09x - 1.67 \\ \text{Choice 3:} & \quad y = 21.89x - 2.04 \\ \text{Choice 4:} & \quad y = 0.04x - 1.08 \\ \end{align*} \][/tex]
Our calculated equation \( y = 0.0427354069024286x - 1.350035506320122 \) is closest to "Choice 4: \( y = 0.04x - 1.08 \)”.
Therefore, the equation of the line that best fits the data is [tex]\( y = 0.04x - 1.08 \)[/tex].
1. Organize the data points:
[tex]\[ \begin{array}{|c|c|} \hline IQ & GPA \\ \hline 117 & 3.9 \\ 93 & 2.7 \\ 102 & 2.9 \\ 110 & 3.1 \\ 88 & 2.4 \\ 75 & 1.9 \\ \hline \end{array} \][/tex]
2. Create the design matrix \( A \):
In linear regression, each row of matrix \( A \) should consist of an \((x, 1)\) pair from your data.
[tex]\[ A = \begin{pmatrix} 117 & 1 \\ 93 & 1 \\ 102 & 1 \\ 110 & 1 \\ 88 & 1 \\ 75 & 1 \\ \end{pmatrix} \][/tex]
3. Construct the \( y \) vector:
The \( y \) vector contains the GPA values corresponding to each IQ.
[tex]\[ y = \begin{pmatrix} 3.9 \\ 2.7 \\ 2.9 \\ 3.1 \\ 2.4 \\ 1.9 \\ \end{pmatrix} \][/tex]
4. Calculate the coefficients \( m \) and \( c \):
Using the least squares method, we solve for the coefficients that best fit the data. The least squares solution seeks to minimize \( \|Ax - y\| \), where \( x \) is a vector containing the coefficients \( m \) and \( c \).
5. Results:
After performing the calculations, the resulting values for the slope \( m \) and the intercept \( c \) are found to be:
[tex]\[ m = 0.0427354069024286 \\ c = -1.350035506320122 \][/tex]
6. Formulate the equation:
Substituting \( m \) and \( c \) into the linear equation \( y = mx + c \), we get:
[tex]\[ y = 0.0427354069024286 \cdot x - 1.350035506320122 \][/tex]
7. Compare with given choices:
We now compare the calculated equation with the given choices:
[tex]\[ \begin{align*} \text{Choice 1:} & \quad y = 1.09x - 2.55 \\ \text{Choice 2:} & \quad y = 0.09x - 1.67 \\ \text{Choice 3:} & \quad y = 21.89x - 2.04 \\ \text{Choice 4:} & \quad y = 0.04x - 1.08 \\ \end{align*} \][/tex]
Our calculated equation \( y = 0.0427354069024286x - 1.350035506320122 \) is closest to "Choice 4: \( y = 0.04x - 1.08 \)”.
Therefore, the equation of the line that best fits the data is [tex]\( y = 0.04x - 1.08 \)[/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. IDNLearn.com has the answers you need. Thank you for visiting, and we look forward to helping you again soon.
