Answer:
The equation of the regression line is: [tex]y~=~88.518 ~-~ 3.068 \cdot x[/tex]
Step-by-step explanation:
Linear regression is a linear approach to modelling the relationship between a dependent variable and one or more independent variables.
Let X be the independent variable and Y be the dependent variable. We will define a linear relationship between these two variables as follows:
[tex]Y=bX+a[/tex]
We have the the following data:
[tex]\begin{array}{c|cccccccccc}No. \:of \:absences,\:x&0&1&2&3&4&5&6&7&8&9\\Final \:grade, y&88.5&85.7&82.8&80.3&77.4&73.1&63.6&68.1&65.2&62.4\end{array}[/tex]
To find the line of best fit for the points, follow these steps:
Step 1: Find [tex]X\cdot Y[/tex] and [tex]X\cdot X[/tex] as it was done in the table.
Step 2: Find the sum of every column:
[tex]\sum{X} = 45 ~,~ \sum{Y} = 747.1 ~,~ \sum{X \cdot Y} = 3108.8 ~,~ \sum{X^2} = 285[/tex]
Step 3: Use the following equations to find intercept a and slope b:
[tex]\begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 747.1 \cdot 285 - 45 \cdot 3108.8}{ 10 \cdot 285 - 45^2} \approx 88.518 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 10 \cdot 3108.8 - 45 \cdot 747.1 }{ 10 \cdot 285 - \left( 45 \right)^2} \approx -3.068\end{aligned}[/tex]
Step 4: Assemble the equation of a line
[tex]\begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~88.518 ~-~ 3.068 \cdot x\end{aligned}[/tex]
The graph of the regression line is:
