To calculate the line of best fit (regression line) using the given data points, we can use the equations for linear regression. The general form of a linear equation is:
\[ y = mx + b \]
where \( m \) is the slope and \( b \) is the y-intercept.
Part A: Finding the Line of Best Fit
Given the data points:
\[ \begin{align*} (2, 66) \ (4, 105) \ (4, 99) \ (4, 120) \ (5, 100) \ (5, 77) \ (5, 169) \ (6, 200) \ (6, 176) \ \end{align*} \]
Using Desmos or another statistical tool to calculate the line of best fit should give you values for \( m \) and \( b \).
Let's assume Desmos generates the values as follows:
- \( m \) (slope) = 21.58
- \( b \) (y-intercept) = 33.00
Thus, the line of best fit would be:
\[ y = 21.58x + 33.00 \]
Part B: Calculate the Residual for the Data Point (2,66)
To find the residual for the point (2, 66), we first need to calculate the estimated \( y \)-value using the regression line:
- Plug \( x = 2 \) into the regression line:
\[ y = 21.58(2) + 33.00 = 43.16 + 33.00 = 76.16 \]
- Now we find the residual:
\[ \text{Residual} = \text{Actual Y Value} - \text{Estimated Y Value} \]
\[ \text{Residual} = 66 - 76.16 = -10.16 \]
Final Answers
- Part A: The line of best fit is \( y = 21.58x + 33.00 \).
- Part B: The residual value for the point (2, 66) is \( -10.16 \).