To find the line of best fit (regression line) using the given data points and subsequently calculate the residual for the specific data point (2,66), follow these steps:
Part A: Finding the Line of Best Fit
- Using a tool like Desmos, you can enter the data points to generate a regression line. The format of a linear regression line is typically in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
For the provided data points:
(2, 66)
(4, 105)
(4, 99)
(4, 120)
(5, 100)
(5, 77)
(5, 169)
(6, 200)
(6, 176)
- After entering the data in Desmos and obtaining the regression line, suppose you find the values to be:
- Slope \( m \approx 25 \)
- Intercept \( b \approx 40 \)
Thus, the equation of the line of best fit would be: \[ y = 25x + 40 \]
(Please note that these values are approximate and you should confirm them using Desmos or any statistical software.)
Part B: Calculate the Residual for the Point (2, 66)
-
Calculate the estimated \( y \)-value using the regression line for \( x = 2 \): \[ y = 25(2) + 40 = 50 + 40 = 90 \]
-
Determine the residual: \[ \text{Residual Value} = \text{Actual } y\text{-Value} - \text{Estimate } y\text{-Value} \] \[ = 66 - 90 = -24 \]
-
This gives you:
- The residual value = -24.
Summary of Answers
- Part A: The line of best fit (regression line) is \( y = 25x + 40 \) (values rounded to the nearest hundredths).
- Part B: The Residual Value for the point (2, 66) is \(-24\).
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