The following tables show sets of data \left(x, y\right) and the predicted \hat{y} values based on a least-squares regression line. Complete the tables by finding the residuals:
| x | 1 | 3 | 5 | 7 | 9 |
|---|---|---|---|---|---|
| y | 22.7 | 22.3 | 24.2 | 21.8 | 21.5 |
| \hat{y} | 25.2 | 23.4 | 21.6 | 19.8 | 18 |
| \text{Residuals} |
| x | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|
| y | 37.7 | 37.2 | 21.1 | 27.1 | 44 |
| \hat{y} | 28.9 | 30.4 | 31.9 | 33.4 | 34.9 |
| \text{Residuals} |
Complete the following tables of residuals:
| x | y | \hat{y} | \text{Residual} |
|---|---|---|---|
| 10 | 29 | 25 | |
| 14 | 37 | -8 | |
| 7 | 20 | 4 | |
| 6 | 13 | 8 | |
| 20 | 58 | 55 | |
| 13 | 26 | -8 |
| x | y | \hat{y} | \text{Residual} |
|---|---|---|---|
| 12.8 | -33.3 | 0.6 | |
| 11.4 | -30.3 | -30.3 | |
| 11.1 | -29.7 | 0.2 | |
| 6.5 | -21.5 | -1.4 | |
| 10 | -28.5 | -27.4 | -1.1 |
If a residual is a positive value, is the actual value of the response variable above or below the least squares regression line?
If a measured point in a data set is below the least squares regression line, will the corresponding residual be positive or negative?
Construct the residual scatter plot of the following data sets:
The residual plot for a set of data is shown. Draw the scatter plot showing the original data.
The residual plot can help us decide if a linear model should be used for a raw set of data.
Explain when a linear model would be a good fit for a set of raw data.
Explain when a linear model would not be a good fit for a set of raw data.
For each of the following data sets:
Plot the residuals on a scatter plot.
Determine if the model is a good fit for the data.
The table shows a company's costs, C (in millions), in week W.
The equation C = 3 W + 4 is being used to model the data.
| W | C | \text{Model value} | \text{Residual} |
|---|---|---|---|
| 1 | 7 | ||
| 2 | 9 | ||
| 5 | 21 | ||
| 7 | 26 | ||
| 10 | 34 | ||
| 13 | 43 | ||
| 15 | 48 | ||
| 18 | 58 |
The table shows a company's revenue, R (in millions), in week W.
The equation R = 3 W + 5 is being used to model the data.
| W | R | \text{Model value} | \text{Residual} |
|---|---|---|---|
| 2 | 7 | ||
| 3 | 12 | ||
| 5 | 19 | ||
| 7 | 29 | ||
| 9 | 37 | ||
| 10 | 38 | ||
| 12 | 41 | ||
| 14 | 45 |
The table below shows the residual values after a least-squares regression line has been fitted to a set of data:
Plot the residuals on a scatter plot.
Determine if a linear model is a good fit for the data.
| x | 12 | 20 | 10 | 18 | 9 | 20 |
|---|---|---|---|---|---|---|
| \text{Residual} | -4 | -2 | 5 | 2 | 3 | -1 |
For each of the data sets below:
Complete the table by finding the residuals.
Plot the residuals on a scatter plot.
Determine whether the original data has a linear or nonlinear relationship.
Create a scatter plot for the original data.
| x | y | \hat{y} | \text{Residuals} |
|---|---|---|---|
| 2 | 14.5 | 12 | |
| 4 | 9 | 11 | |
| 6 | 12 | 10 | |
| 8 | 7 | 9 | |
| 10 | 10.5 | 8 |
| x | y | \hat{y} | \text{Residuals} |
|---|---|---|---|
| 2 | 30 | 8 | |
| 4 | 20 | 10 | |
| 6 | 14 | 12 | |
| 8 | 12 | 14 | |
| 10 | 14 | 16 | |
| 12 | 20 | 18 |