Worksheet

1

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:

a

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} |

b

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} |

2

Complete the following tables of residuals:

a

x | y | \hat{y} | \text{Residual} |
---|---|---|---|

10 | 29 | 25 | |

14 | 37 | -8 | |

7 | 20 | 4 | |

6 | 13 | 8 | |

20 | 58 | 55 | |

13 | 26 | -8 |

b

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 |

3

If a residual is a positive value, is the actual value of the response variable above or below the least squares regression line?

4

If a measured point in a data set is below the least squares regression line, will the corresponding residual be positive or negative?

5

Construct the residual scatter plot of the following data sets:

a

b

6

The residual plot for a set of data is shown. Draw the scatter plot showing the original data.

7

The residual plot can help us decide if a linear model should be used for a raw set of data.

a

Explain when a linear model would be a good fit for a set of raw data.

b

Explain when a linear model would not be a good fit for a set of raw data.

8

For each of the following data sets:

i

Complete the table of residuals.

ii

Plot the residuals on a scatter plot.

iii

Determine if the model is a good fit for the data.

a

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 |

b

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 |

9

The table below shows the residual values after a least-squares regression line has been fitted to a set of data:

a

Plot the residuals on a scatter plot.

b

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 |

10

For each of the data sets below:

i

Complete the table by finding the residuals.

ii

Plot the residuals on a scatter plot.

iii

Determine whether the original data has a linear or nonlinear relationship.

iv

Create a scatter plot for the original data.

a

x | y | \hat{y} | \text{Residuals} |
---|---|---|---|

2 | 14.5 | 12 | |

4 | 9 | 11 | |

6 | 12 | 10 | |

8 | 7 | 9 | |

10 | 10.5 | 8 |

b

x | y | y | \text{Residuals} |
---|---|---|---|

2 | 30 | 8 | |

4 | 20 | 10 | |

6 | 14 | 12 | |

8 | 12 | 14 | |

10 | 14 | 16 | |

12 | 20 | 18 |

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use a residual plot to assess the appropriateness of fitting a linear model to the data