AP Statistics 5.4 Residuals Overview

Part (a):
For a car with length 175 inches, the predicted value for the car's FCR, based on the least squares regression line, is

The actual FCR for the car is 5.88, so the residual is 5.88-4.92=0.96. The residual value means that the car's FCR is 0.96 gallons per 100 miles greater than would be predicted for a car of its length.

Part (b):

Part (c):

Graph II reveals a moderate association that is positive and linear. In contrast, there is a weak association that is positive and linear in graph III. The association between engine size and residual (from predicting FCR based on length) is stronger than the association between wheel base and residual (from predicting FCR based on length).

Part (d):

Engine size is a better choice than wheel base for including with length in a regression model for predicting FCR. The stronger association between engine size and residual (from predicting FCR based on length) indicates that engine size is more useful than wheel base for reducing the variability in FCR values that remains unexplained (as indicated by residuals) after predicting FCR based on length.

Scoring

Parts (a), (b), (c), and (d) were scored as essentially correct (E), partially correct (P), or incorrect (I).
Part (a) is scored as follows:
Essentially correct (E) if the response provides the following two components:

A correct residual value with supporting calculation.

A correct interpretation of the residual value, in context.

Partially correct (P) if the response includes only one of the two components listed above.
Incorrect (I) if the response does not meet the criteria for E or P.
Notes:

- If the residual value is incorrect, the interpretation should be considered correct if it follows from the incorrect residual value.

- Correct interpretation of the residual must include the correct direction and magnitude of the FCR value away from the predicted FCR value.

- A calculated residual value which is slightly different from 0.96 due to the number of significant digits is acceptable.

Part (b) is scored as follows:
Essentially correct (E) if the response provides the following two components:

Circles the correct point in graph III.

Provides a reasonable interpretation of the car associated with point B having a residual near 0 that refers to predicting FCR based on length.

Partially correct (P) if the response correctly provides only one of the two components listed above.
Incorrect (I) if the response does not meet the criteria for E or P.

Note: A correct response for the second component must include reference to the observed FCR value of the car represented by point B, not the point B itself.

Part (c) is scored as follows:
Essentially correct (E) if the response correctly provides the following three components:

A description of form AND direction for both graphs.

A description of the strength of association for both graphs.

A comparison between the two graphs.

Partially correct (P) if the response correctly provides only two of the three components listed above.
Incorrect (I) if the response does not meet the criteria for E or P.
Notes:

- Part (c) is focused on the comparison of graph II and graph III. Inferences drawn from patterns in these graphs are considered in part (d).

- Linear is needed for form in graph II.

- Graph III may be described as having no association between wheel base and the residuals of FCR based on length, which is sufficient for describing the form, direction and strength of association of graph III.

Part (d) is scored as follows:
Essentially correct (E) if the response indicates the correct choice with a sound justification based on the following two components:

The strong(er) association.

Reducing the variability that remains unexplained in the model which predicts FCR based on length.

Partially correct (P) if the response indicates the correct choice and provides a justification based on only one of the two components which are listed above.

Incorrect (I) if the response indicates the incorrect choice;
OR
if the response indicates the correct choice but does not mention either of the two components which are listed above.

Note: Describing the variables in graph II and graph III as residuals is not required but can be used positively in holistic scoring. Incorrect descriptions of graph II or graph III or the variables in graphs are not acceptable.

Each essentially correct (E) part counts as 1 point. Each partially correct (P) part counts as $\frac{1}{2}$ point.

Complete Response

Substantial Response

Developing Response

Minimal Response

If a response is between two scores (for example, $2 \frac{1}{2}$ points), use a holistic approach to decide whether to score up or down, depending on the overall strength of the response and communication.