AP Statistics 5.4 Residuals

[Maximum number: 4]

A biologist gathered data on the length, in millimeters (mm), and the mass, in grams (g), for 11 bullfrogs. The data are shown in Plot 1.

From the data, the biologist calculated the least-squares regression line for predicting mass from length. The least-squares regression line is shown in Plot 2.

Official question visual

Scoring for Question 1 Each essentially correct (E) part counts as 1 point, and each partially correct (P) part counts as $\frac{1}{2}$ point.
	Score
Complete Response	4
Substantial Response	3
Developing Response	2
Minimal Response	1
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 strength of the response and quality of the communication.

(a)

From Plot 2, consider the residuals of the 11 bullfrogs.

(i)

Based on the plot, approximately what is the length and mass of the bullfrog with the largest absolute value residual?

(ii)

Does the least-squares regression line overestimate or underestimate the mass of the bullfrog identified in part (d-i)? Explain your answer.

[Maximum number: 4]

Jamal is researching the characteristics of a car that might be useful in predicting the fuel consumption rate (FCR); that is, the number of gallons of gasoline that the car requires to travel 100 miles under conditions of typical city driving. The length of a car is one explanatory variable that can be used to predict FCR. Graph I is a scatterplot showing the lengths of 66 cars plotted with the corresponding FCR. One point on the graph is labeled A.

GRAPH I

Jamal examined the scatterplot and determined that a linear model would be a reasonable way to express the relationship between FCR and length. A computer output from a linear regression is shown below.
Linear Fit
$\mathrm{FCR}=-1.595789+0.0372614 * \mathrm{Length}$
Summary of Fit

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.

(a)

The point on the graph labeled A represents one car of length 175 inches and an FCR of 5.88. Calculate and interpret the residual for the car relative to the least squares regression line.

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

\text { predicted FCR }=-1.595789+0.0372614(175) \approx 4.92 \text { gallons per } 100 \text { miles. }

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.

(b)

Jamal knows that it is possible to predict a response variable using more than one explanatory variable. He wants to see if he can improve the original model of predicting FCR from length by including a second explanatory variable in addition to length. He is considering including engine size, in liters, or wheel base (the length between axles), in inches. Graph II is a scatterplot showing the engine size of the 66 cars plotted with the corresponding residuals from the regression of FCR on length. Graph III is a scatterplot showing the wheel base of the 66 cars plotted with the corresponding residuals from the regression of FCR on length.

In graph II, the point labeled A corresponds to the same car whose point was labeled A in graph I. The measurements for the car represented by point A are given below.

(i)

Circle the point on graph III that corresponds to the car represented by point A on graphs I and II.

(ii)

There is a point on graph III labeled B. It is very close to the horizontal line at 0. What does that indicate about the FCR of the car represented by point B?

(c)

Write a few sentences to compare the association between the variables in graph II with the association between the variables in graph III.

(d)

Jamal wants to predict FCR using length and one of the other variables, engine size or wheel base. Based on your response to part (c), which variable, engine size or wheel base, should Jamal use in addition to length if he wants to improve the prediction? Explain why you chose that variable.

AP Statistics 5.4 Residuals Questions

Exam points

Covered syllabus points

Covered syllabus points

Question 1

Question (a)

Question (i)

Question (ii)

Question 2

Question (a)

Question (b)

Question (i)

Question (ii)

Question (c)

Question (d)