AP Statistics Unit 5: Regression Analysis

Use a correctly labeled scatterplot of the twelve temperature–therm pairs.

There is a strong, negative, somewhat curved relationship: colder months use more therms.

Part (a):
The scatterplot supports the newspaper report about number of semesters needed to complete an academic program and starting salary because it shows a positive association between these two variables.

Part (b):
The slope is 1.1594. For each additional semester needed to complete an academic program, the predicted starting salary increases by €1,159.40.

Part (c):
For the business majors alone, there is a strong, negative, linear association between number of semesters and starting salary. Business majors who need a greater number of semesters to complete an academic program tend to have lower starting salaries.

Part (e):
The newspaper report should be modified to account for major. Overall, majors that take longer to complete tend to have higher starting salaries, with chemistry the highest, physics the next highest, and business the lowest. However, within a major, students who take a greater number of semesters tend to have lower starting salaries.

Scoring
This question is scored in three sections. Section 1 consists of parts (a) and (b), section 2 consists of parts (c) and (d), and section 3 consists of part (e). Sections 1, 2, and 3 are scored as essentially correct (E), partially correct (P), or incorrect (I).

Section 1 is scored as follows:
Essentially correct (E) if the response includes the following five components:

In part (a) the response addresses the positive association.

In part (a) the response uses the positive association to justify that the scatterplot supports the newspaper report.

In part (b) the response correctly identifies the numerical value of the slope from the computer output.

In part (b) the response interprets the slope as the change in starting salary for each additional semester, in context.

In part (b) the interpretation of slope includes nondeterministic language (e.g., "predicted starting salary" or equivalent) when interpreting the slope.

Partially correct (P) if the response includes three or four of the five components.
Incorrect (I) if the response does not meet the criteria for E or P.
Notes:

- In part (a) the response can use phrases such as "positive association (correlation, relationship)," "increasing relationship," or describe a positive association (e.g., "the starting salaries are higher when there is a greater number of semesters") to satisfy component 1. However, describing the relationship between only two points does not satisfy this component.

- In part (a) comments about linearity and strength should be ignored, even if the response implies these are required (e.g., "Yes, because the relationship is strong, positive, and linear.").

- In part (a) responses that answer "no," "maybe," "kind of," "somewhat," or equivalent do not satisfy component 2.

- In part (a) a response that says "no, because the three clusters of points each have a negative association" does not satisfy component 1 or component 2.

- In part (a) no context is required, but variable names are required in part (b) to satisfy component 4.

- In part (b) a response that states the equation $\hat{y}=34.018+1.1594 x$ does not satisfy component 3 unless the slope is specifically identified or used in the interpretation.

- In part (b) a response that incorrectly identifies the numerical value of the slope can still satisfy components 4 and 5 using the incorrect value.

- In part (b) the 1-unit increase in number of semesters must be stated or implied to satisfy component 4 (e.g., for each semester, for every semester). A response that states or implies an unspecified number of semesters does not satisfy this component (for example, as semesters increase).

- In part (b) examples of nondeterministic language include "predicted starting salary," "expected starting salary," "estimated starting salary," "typical starting salary," "average starting salary," "starting salary, on average," "our model predicts," and so on. However, "about," "approximately," and "according to the model" do not satisfy component 5.

- In part (b) no units are required for the change in predicted starting salary (which means it is OK to say 1.1594 euros, 1159.40, 1.1594, 1,159.40 dollars, and so on).

- In all parts it is acceptable if a response refers to salary rather than starting salary.

Section 2 is scored as follows:
Essentially correct (E) if the response includes the following five components:

In part (c) the response states that the association is negative.

In part (c) the response states that the association is strong OR linear OR both.

In part (c) the response refers to both variables (semesters, salary) in context.

In part (d) the response correctly compares the three majors.

In part (d) the response provides reasonable values for the median salaries or refers to "median starting salaries" when describing a characteristic of the graph.

Partially correct (P) if the response includes three or four of the five components.
Incorrect (I) if the response does not meet the criteria for E or P.
Notes:

- In part (c) the response can use phrases such as "negative association (correlation, relationship)," "decreasing relationship," "inversely related," or describe a negative association (for example, "the starting salaries are smaller when there is a greater number of semesters") to satisfy component 1. However, describing the relationship between only two points does not satisfy this component.

- In part (c) responses that describe the relationship incorrectly as weak or nonlinear do not satisfy component 2, even if the other characteristic is described correctly.

- Only drawing a line on the scatterplot does not satisfy component 2.

- In part (d) a response that says "chemistry has the highest median starting salary and business has the lowest median starting salary" (or the equivalent) implies that physics is in the middle, and satisfies component 4.

- In part (d) if no values are provided for the medians, the response must use the phrase "median starting salary" at least once to satisfy component 5.

- In all parts it is acceptable if a response refers to salary rather than starting salary.

Section 3 is scored as follows:
Essentially correct (E) if the response states that there is a negative association for each of the majors AND the response notes the overall positive association.

Partially correct (P) if the response states that there is a negative association for each of the majors BUT does not note the overall positive association;

OR
if the response states that there is a negative association for one or two specific majors (for example, for business majors) AND the response notes the overall positive association.

Incorrect (I) if the response does not meet the criteria for E or P.
Notes:

- Additional ways to note the overall positive association include stating that the original report was correct, stating that majors that take longer to complete tend to have higher starting salaries, or the equivalent.

- A response that does not explicitly name the majors or refer to the three majors but makes a general statement such as "However, for an academic major, there is a negative association" satisfies the requirement to state the negative association for each of the majors.

- A response that states the original report was wrong, or incorrect, or should be retracted, etc., cannot satisfy the requirement to note the overall positive association. However, a response that says the original report might be misleading (or the equivalent) but still notes the overall positive association satisfies the requirement to note the overall positive association.

- In all parts it is acceptable if a response refers to salary rather than starting salary.

Complete Response
All three sections essentially correct

Substantial Response
Two sections essentially correct and one section partially correct

OR
Sections 1 and 2 partially correct and section 3 essentially correct

OR
Sections 1 and 3 essentially correct and section 2 incorrect

OR
Sections 2 and 3 essentially correct and section 1 incorrect

Developing Response
Sections 1 and 2 essentially correct and section 3 incorrect

OR
Section 1 essentially correct and sections 2 and 3 partially correct

OR
Sections 1 and 3 partially correct and section 2 essentially correct

OR
One section essentially correct and one section partially correct

OR
All three sections partially correct

Minimal Response
One section essentially correct and two sections incorrect

OR
Two sections partially correct and one section incorrect

OR
One section partially correct and two sections incorrect

Part (a):
There is a moderately strong, positive, linear relationship between height and arm span so that taller students tend to have longer arm spans.

Part (c):
The predicted arm span is $\hat{y}=11.74+0.8247 x=11.74+0.8247(61)=62.05$ inches.

The scatterplot reveals a strong, positive, roughly linear association between the mass and length of bullfrogs. There are no points that seriously deviate from the straight-line pattern of the points in the plot.

Essentially correct (E) if the response provides
a description that includes at least three of
components 1-4 and component 5:
1. Direction of association (positive or
increasing)
3. Form of association (linear or approximately
linear)
4. Unusual features (no points with large
discrepancies from the pattern (straight line) exhibited by most of the points on the plot)
5. Context (association between length and mass of bullfrogs)
Partially correct (P) if the response satisfies only one or two components out of components 1-4 and component 5
OR
if the response satisfies at least three out of
components 1-4 but does not satisfy
component 5.
Incorrect (I) if the response does not meet the
criteria for E or P.

Additional Notes:

- To satisfy component 4, it is sufficient to simply indicate that there are no unusual features.

- To satisfy component 5, it is minimally sufficient for the response to refer to the association or relationship between mass and length without explicitly mentioning bullfrogs.

- The strength of the response in part (a) may be considered if holistic scoring is needed.

Model Solution

Scoring

The value of the slope of the least-squares regression line is 6.086. This value indicates that the predicted mass of a bullfrog increases by 6.086 grams for each additional millimeter of length.

Essentially correct (E) if the response satisfies the following three components:
2. Provides an interpretation that references an
increase of a number of grams of mass for each one-millimeter increase in length
3. Indicates that the slope represents a change in
a prediction using non-deterministic
language such as "predicted," "estimated,"
"expected," or "average"
Partially correct (P) if the response satisfies only two of the three components.
Incorrect (I) if the response does not meet the
criteria for E or P.

Additional Notes:

- The value of the slope, 6.086, may be rounded to 6.09 or 6.1, but not to 6, to satisfy the numerical requirement in component 1.

- A response that only contains 6.086 in the interpretation satisfies component 1.

- A calculation of slope may satisfy component 1, provided that two points from the line are used in the calculation.

- Units of measurements must be correctly specified for both mass and length to satisfy component 2.

- It is not required to refer specifically to the "least-squares regression line."

Model Solution

Scoring

The coefficient of determination is $r^{2} \approx 0.819$.
This value indicates that 81.9\% of the variation
in bullfrog mass can be explained by variation in bullfrog length as described by the least-squares
line.

Essentially correct (E) if the response provides a correct interpretation of $r^{2}$ in context.

Partially correct (P) if the response provides a generic interpretation (no context) OR if the response provides a reasonable but incorrect interpretation of $r^{2}$ in context.

Incorrect (I) if the response does not satisfy the criteria for E or P.

Additional Notes:

- Correct interpretations of $r^{2}$ include the concept that part of the variation in the response (dependent or y ) variable is explained by the linear relationship with the explanatory (independent or x ) variable. The response can take any of several equivalent forms, such as:

The proportion of the total variability in the dependent (response) variable y that is explained by the independent (explanatory) variable x.

The proportion of variation in y that is accounted for by the linear model.

The proportionate reduction of the total variation of the y-values that is associated with the use of the independent variable x.

The proportionate reduction in the sum of the squares of vertical deviations obtained by using the least-squares line instead of the sample mean to predict values of y.

- Correct interpretation of $r^{2}$ must explicitly relate to the dependent variable. Mention of the data, predicted values, or no mention of the dependent variable are incorrect interpretations. Common incorrect interpretations include:

The percent (or proportion or part of the total) variability in the predicted y-values that is explained by the linear relationship between y and x.

The percent (or proportion or part of the total) variability in the data that is explained by the linear relationship between y and x.

The percent (or proportion or part of the total) variability that is explained by the linear relationship between y and x.

The percent (or proportion or part of the total) variability in y that is on average explained by the linear relationship between y and x.

- A reasonable but incorrect interpretation of $r^{2}$ with context might include the following responses:

81.9\% of the variation in mass and length can be accounted for by the least-squares regression line.

81.9\% of the variability in predicted mass is accounted for by the length.

- For context, the response variable (y) must be identified as mass, and the explanatory variable (x) must be identified as length.

- An interpretation of the correlation between mass and length, $r=\sqrt{0.819}=0.905$, is not considered a reasonable interpretation of $r^{2}$.

- The value of the percentage (81.9\%) or proportion (0.819) of variation does not need to be specified, but if an incorrect value is specified, the score is lowered by one level, from E to P or from P to I.

- The strength of the response in part (c) may be considered if holistic scoring is needed.

Model Solution

Scoring