AP Statistics 5 Regression Analysis Questions

4 Complete Response
Three parts essentially correct

3 Substantial Response
Two parts essentially correct and one part partially correct

2 Developing Response
Two parts essentially correct and no parts partially correct
OR
One part essentially correct and one or two parts partially correct
OR
Three parts partially correct

1 Minimal Response
One part essentially correct
OR
No parts essentially correct and two parts partially correct

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

Scoring

Parts (a), (b), and (c) are scored as essentially correct (E), partially correct (P), or incorrect (I).
Part (a) is scored as follows:
Essentially correct (E) if the response includes the following four components:
1. The relationship is approximately linear.
2. The relationship is positive.
3. There is a moderate to strong association (or relationship).
4. The response is given in context.

Partially correct (P) if the response includes only two or three of the four components.
Incorrect (I) if the response includes at most one of the four components.

Notes:
- Correlation alone is not sufficient to indicate a linear relationship. However, linear correlation is sufficient to indicate a linear relationship.
- Reporting that the correlation coefficient is 0.81, the correlation coefficient is close to 1.0, or some other value or range of values for the correlation coefficient is not sufficient to satisfy component 3.

Part (b) is scored as follows:
Essentially correct (E) if the response contains the following four components:
1. The line in Graph 2 is selected.
2. A reasonable explanation for selecting the line in Graph 2 that links the body shapes to the regions defined by the line is given. The explanation should indicate that square shapes (arm span is equal to height) are represented by points on the line, tall rectangle shapes (arm span less than height) are represented by points below the line, and short rectangle shapes (arm span is greater than height) are represented by points above the line.
3. Correct counting of body types demonstrated by reporting correct frequencies (3, 4, and 5) or reporting correct proportions $\left(\frac{1}{4}=\frac{3}{12}, \frac{1}{3}=\frac{4}{12}\right.$, and $\left.\frac{5}{12}\right)$ for the square, tall rectangle, and short rectangle body types, respectively.
4. The correct frequencies (3, 4, and 5) are reported in the table.

Partially correct (P) if the response includes only two or three of the four components.
Incorrect (I) if the response includes at most one of the four components.
Notes:
- Selecting the regression line on Graph 1 cannot satisfy either component 1 or component 2.
- To satisfy component 1 , it is sufficient to refer to the y=x line without explicitly mentioning Graph 2.
- Use of incorrect labels, such as regression line or least-squares regression line, in referring to the y=x line in Graph 2 is an incorrect use of terminology that should be strongly discouraged, but it is ignored in this scoring rubric.
- The explanation for selecting Graph 2 is acceptable if it explicitly includes at least two of the following: (i) square body shapes (arm span equal to height) are represented by points on the line, (ii) tall rectangle body shapes (arm span less than height) are represented by points below the line, or (iii) short rectangle body shapes (arm span greater than height) are represented by points above the line.
- Incorrectly counting the points on the y=x line in Graph 2 is considered a minor error. Frequencies (2, 4, and 5) are accepted for both component 3 and component 4. Reporting corresponding proportions $\left(\frac{2}{11}, \frac{4}{11}, \frac{5}{11}\right)$ or $\left(\frac{2}{12}, \frac{4}{12}, \frac{5}{12}\right)$ in the table satisfies component 3 but does not satisfy component 4.
- Frequencies reported in the table as (2, 5, 4) or (3, 5, 4), interchanging the counts for tall and short rectangle body shapes, satisfies component 3, but does not satisfy component 4. Reporting corresponding proportions, $\left(\frac{2}{11}, \frac{5}{11}, \frac{4}{11}\right)$ or $\left(\frac{3}{12}, \frac{5}{12}, \frac{4}{12}\right)$, in the table does not satisfy either component 3 or component 4.

Part (c) is scored as follows:
Essentially correct (E) if the response contains the following three components:
1. Correct formula for predicting arm span with 61 inserted for x.
2. Correct value for the predicted arm span.
3. Units for the predicted arm span given as inches.

Partially correct (P) if the response includes only two of the three components.
Incorrect (I) if the response includes at most one of the three components.
Notes:
- Any value for the predicted arm span between 61.5 and 62.5 is acceptable; values outside the interval do not satisfy component 2. This allows for inaccuracy in obtaining a value for the predicted arm span directly from the line displayed on Graph 1 and for rounding in applying the prediction formula; but it excludes predictions based on the y=x line and other unreasonable predictions.
- Reporting a prediction equation that is similar to the equation given in the stem, with 61 inserted for x, satisfies components 1 and 2 if the value of the predicted arm span is between 61.5 and 62.5 inches. Otherwise, neither of those two components is satisfied. This could occur if a student enters the data from the graph into a calculator to compute the least squares regression line, instead of using the formula provided in the stem.
4 Complete Response
All three parts essentially correct
3 Substantial Response
Two parts essentially correct and one part partially correct
2 Developing Response
Two parts essentially correct and one part incorrect
OR
One part essentially correct and two parts partially correct
OR
One part essentially correct, one part partially correct, and one part incorrect
OR
Three parts partially correct
1 Minimal Response
One part essentially correct and two parts incorrect
OR
Two parts partially correct and one part incorrect