AP Statistics 5.5: Least-Squares Regression

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