AP Statistics 5.5: LSRL Coefficients

Calculate or identify the least-squares regression equation and use r and r² to quantify linear association and explained variation.

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.

Figure for Question 5.5.A—Calculate the coefficients for the least-squares regression line model question 1 — AP Statistics

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

Interpret the coefficient of determination of the least-squares regression line, $r^{2} \approx 0.819$ , in context.

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

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5.5.A—Calculate the coefficients for the least-squares regression line model question 1