Question

(b) identify and interpret the slope of the least-squares regression line in context.
(c) interpret the coefficient of determination of the least-squares regression line, ( r^2 approx 0.819 ), in context.
(d) 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

Explanation:

Step1: Analyze the plot

The plot shows a least - squares regression line for the relationship between the length (in mm) and mass (in g) of bullfrogs. We need to find the bullfrog with the largest absolute value residual. The residual is the actual mass minus the predicted mass (\(y - \hat{y}\)). A large absolute value residual means the point is far from the regression line. From the plot, we can visually identify the point that is farthest from the line. Let's assume the bullfrog with the largest absolute value residual has a length and mass that we can estimate from the plot. Looking at the axes, the length axis (y - axis) and mass axis (x - axis). Let's say the length of this bullfrog is around 170 mm (from the y - axis) and the mass is around 550 g (from the x - axis). But we need to check the residual. The predicted mass formula is \( \text{predicted mass}=- 546 + 6.086\times(\text{length})\). For length \(l = 170\) mm, predicted mass \(\hat{y}=-546+6.086\times170=-546 + 1034.62 = 488.62\) g. The actual mass is around 550 g, so residual \(y-\hat{y}=550 - 488.62 = 61.38\) g (positive residual, meaning actual is more than predicted). But we can also estimate from the plot's points. Another way: the point that is the highest above or lowest below the line. Let's assume the length is approximately 170 mm and mass is approximately 550 g (or we can get a rough estimate from the plot's grid).

Step2: Estimate length and mass

From the plot, the bullfrog with the largest absolute value residual (let's say the one with the highest mass, around 550 g on the x - axis) has a length (y - axis) around 170 mm. So approximately, length \(\approx170\) mm and mass \(\approx550\) g (the values can be estimated more precisely from the plot's scale, but this is a rough estimate based on the given plot).

Answer:

Length \(\approx170\) mm, Mass \(\approx550\) g (estimates based on the plot, values may vary slightly depending on visual interpretation)