Question

The paper "Postmortem Changes in Strength of Gastropod Shells" (Paleobiology [1992]: \(367-377\) ) included scatterplots of data on \(x=\) shell height (in centimeters) and \(y=\) breaking strength (in newtons) for a sample of \(n=38\) hermit crab shells. The least-squares line was \(\hat{y}=-275.1+244.9 x\) a. What are the slope and the intercept of this line? b. When shell height increases by \(1 \mathrm{~cm}\), by how much does breaking strength tend to change? c. What breaking strength would you predict when shell height is \(2 \mathrm{~cm} ?\) d. Does this approximate linear relationship appear to hold for shell heights as small as \(1 \mathrm{~cm} ?\) Explain.

Short answer

a) The slope of the line is 244.9 and the intercept is -275.1. b) The breaking strength tends to increase by 244.9 newtons when the shell height increases by 1 cm. c) The breaking strength would be 214.7 newtons when shell height is 2 cm. d) The applicability of this model for smaller shell heights can only be determined with data on shells of such sizes.

Step-by-step solution

Identify the slope and intercept

The given least-squares line equation is \(\hat{y} = -275.1 + 244.9x\). In this equation, -275.1 is the intercept and 244.9 is the slope of the line.

Interpret the slope

The slope indicates the change in the dependent variable (\(y\)) for a one unit increase in the independent variable (\(x\)). In this case, for each 1 cm increase in shell height, the breaking strength increases by 244.9 newtons.

Predict breaking strength for a specific shell height

To predict the breaking strength when the shell height is 2 cm, we can substitute \(x = 2\) in the equation to get: \(\hat{y} = -275.1 + 244.9 * 2 = 214.7\) newtons.

Evaluate the applicability of the line for smaller shell heights

The approximation holds if our model consistently predicts accurate strengths for smaller shell heights. However, this can't be determined without data on shells with smaller heights. If the data indicates a linear relationship between shell height and breaking strength across all heights, then we can cautiously apply this model to smaller shells.