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Chapter 5: Problem 22

In the article "Reproductive Biology of the Aquatic Salamander Amphiuma tridactylum in Louisiana" (Journal of Herpetology [1999]: \(100-105\) ), 14 female salamanders were studied. Using regression, the researchers predicted \(y=\) clutch size (number of salamander eggs) from \(x=\) snout-vent length (in centimeters) as follows: $$ \hat{y}=-147+6.175 x $$ For the salamanders in the study, the range of snout-vent lengths was approximately 30 to \(70 \mathrm{~cm}\). a. What is the value of the \(y\) intercept of the least-squares line? What is the value of the slope of the least-squares line? Interpret the slope in the context of this problem. b. Would you be reluctant to predict the clutch size when snout-vent length is \(22 \mathrm{~cm}\) ? Explain.

Short Answer

Expert verified
The y-intercept is -147 and represents the predicted clutch size for a snout-vent length of 0 cm. The slope is 6.175, it means for every centimeter increase in the snout-vent length, the clutch size increases by 6.175. We must be hesitant to predict the clutch size for a snout-vent length of 22 cm as it is outside the observed snout-vent length range (30-70 cm). Extrapolation can lead to unreliable predictions.

Step by step solution

01

Interpret y-intercept of the Least-squares Line

The y-intercept of the equation is -147. This is the predicted clutch size when the snout-vent length, \(x\), is 0 cm.
02

Interpret Slope of the Least-squares Line

The slope of the equation is 6.175. This indicates that for each centimeter increase of the snout-vent length, the number of eggs in a clutch (clutch size) increases by 6.175.
03

Prediction Reluctance.

As the snout-vent length range from the study was 30 to 70 cm, one should be reluctant to predict the clutch size for a snout-vent length of 22 cm. Predictions outside the range of the original data are often unreliable. This principle is called extrapolation.

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