Here is one way in which nature regulates the size of animal populations: high population density attracts predators, which remove a higher proportion of the population than when the density of the prey is low. One study looked at kelp perch and their common predator, the kelp bass. On each of four occasions, the researcher set up four large circular pens on sandy ocean bottoms off the coast of southern California. He randomly assigned young perch to 1 of 4 pens so that one pen had 10 perch, one pen had 20 perch, one pen had 40 perch, and the final pen had 60 perch. Then he dropped the nets protecting the pens, allowing bass to swarm in, and counted the number of perch killed after two hours. A regression analysis was performed on the 16 data points using x = number of perch in pen and y = proportion of perch killed. Given is computer output from the least-squares regression analysis of the perch data.
Predictor Coef Stdev. t-ratio p
Constant 0.12049 0.09269 1.30 0.215
Perch 0.008569 0.002456 3.49 0.004
S 0.1886 R-Sq 46.5% R-Sq (adj) = 42.78
Give the standard error of the slope SE. Interpret this value.
a. The standard error of the slope, SE, 0.002456. The slope of this regression line is 0.002456 from the slope of the true regression line for predicting proportion of perch killed from number of perch.
b. The standard error of the slope, SE, = 0.002456. If we repeated the random assignment many times, the slope of the sample regression line would typically vary by about 0.002456 from the slope of the true regression line for predicting proportion of perch killed from number of perch.
c. The standard error of the slope, SE, = 0.002456. If we repeated the random assignment many times, the number of perch will typically vary by about 0.0024 from the predicted slope of 0.002456.
d. The standard error of the slope, SE, 0.09269. If we repeated the random assignment many times, the slope of the sample regression line would typically vary by about 0.09269 from the slope of the true regression line for predicting proportion of perch killed from number of perch.
e. The standard error of the slope, SE, = 0.09269. If we repeated the random assignment many times, the number of perch will typically vary by about 0.0024 from the predicted slope of 0.09269.