Question

The American black bear is the smallest North American bear and the most common bear species on the planet. In 1969, Dr. Michael R. Pelton initiated a long-term study of the population of bears in the Great Smoky Mountains National Park. One aspect of the study was to develop a model that could be used to predict a bear’s weight from their length. Through the data he found that the mean bear length was 143.48 cm (standard deviation 143 cm) and the mean weight was 99.68 kg (standard deviation = 28 kg). The correlation between length and weight was .74.

a) Find the equation for the regression line for predicting weight from length.

b) Interpret slope and y-intercept.

c) Suppose a 162 cm bear is captured in the field, use the regression equation to predict the weight of the bear.

d) The researchers were able to weigh the bear and his weight turned out to be 120 kg. Interpret the residual.

Answer 1

x = length, y = weight

x̅ = 143.48

y̅ = 99.68

sx = 14.3

sy = 28

r = 0.74

a) Slope, b = r*sy/sx = .74*28/14.3 = 1.448951

Intercept, a = y̅ - b* x̅ = -108.2155

Regression equation :  

ŷ =-108.2155 + (1.448951) x

b) Slope: with a unit increase in length the weight of the bear increases by 1.448951 units.

Y-intercept. It is value of y at x = 0. in this case it is not reasonable as weight cannot be negative.

c) Predicted value of y at x = 162

ŷ =-108.2155 + (1.448951) *162=126.5146

d) Residual = y - ŷ = 120 - 126.5146 = -6.5146

The original value is less than the predicted value.