Question

For a sample of eight​ bears, researchers measured the distances around the​ bears' chests and weighed the bears. Minitab was used to find that the value of the linear correlation coefficient is r=0.931 Using alphaα=​0.05, determine if there is a linear correlation between chest size and weight. What proportion of the variation in weight can be explained by the linear relationship between weight and chest​ size?

a. Is there a linear correlation between chest size and​ weight?

A.​Yes, because the absolute value of r exceeds the critical value of 0.707

B.​No, because the absolute value of r exceeds the critical value of 0.707.

C.​Yes, because r falls between the critical values of −0.707 and 0.707.

D. The answer cannot be determined from the given information.

b. What proportion of the variation in weight can be explained by the linear relationship between weight and chest​ size?

Answer 1

Solution:

Given:

Sample size =n = 8

the linear correlation coefficient is r = 0.931

α=​0.05

We have to test if there is a linear correlation between chest size and weight.

Part a. Is there a linear correlation between chest size and​weight?

Vs

Find r critical value:

df = n - 2 =8 -2 = 6

Two tailed area = α=​0.05

r critical value = 0.707

Since r calculated = 0.931 > r critical value = 0.707, we reject null hypothesis H0.

Thus  there is a linear correlation between chest size and​weight.

Thus correct option is:
A.​Yes, because the absolute value of r exceeds the critical value of 0.707

Part b) What proportion of the variation in weight can be explained by the linear relationship between weight and chest​ size?

Coefficient of determination = r² gives the proportion of the variation in dependent variable by the regression equation.

Thus

Coefficient of determination = r² = 0.931²

Coefficient of determination = r² = 0.866761

Coefficient of determination = r² = 0.8668

Coefficient of determination = r² = 86.68%

Thus 86.68% or 0.8668 proportion of the variation in weight can be explained by the linear relationship between weight and chest​ size