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=0.992 Using alpha=​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.

​No, because the absolute value of the test statistic exceeds the critical value

of 0.707of 0.707.

B.

​Yes, because the absolute value of the test statistic exceeds the critical value

of 0.707of 0.707.

C.

​Yes, because the test statistic falls between the critical values of

negative 0.707−0.707

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?

​(Round to three decimal places as​ needed.)

Answer 1

a. The sample size is n=8, so then the number of degrees of freedom is df=n−2=8−2=6

The corresponding critical correlation value rc​ for a significance level of α=0.05, for a two-tailed test is:

rc​=0.707

Observe that in this case, the null hypothesis is rejected if ∣r∣>rc​=0.707.

Here r=0.992

So answer here is

B.​Yes, because the absolute value of the test statistic exceeds the critical value of 0.707.

b. Here r=0.992, so r^2=0.992^2=0.984

So 98.4% of variation in weight can be explained by the linear relationship between weight and chest​ size.