Critical Values for the Correlation Coefficient
n alpha = .05 alpha = .01
4 0.95 0.99
5 0.878 0.959
6 0.811 0.917
7 0.754 0.875
8 0.707 0.834
9 0.666 0.798
10 0.632 0.765
11 0.602 0.735
12 0.576 0.708
13 0.553 0.684
14 0.532 0.661
15 0.514 0.641
16 0.497 0.623
17 0.482 0.606
18 0.468 0.59
19 0.456 0.575
20 0.444 0.561
25 0.396 0.505
30 0.361 0.463
35 0.335 0.43
40 0.312 0.402
45 0.294 0.378
50 0.279 0.361
60 0.254 0.33
70 0.236 0.305
80 0.22 0.286
90 0.207 0.269
100 0.196 0.256
Note: To test H subscript 0: rho equals 0 against H subscript 1: rho not equal to 0, reject H subscript 0 if the absolute value of r is greater than the critical value in the table.
For a sample of eight bears, researchers measured the distances around the bear’s chests and weighed the bears. Minitab was used to find that the value of the linear correlation coefficient is r = 0.894. Using a= 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?
(b) What proportion of the variation in weight can be explained by the linear relationship between weight and chest size? (round to three decimals places as needed)
.707
To determine if there is a linear correlation between chest size and weight, we need to compare the absolute value of the correlation coefficient to the critical value for a given significance level (alpha).
In this case, the significance level (alpha) is 0.05.
Looking at the table given, for a sample size of 8, the critical value for alpha = 0.05 is 0.707.
The given correlation coefficient (r) is 0.894, which is greater than the critical value of 0.707.
Since the absolute value of r (0.894) is greater than the critical value (0.707), we reject the null hypothesis (H0) that there is no linear correlation between chest size and weight. This means there is evidence of a linear correlation between the two variables.
To determine the proportion of the variation in weight that can be explained by the linear relationship with chest size, we need to calculate the coefficient of determination (r^2).
The coefficient of determination (r^2) represents the proportion of the total variation in the dependent variable (weight) that can be explained by the independent variable (chest size).
In this case, r^2 = 0.894^2 = 0.799.
Therefore, approximately 79.9% (0.799) of the variation in weight can be explained by the linear relationship between weight and chest size.