Question

Measuring Seals from Photos Listed below are the overhead widths (in cm) of seals measured from photographs and the weights of the seals (in kg). The data are based on “Mass Estimation of Weddell Seals Using Techniques of Photogrammetry,” by R. Garrott of Montana State University. The purpose of the study was to determine if weights of seals could be determined from overhead photographs. Is there sufficient evidence to conclude that there is a correlation between overhead widths and the weights of the seals?

Overhead width (cm)	7.2	7.4	9.8	9.4	8.8	8.4
Weight (kg)	116	154	245	202	200	191

Short answer

The rank correlation coefficient between overhead width (cm) and the weight of the seals (kg) is equal to 1.

There is enough evidence to conclude that there is a significant correlation between overhead width (cm) and the weights of the seals (kg).

Step-by-step solution

Given information

Data are provided on the samples of overhead widths (cm) and the weights of seals (kg).

The significance level is 0.05.

Determine the rank correlation test and identify the statistical hypothesis

Rank correlation coefficient is used to test the significance of the correlation between the two ordinal variables.

The null hypothesis is set up as follows:

There is no correlation between overhead widths and the weights of seals.

\({\rho _s} = 0\)

The alternative hypothesis is set up as follows:

There is a significant correlation between overhead widths and the weight of the seals.

\({\rho _s} \ne 0\)

The test is two tailed.

Assign ranks

Here, the data is not given in the form of ranks; so first, determine the ranks of each of the two samples.

Ranks are assigned from the smallest to the largest observations for both samples.

In case some of the values are the same in the sample, the mean of the ranks is assigned to each of the observations.

The following table shows the ranks of the two samples:

Ranks of overhead widths	1	2	6	5	4	3
Ranks of weights	1	2	6	5	4	3

Calculate the Spearman rank correlation coefficient

Since there are no ties present, the following formula is used to compute the rank correlation coefficient:

\({r_s} = 1 - \frac{{6\sum {{d^2}} }}{{n\left( {{n^2} - 1} \right)}}\)

The following table shows the differences (d) between the ranks of two values for every pair:

Ranks of overhead widths	1	2	6	5	4	3
Ranks of weights	1	2	6	5	4	3
d	0	0	0	0	0	0
\({d^2}\)	0	0	0	0	0	0

Substituting the differences in the formula, the value of\({r_s}\)is obtained as follows:

\(\begin{array}{c}{r_s} = 1 - \frac{{6\sum {{d^2}} }}{{n\left( {{n^2} - 1} \right)}}\\ = 1 - \frac{{6\left( 0 \right)}}{{6\left( {{6^2} - 1} \right)}}\\ = 1\end{array}\)

Therefore, the value of the Spearman rank correlation coefficient is equal to 1.

Determine the critical value and the conclusion of the test

The critical values of the rank correlation coefficient for n=6 and\(\alpha = 0.05\)are -0.886 and 0.886.

Since the value of the rank correlation coefficient does not fallin the interval bounded by the critical values, the null hypothesis is rejected.

There is enough evidence to conclude that there is a correlation between the overhead width and the weights of the seal.