Question

Crickets and Temperature The association between the temperature and the number of times a cricket chirps in 1 min was studied. Listed below are the numbers of chirps in 1 min and the corresponding temperatures in degrees Fahrenheit (based on data from The Song of Insects by George W. Pierce, Harvard University Press). Is there sufficient evidence to conclude that there is a relationship between the number of chirps in 1 min and the temperature?

Chirps in 1 min	882	1188	1104	864	1200	1032	960	900
Temperature\(\left( {^ \circ F} \right)\)	69.7	93.3	84.3	76.3	88.6	82.6	71.6	79.6

Short answer

The rank correlation coefficient between the number of chirps in one minute and temperature is equal to 1.

There is enough evidence to conclude that there is a significant correlation between the number of chirps in one minute and temperature.

Step-by-step solution

Given information

Data are provided on the samples of the number of chirps in one minute and temperature.

The significance level is 0.05.

The sample size (n) is 8.

Identify the statistical hypothesis

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

The null hypothesis is set up as follows:

There is no correlation between the number of chirps in one minute and temperature.

\({\rho _s} = 0\)

The alternative hypothesis is set up as follows:

There is a correlation between the number of chirps in one minute and temperature.

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

The test is two tailed.

Assign ranks

Compute the ranks of each of the two samples.

For the first sample, assign rank 1 for the smallest observation, rank 2 to the next smallest observation, and so on until the largest observation.If some observations are equal, the mean of the ranks isassigned to each of the observations.

In a similar pattern, assign ranks to the second sample.

The following table shows the ranks of the two samples:

Ranks of chirps in 1min	2	7	6	1	8	5	4	3
Ranks of temperature	1	8	6	3	7	5	2	4

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 between the ranks of the two values for every pair:

Ranks of chirps in 1min	2	7	6	1	8	5	4	3
Ranks of temperature	1	8	6	3	7	5	2	4
d	1	-1	0	-2	1	0	2	-1
\({d^2}\)	1	1	0	4	1	0	4	1

Here,

\(\begin{array}{c}\sum {{d^2}} = 1 + 1 + .... + 1\\ = 12\end{array}\)

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

\(\begin{array}{c}{r_s} = 1 - \frac{{6\sum {{d^2}} }}{{n\left( {{n^2} - 1} \right)}}\\ = 1 - \frac{{6\left( {12} \right)}}{{8\left( {{8^2} - 1} \right)}}\\ = 0.857\end{array}\)

Therefore, the value of the Spearman rank correlation coefficient is equal to 0.857.

Determine the critical value and the conclusion of the test

The critical values of the rank correlation coefficient for n=8 and\(\alpha = 0.05\)are -0.738 and 0.738.

Since the value of the rank correlation coefficient does not fall in 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 number of chirps in one minute and temperature.