617. In problem 540, for a sample size of $n=10$, the sample Spearman rank correlation coefficient $\rho_{\mathrm{e}}=0.64$ was calculated between the test scores of students in two tests. At a significance level of 0.01, test the null hypothesis that the population Spearman rank correlation coefficient is zero. In other words, it is required to check whether the rank correlation relationship between the test scores is significant.
Solution. We will find the critical point of the two-sided critical region of the Student's distribution at the significance level $\alpha=0.01$ and the number of degrees of freedom $k=n-2=10-2=8$ (see Appendix 6$): t_{\text {cr }}(0.01 ; 8)=3.36$.

We will find the critical point:

$$
T_{\mathrm{kp}}=t_{\mathrm{kp}}(\alpha ; k) \sqrt{\frac{1-\rho_{\mathrm{B}}^{2}}{n-2}}=3.36 \sqrt{\frac{1-0.64^{2}}{10-2}}=0.92
$$

Thus, $T_{\text {cr }}=0.92, \rho_{\mathrm{B}}=0.64$. Since $\rho_{\mathrm{B}}<T_{\text {cr }}$ - there is no reason to reject the null hypothesis of the equality to zero of the population coefficient of rank correlation of Spearman. In other words, the rank correlation relationship between the scores of the two tests is not significant.