610. Based on a sample of size $n=100$, extracted from a two-dimensional normal population ( $X, Y$ ), the sample correlation coefficient $r_{\mathrm{B}}=0.2$ has been found. It is required to test the null hypothesis of the population correlation coefficient being zero against the alternative hypothesis $H_{1}: r_{\mathrm{r}} \neq 0$ at a significance level of 0.05.
Solution. Let's find the observed (empirical) value of the criterion:

$$
T_{\text {na } 6 \pi}=r_{\mathrm{B}} \sqrt{n-2} / \sqrt{1-r_{\mathrm{B}}^{2}}=0.2 \sqrt{100-2} / \sqrt{1-0.2^{\mathrm{z}}}=2.02
$$

According to the condition, the alternative hypothesis is $r_{\mathbf{r}} \neq 0$, so the critical region is two-tailed.

From the table of critical points of the Student's distribution (see Appendix 6), at the significance level $\alpha=0.05$, placed in the top row of the table, and the number of degrees of freedom $k=n-2=100-2=98$, we find the critical point of the two-tailed critical region $t_{\text {cr }}(0.05 ; 98)=1.99$.

Since $T_{\text {publ }}>t_{\text {cr }}$-we reject the null hypothesis of the equality to zero of the population correlation coefficient. In other words, the correlation coefficient significantly differs from zero; therefore, $X$ and $Y$ are correlated.