556. For two independent samples, the sizes of which are $n_{1}=14$ and $n_{2}=10$, extracted from normal general populations $X$ and $Y$, the corrected sample variances are $s_{X}^{2}=0.84$ and $s_{Y}^{2}=2.52$. At the significance level $\alpha=0.1$, test the null hypothesis $H_{0}: D(X)=$ $=D(Y)$ about the equality of the general variances against the alternative hypothesis $H_{1}: D(X) \neq D(Y)$.
Solution. Let's find the ratio of the larger corrected variance to the smaller one:

$$
F_{\text {obs }}=2.52 / 0.84=3 .
$$

According to the condition, the alternative hypothesis is $D(X) \neq D(Y)$, so the critical region is two-tailed. According to rule 2, when finding the critical point, the significance level should be half of the given one.

From the table in Appendix 7, at the significance level $\alpha / 2=0.1 / 2=$ $=0.05$ and degrees of freedom $k_{1}=10-1=9$ and $k_{2}=14-1=13$, we find the critical point

$$
F_{\kappa p}(0.05 ; 9 ; 13)=2.72
$$

Since $F_{\text {obs }}>F_{\text {crit }}$ - we reject the null hypothesis of equal population variances.