554. From two independent samples, with sizes $n_{1}=11$ and $n_{2}=14$, extracted from normal general populations $X$ and $Y$, the corrected sample variances $s_{X}^{2}=0.76$ and $s_{Y}^{2}=0.38$ were found. At the significance level $\alpha=0.05$, test the null hypothesis $H_{0}: D(X)=$ $=D(Y)$ about the equality of the population variances, against the alternative hypothesis $H_{1}: D(X)>D(Y)$.
Solution. Let's find the ratio of the larger corrected variance to the smaller one:

$$
F_{\text {obs }}=0.76 / 0.38=2
$$

According to the condition, the alternative hypothesis is $D(X)>D(Y)$, so the critical region is right-tailed.

From the table in Appendix 7, at the significance level $\alpha=0.05$ and degrees of freedom $k_{1}=n_{1}-1=11-1=10$ and $k_{2}=n_{2}-1=14-1=13$, we find the critical point

$$
F_{\mathbf{cp}}(0.05 ; 10 ; 13)=2.67 .
$$

Since $F_{\text {obs }}<F_{\text {crit }}$ - there is no reason to reject the hypothesis of equality of the population variances. In other words, the sample corrected variances do not differ significantly.