Let $X_{1},X_{2},\cdots,X_{n_{1}}$ and $Y_{1},Y_{2},\cdots,Y_{n_{2}}$ denote independent random sample from the normal distributions $N(\mu_{1},\sigma^2_{1})$ and $N(\mu_{2},\sigma_{2}^{2})$, respectively. The sample means and sample variances are denoted by $\bar{X}$, $\bar{Y}$, $S_{1}^{2}$, $S_{2}^{2}$, respectively.
Show that the distribution of the random quantity
$Q=\frac{\frac{S_{1}^{2}}{n_{1}}+\frac{S_{2}^{2}}{n_{2}}}{\frac{\sigma^2_{1}}{n_{1}}+\frac{\sigma^{2}_{2}}{n_{2}}}$
can be approximated by a chi-squared distribution, and obtain its corresponding effective degree of freedom $\delta$.