I have a problem that I am struggling with:
Suppose $X$ and $Y$ are independent normally distributed random variables with respective unknown means $\mu _x$ and $\mu _y$ and respective unknown variances $\sigma^2 _x$ and $\sigma^2 _y$. Random samples of sizes $n_x$ for $X$ and $n_y$ for $Y$ are taken, and the following hypotheses are tested: $H_0:\mu _x = \mu _y$, $H_1:\mu _x \neq \mu _y$. The test used consisted of a critical region $C = \{(x_1, ..., x_{n_x}, y_1, ..., y_{n_y}): 0 \notin I\}$, where $I$ is the two-sided $90$% confidence interval for $\mu _x - \mu _y$. Determine the significance level, $\alpha$, of this test.
All I know is that
$$C = \{(x_1, ..., x_{n_x}, y_1, ..., y_{n_y}): 0 \notin I\}: |X-Y| \geq t_{\alpha/2}(n_x + n_y - 2) \cdot \sqrt{\frac{(n_x - 1)S_x^2 + (n_x - 1)S_x^2}{n_x + n_y - 2}} \cdot \sqrt{\frac{1}{n_x} + \frac{1}{n_y}}$$
It doesn't seem like there is enough information and so I am unsure how to use this information to find $\alpha$. I would appreciate any help I can get. Thank.