Derive a formula for a $(1-\alpha)100\%$ C.I. for $\mu_x -\mu_y $ for data that has the following properties:
A random sample $X_1,X_2...X_n \ are \ i.i.d ~N(\mu_x, \sigma^2 ) $
Another random ] sample $Y_1,Y_2...Y_n \ are \ i.i.d ~N(\mu_y, 2\sigma^2 ) $
Two random samples are independent and $\sigma$ unknown
How might I approach this? I believe I need to use
$$\bar{x}-\bar{y} \pm t_{\alpha/2}(df)\sqrt{\frac{s^2_p}{n}+ \frac{s^2_p}{m}} $$
But I'm not sure how to go about finding $ s^2_p $ and $ df $
I know normally $$ s^2_p = \frac{(n-1)s^2_x+(m-1)s^2_y}{n+m-2} $$
And $ df= n + m -2 $
But that can't be the case here, I don't think. How could I approach this? I've been beating my head into the wall the whole evening over this. Thanks!