Let $X_1,\ldots ,X_N,i=1,\ldots,N$ i.i.d. $\mathcal{N}(\mu_X,\sigma_1^2)$ distributed random variables and $Y_1,\ldots,Y_N,i=1,\ldots,N$ i.i.d. $\mathcal{N}(\mu_Y,\sigma_2^2)$ distributed random variables. Furthermore, let $\mathbf{X}=(X_1,\ldots,X_N)$ be independent from $\mathbf{Y}=(Y_1,\ldots,Y_N)$.
Find a confidence interval of minimal length for $\mu_X-\mu_Y$ to the level $1-\alpha$.
My approach: It seems that the statistical model to be considered here is the product model with family of probability measures $\mathcal{N}(\mu_X,\sigma_1^2)^{\otimes N}\otimes\mathcal{N}(\mu_Y,\sigma_2^2)^{\otimes N}$ and that the confidence interval is to be constructed wrt. this family of measures. I'm not sure how to continue from here and the case of ratios of two means seems to be quite different.
How do I continue?
EDIT: By one of the comments below, I have now done the following: Observe that $\bar{X}-\bar{Y} \sim \mathcal{N}(\mu_X-\mu_Y,\sigma^2)$, with $\sigma^2$ having some ancillary and unknown (but expressible in terms of $\sigma_1^2$ and $\sigma_2^2$) value. This means that we are now dealing with the problem of finding a confidence interval for the normal distribution in the case of unknown mean and unknown variance and thus
$C(X_1,\ldots,X_n)=(\bar{X} -t^*\sqrt{s^2/N},\bar{X} +t^*\sqrt{s^2/N})$,
with $t^*$ the $1-\alpha/2$ quantile of the $t_{N-1}$ distribution and $s^2$ the unbiased sample variance. I would very much appreciate any feedback regarding the correctness of this.
