I have $n_X$ observations of variable X, $n_Y$ of variable Y, and $n_Z$ of variable Z. I'd like to test the hypothesis that the true mean of $X$ is equal to the sum of the means of Y and Z. $$H_0: \mu_X - (\mu_Y+\mu_Z) = 0$$
Initial thoughts
I can use the sample means to define an estimator $\hat{\gamma} = \bar{y}_X - (\bar{y}_Y+\bar{y}_Z)$. Can I estimate the variance using $$\hat{\sigma}^2(\hat{\gamma}) = \sigma^2 (1/n_1+1/n_2+1/n_3)?$$ If so, is the test statistic $\hat{\gamma}/\sqrt{\hat{\sigma}^2(\hat{\gamma})}$ t-distributed?