Say, I have two normally distributed random variables: $X \sim \mathcal N(\mu_x, \sigma_x^2), Y \sim \mathcal N(\mu_y, \sigma_y^2)$.

I want to test if $\mu_x = \mu_y$. Further, I assume that these variables are independent of each other. Then, I believe the test statistic would be:

${\mu_x - \mu_y}\over{\sqrt{\sigma_x^2 + \sigma_y^2)}}$

Is my intuition correct?

Say, I have two normally distributed random variables: $X \sim \mathcal N(\mu_x, \sigma_x^2), Y \sim \mathcal N(\mu_y, \sigma_y^2)$.

I want to test if $\mu_x = \mu_y$. Further, I assume that these variables are independent of each other. Then, I believe the test statistic would be:

${\mu_x - \mu_y}\over{\sqrt{\sigma_x^2 + \sigma_y^2)}}$

EDIT:

I want to be a bit more specific. In a journal article they test if two regression coefficients from two DIFFERENT regressions are equal.

So, I want to test if two estimated OLS coefficients from DIFFERENT regression models (the sample size is identical) are different from each other. I assume that the coefficients are independent.

Say, I have two normally distributed random variables: X ~ N(mu_x, sigma_x^2), Y ~ N(mu_y, sigma_y^2)

I want to test if mu_x = mu_y. Further, I assume that these variables are idependent of each other. Then, I believe the test statistic would be:

mu_x - mu_y / sqrt(sigma_x^2 + sigma_y^2)

Is my intutition correct?

Say, I have two normally distributed random variables: $X \sim \mathcal N(\mu_x, \sigma_x^2), Y \sim \mathcal N(\mu_y, \sigma_y^2)$.

I want to test if $\mu_x = \mu_y$. Further, I assume that these variables are independent of each other. Then, I believe the test statistic would be:

${\mu_x - \mu_y}\over{\sqrt{\sigma_x^2 + \sigma_y^2)}}$

Is my intuition correct?