Suppose I have the following data: $$\bar x_1=15$$ $$\bar x_2=12.5$$ $$\sigma_1=1.8$$ $$\sigma_2=1.75$$ $$n_1=800$$ $$n_2=3000$$ And I want to test: $$H_0:\mu_1-\mu_2>0$$ $$H_a:\mu_1-\mu_2<0$$
Lets say at a 0.05 level. So I find SE: $$SE^2=\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}$$ This gives SE = 0.07 So I calculate my t stat: $$t=\frac{\bar x_1-\bar x_2}{SE}=35$$ So my p-value is 0. This means I reject my null hypothesis, as the corresponding p value ~0.
I want to make sure I am doing this right, because with these numbers, I would expect that we would fail to reject the null because x1 will be much greater than x2. Is my analysis of this correct?
Your calculations are correct. Consider that the high level of significance in this case mostly depends on the relatively large sizes of the two groups.