Changing the null hypothesis in a t-test

Question

Hello.

I am working on a problem and am confused.

What I know so far is that the two-sided t-test with $\alpha=.05$ will reject the null since the $p-$value is smaller.

Here is where I am lost. The question asks me what to do when the null hypothesis is

$$H_0: \mu_1 - \mu_2 = 2$$

Intuitively

$$\bar{Y_1} - \bar{Y_2} = -2$$

so I instantly want to reject the null. However, I have never seen a situation where the difference between the means are not equal to 0.

How would one approach this problem?

Answer 1

Define $Y_3:=Y_2-2$ so the null hypothesis is $\mu_1=\mu_3,\,\mu_3:=\bar{Y}_3$. Note that subtracting $2$ from one variable shifts its mean while leaving the standard deviation and mean's standard error unchanged. Now you can use an equal-means test.