# Null Hypothesis in Paired t-test

Suppose that I was tasked to find if the mean difference between the current and starting salary of employees is greater than 15,000. How will my null and alternative hypotheses be?

Can it be: Ho: Current - Starting is less than or equal to 15,000. Ha: Current - Starting is greater than 15,000.

Will it be a right-tailed distribution? Also, does the order of difference matter? (current minus starting and starting minus current). I was told that paired t-tests should always have a null of equal to zero so I'm confused.

• The hypothesis are right as long as you apply the test to the difference ‘current income -starting income’ Commented Apr 24, 2023 at 5:09
• As writtn your null and alternate are not explicitly about population quantities (as opposed to sample quantities). Commented Apr 24, 2023 at 20:45

Mostly an expanded version of my comment. To

find if the mean difference between the current and starting salary of employees is greater than 15,000

you have to apply a simple $$t$$-test for $$H_0: \mu \leq 15k$$ against $$H_1:\mu>15k$$ to the variable

current income - starting income

This would be a test where the rejection region is single-tailed and situated on the right tail.

I was told that paired t-test should always have a null of equal to zero so I'm confused.

No, that's not right. Under the null goes whatever value is meaningful for the problem at hand$$^{(*)}$$.

(*) As per Alexis's comment, the null and the alternative can also be written as

$$H_0: \mu- 15k\leq 0$$ against $$H_1:\mu-15k>0.$$

• +1 I will also point out in response to ash's question about "zero" that $H_0: \mu \le 15k$ is equivalent to $H_0 : \mu - 15k \le 0$. Commented Apr 24, 2023 at 20:47
• @Alexis agreed! Commented Apr 24, 2023 at 21:04