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Does this hypothesis test:

H0: μd = 0 H1: μd ≠ 0

Represent the same as this test:

H0: μ1 = μ2 H1: μ1 ≠ μ2

Since the first is asking if there is a difference between the true mean of the two samples (is it 0), whereas the second is asking whether there is any difference in means between the two sample means.

Does this mean the two hypothesis test statements are equivalent in their meaning?

If I want to find if the mean of two samples is different, which would I use? Example being whether smokers vs non-smokers have the same mean lung-capacity.

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  • $\begingroup$ Both sets of hypotheses are about population means. The first looks more suitable for a paired test and the second for an unpaired test $\endgroup$
    – Henry
    Feb 28, 2019 at 6:52

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Hypothesis 1 is a comparison of a sample mean (the mean of $X_1-X_2$) to a theoric value of $0$:

H0: μd = 0 H1: μd ≠ 0

The test variable will therefore be $\frac{m_d}{\sqrt{\frac{\sigma_d^2}{n_1}}}$

Hypothesis 2 is a comparison of 2 sample means (the mean of $X_1$ and the mean of $X_2$):

H0: μ1 = μ2 H1: μ1 ≠ μ2

The test variable will therefore be $\frac{m_1-m_2}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}$

Please note that the 2 hypothesis can only be compared if $n_1=n_2$, else the mean of $X_1-X_2$ have no sense.

In the numerator, we have the same thing since $Mean(A-B) = Mean(A)-Mean(B)$.
But in the denominator though, things may be different because $var(A-B) = Var(A)+Var(B)+2\times Covar(A, B)$.

Therefore, tests will be the same only if $A$ and $B$ are independant, so $Covar(A, B)=0$.

I hope this helps you to understand why (and moslty when) these hypothesis are different.

Maybe thinking in terms of distributions instead of means will help too: the first hypothesis is testing if the distribution of a variable (which is the difference between 2 variables) is centered on a value significantly different from $0$, while the second is testing if there is a significant difference between the 2 distributions of 2 variables.

As @Henry said, the hypothesis #1 is especially adapted to the paired scheme, where $n_1=n_2$ by concept and the very variable of interest if the difference in the pair more than the difference between the groups.

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  • $\begingroup$ Thank you for the detailed response. I have two categories of sample sets to compare whether the population means of their attributes differ. They have the same n. The hypothesis effectively asks whether those with in Group A having differing means from those in Group B, given that they share attributes x1, x2,...xn. Thus I'll use the second hypothesis test type, is how I'm interpreting your response. $\endgroup$
    – statTesting
    Feb 28, 2019 at 21:09
  • $\begingroup$ Then you have 2 populations so you should indeed use the hypothesis 2 (that was not crystal clear from your question since you can take 2 samples of the same population). If you had one population observed at different moments/conditions, then you would be interested in the difference and use the hypothesis 1. $\endgroup$
    – Dan Chaltiel
    Mar 1, 2019 at 7:26

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