# Hypothesis Directionality

To see why:
Ho: µ1 >= µ2 is the same as Ho: µ2 <=µ1

This example is of a null hypothesis. In it we are asking if these two statements are the same:

* the sample mean of the first group is greater than or equal to the second group * the sample mean of the second group is less than or equal to the first group

This is true because you're reversing the entire equation. In mathematics an equation should work in both directions.

Here is an example. Say you had origami cranes you were folding for a wedding in two colors - pink and blue. Say you were wondering about the mean weight of these cranes. You found out that the blue ones had an average weight of .6g. You also found the pink ones had an average weight of .6g. Your two statements would be:

* blue cranes at .6g >= pink cranes at .6g - true
* pink cranes at .6g <= blue cranes at .6g - true

Now let's say the blue cranes actually came out to a mean of 1g. The pink cranes were still .6g. This would read -

* blue cranes at 1g >= pink cranes at .6g - true
* pink cranes at .6g <= blue cranes at 1g - true

Again it does work, and it needs to work because an equation has to be valid both ways.

Even if you ended up with results of the pink cranes being 1g and the blue cranes being .6g, it works -

* blue cranes at .6g >= pink cranes at 1g - false
* pink cranes at 1g <= blue cranes at .6g - false

So the results are equivalent whichever way you phrase it.

To see why:
H1: µ1 < µ2 is the same as H1: µ2 > µ1

This example involves a H1 - an alternate hypothesis. In this case there cannot be an equal sign. So again using our crane example, let's say both blue and pink cranes come out with a mean of .6. So you would have

* blue cranes at .6g > pink cranes at .6g - false
* pink cranes at .6g < blue cranes at .6g - false

In neither direction are they having one larger than the other. They are equal.

Now let's say the blue cranes actually came out to a mean of 1g. The pink cranes were still .6g. This would read -

* blue cranes at 1g > pink cranes at .6g - true
* pink cranes at .6g < blue cranes at 1g - true

Again it does work, and it needs to work because an equation has to be valid both ways.

Even if you ended up with results of the pink cranes being 1g and the blue cranes being .6g, it works -

* blue cranes at .6g > pink cranes at 1g - false
* pink cranes at 1g < blue cranes at .6g - false

So again just like with the top example the equation has to work in both direction if you flip the entire equation around in a mirror image.

Statistics Basics