Complaint about AP stats: one tail hypothesis testing

Apologies for people who don’t care…

But DC1 needed help on hypothesis testing, which is totally understandable since it’s non-intuitive for a lot of people (including me!), particularly the one-tailed tests.

So, most of us who use stats regularly are used to the idea of two-tail t-testing– that’s when we want to know if there is actually a relationship between two variables or if it’s just random chance making it look like two variables are related.  This is kind of the essence of regression analysis– we want to know if we can be 95% ok with the idea that the coefficient we are getting is different from zero.  Two-tailed t-tests are used for a lot of other things besides regression, like seeing if a variable is different from a mean or two means are different from each other, but the basic idea is we want to know if things are the same or different.  We don’t assign any direction to the difference– we don’t say which is bigger or smaller, just that the sample was drawn from a distribution with the same/different mean, or the two samples were drawn from one distribution or from two separate distributions.

Hypotheses then look like:

H1: μ1≠μ2

H0: μ1=μ2

With a one-tailed test you are applying a direction– you’re saying that it could be equal, but it could also be going the other direction.  Your alternative (H1) hypothesis is that μ1 is bigger than μ2.  (Or that μ1 is smaller than μ2, depending on what you’re trying to show.)

So the hypothesis should look like:

H1: μ1>μ2

H0: μ1≤μ2

There are a number of different ways to write out H0.  You could write > but then cross it out (I couldn’t find that character in word).  I tell my students they can write “We cannot say that μ1>μ2,” because that’s really the point.  We don’t know if μ1=μ2 or if μ1<μ2 or… even if μ1>μ2 and we just have too small of a sample size to say for sure.  (My students generally will be doing practical t-testing for non-academic employers rather than writing up fancy scientific papers that require formal hypotheses, so it’s more important for them to understand what they’re doing and what their outputs mean than it is for them to follow a specific jargon structure.)

Imagine my astonishment when the AP review sheet DC1 had showed this instead:

H1: μ1>μ2

H0: μ1=μ2

Surely the teacher made a mistake, I thought.  But no!  This is how all the AP stuff is.  This is genuinely what they’re teaching and testing for AP stats.

The problem is that when you formulate it like this, you’re not allowing for the possibility of μ1<μ2, and you’re making kids think that you’ve actually proven the null hypothesis, that you can really say that μ1=μ2 when really all you can do is say you’re not sure if μ1>μ2.

I took a picture and texted it to my friends without comment and got replies like, “What fresh hell is this?” and “!!, No!  less than or equal.” And they were astonished to find it was AP Stats and for real, not just a mistake in the notes.

No wonder some of my students who come in to my class after having taken AP Stats never really got hypothesis testing the first time around.

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