# How to represent statistical power graphically for a given hypothesis test?

Disclosure: I'm preparing my oral exam for statistics (so I'm tagging as homework, hope it's appropriate).

I think that I'm failing to understand the concept of power of a test (under a graphical point of view). I'll try to explain what I have in mind:

I have a test for which I have

H0: Mu = 52
H1: Mu > 52
Confidence: alpha


And I'm asked to find the power of the test when the true mean is 50.

What I know is that power = 1 - beta = P(rej H0 | H0 is false); and that beta = P(not rej H0 | H0 is false) aka type II error.

Is this figure a correct representation of my reasoning? How would the picture look like if my test was something like:

H0: Mu = 50
H1: Mu < 50
Confidence: alpha


And I was given the true mean = 52 for checking the power of the test? Would it be the same figure but with alpha and beta swapped?

• If $H_0 : \mu = 52$ and $H_1 : \mu \geq 52$, then $H_1$ is true whenever $H_0$ is true? So one can fail to reject the null and still reject it? – Dilip Sarwate Apr 19 '12 at 11:38
• Whops, I put the equal signs in the alternative hypo! They shouldn't be there – luke14free Apr 19 '12 at 11:52
• A while ago I posted this question regarding power & graphical representation that might be of some help – Max Gordon Apr 19 '12 at 21:39

Forget the question about power for now, delete all thoughts about $\mu = 50$ from your mind, and concentrate on getting the test set up correctly.

Your figure is incorrect. With your corrected version of the hypotheses, the red curve is the density of the statistic when $H_0$ is true, and $H_0$ is not $\mu \geq 52$ as marked in your figure. Furthermore, the threshold $\bar{X}_c$ for the test is larger than $52$, not smaller, and the test should be rejecting $H_0$ if the statistic exceeds the threshold. Thus, the area to the right of the threshold should be $\alpha$, not the area to the left as you have it.

Added in response to OP's query as to what is the test illustrated by the figure.

So,your figure (did you draw it yourself or copy it from somewhere?) corresponds to a null hypothesis $H_0 : \mu \geq 52$ and an alternative hypothesis $H_a : \mu < 52$. The red curve is the density of the statistic $\bar{X}$ when $H_0$ is true, and the boundary $\bar{X}_c$ of the decision region is determined by the requirement that the area under the red curve to the left of $\bar{X}_c$ is $\alpha$ (or less, but let's not nitpick for now). Your decision is correctly indicated: the null hypothesis is rejected if the test statistic $\bar{X}$ is smaller than $\bar{X}_c$, and not rejected if $\bar{X}$ is larger than $\bar{X}_c$.

The blue curve is one of many possible distributions of $\bar{X}$ when the null hypothesis is not true. Let us assume that it is drawn for the case of $\mu = 50$. Then, the shaded region under the blur curve is the probability of falsely failing to reject the null when the null is in fact not true (because $\mu = 50$). The complementary probability is the power of the test in this instance.

The figure DOES NOT match the description of the test that you have (as of now) in the sentence beginning "I have a test for..." and the answers that I have given above are not an answer to the question you have asked in words. But, then, your question has less than a thousand words...

• Ok, therefore my test should be H0: mu >= 52 and H1: mu < 52, for the picture to be correct in the first case? – luke14free Apr 19 '12 at 12:31
• This is good practice for your upcoming oral exam! Do you want to learn how to draw a figure that matches the test you are being asked to design, or do you want to learn how to construct a test that fits whatever is shown on the figure? – Dilip Sarwate Apr 19 '12 at 12:53
• Well.. I'd say both things! You never know what you are going to be asked. – luke14free Apr 19 '12 at 13:17