# What is the interpretation of alpha and beta within the plots of a H0 and H1?

and say:

Illustration of the power and the significance level of a statistical test, given the null hypothesis (sampling distribution 1) and the alternative hypothesis (sampling distribution 2).

I don't understand how $$\alpha$$ and $$\beta$$ relate to the plots. Can you give me some hints please?

• Are those Gaussian probability distribution curves (could be any other.) each for a single group of people -say for height, or are they a distribution of the means of many groups of people ? (just taking the mean as an example.)

The graph is correct, but probably more confusing than it is helpful.

The 2 gaussians are meant to represent 2 different scenarios: the first, when $$H_0$$ is true (sampling distribution 1), the other, when $$H_a$$ is true (sampling distribution 2).

The term "sampling distribution" is properly used, referring for example to a single mean (for a 1-sample t test), a difference of means (for a 2-sample t-test), or to some other sampling distribution, for any given statistic $$t$$, which estimates a parameter $$\theta$$.

In the case of the figure, the null hypothesis is that the parameter in question is $$0$$ (a common, but not universal situation).

A better figure would be to stack the 2 scenarios as in the figures below:

How alpha and beta relate to the plots should be clearer:

• Alpha errors can only occur when $$H_0$$ is true, while beta errors can only occur when $$H_a$$ is true
• Alpha errors are when you incorrectly reject $$H_0$$ when it is in fact true (scenario 1), beta errors when you fail to reject $$H_0$$ when it is in fact false (scenario 2)
• If one increases alpha (at the cost of more chances of alpha errors), this will decreases beta (i.e. increase your power)
• Similarly, if one decreases alpha, this will increase beta, thus decreasing your power
• You risk being misunderstood by referring to the parameter $\theta$ as a "statistic." I believe this is certain to confuse anyone who isn't already deeply familiar with estimation and hypothesis testing.
– whuber
Commented May 20 at 2:32
• I apologize in advance if my suspicions are misplaced: Did you make these plots? Or are you borrowing the plots from someone? If the latter, please add a reference to the source. Commented May 20 at 8:41
• ok, i got it finally, at least in broad terms, thank you, that's a master explanation ! Commented May 20 at 14:49
• dipetkov, this is indeed my graph. @whuber, yes indeed, I misused statistic vs. parameter.; Post has been edited accordingly. Commented May 20 at 16:39