# Test hypothesis against true population distribution

I have drawn a sample from a population. For an hour I have asked all the visitors to a website their age.

population = people in Germany
sample = 32 German visitors to the website

(international visitors were excluded)

Now I want to know if there are any abnormalities in the age distribution of this sample. When I look at the data, the ages seem not to be represented uniformly, so I could form the hypothesis that in my sample a certain age is overrepresented (e.g. middle-aged men and women visit the website more often than people of other ages):

I could test this hypothesis with a statistical test, assuming an underlying normal distribution and would probably find that my sample doesn't deviate significantly from a normal distribution.

But when I look at the population pyramid of Germany, I notice that ages aren't distributed normally in my population (men are blue, women violet in this diagram):

Visually comparing my sample distribution to this population distribution I notice that my sample is missing the second modus ("summit") around age 30, so there might be a significant deviation of my sample from the population that does not become apparent when I test the sample against an assumed normal distribution.

In a case like this, where we know the true distribution of the population (age data for Germany are publicly available) and don't have to work with possibly wrong assumptions:

### How do I test a hypothesis against the true population distribution?

Is there a statistical test that allows me to input true, irregularly distributed population data as the comparison distribution?

Note.

I do understand that my sample is too small for a test to have sufficient statistical power. Please ignore this for the purpose of this question.

• Can you generate a PDF / CDF of the population distribution? Jan 28, 2019 at 13:07
• @user2974951 Yes, of course.
– user235489
Jan 28, 2019 at 13:30

Nonetheless, if the points mentioned above do not apply to you and you would still like to perform some sort of test... you could try a Goodness of fit $$\chi^2$$ test... if you would be able to describe your population distribution accurately enough with a PDF / CDF.