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I just created a facebook page for residents in greenland and denmark and I'm wondering if the distribution of likes.

The population of greenland is:    56.483
The population of denmark is:   5.614.000

I have the following likes so far:

Greenland: 11
Denmark:   32

It is obviously skewed in some way. If the hypothesis is that people from greenland and denmark would have the same ability and reason to like the page, what is the probability that I get this distribution by chance?

Note: This is not homework I have a facebook page and I'm genuinely interested in why the distribution is so skewed. And even though it isn't homework I'm also interested in how you would calculate this.

Finally... Sorry if this problem is too simple.

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    $\begingroup$ It's skewed because your sample is terrible. 43 people out of over five million? $\endgroup$
    – Nij
    Sep 13, 2016 at 20:35
  • $\begingroup$ @Nij how would you show that? How many samples would I need? $\endgroup$
    – JohannesH
    Sep 13, 2016 at 20:37
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    $\begingroup$ You need one sample that is big enough to be justified as representative. On the order of $\sqrt{|population|}$ is a good benchmark, but it depends on several things: actual population size, desired accuracy, accessibility to data, and others. $\endgroup$
    – Nij
    Sep 13, 2016 at 20:42
  • $\begingroup$ @Nij thanks for elaborating :) $\endgroup$
    – JohannesH
    Sep 13, 2016 at 20:55

2 Answers 2

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The classical statistics treatment of this problem would be something like this.

Null hypothesis: every member of the population of Greenland and Denmark, independently of everybody else, has the same unknown probability $p$ of liking your page. Nobody who is not a resident of Greenland or Denmark can like the page.

Given a total of $43$ likes for the two pages, every set of $43$ people from the combined population has the same probability of being the ones who liked the page. The conditional probability of getting exactly $x$ likes from Greenland is then

$$ p(x) = \dfrac{{56483 \choose x} {5614000 \choose 43-x}}{5614000+56483 \choose 43} $$

The conditional probability of at least $11$ likes from Greenland would be $$ \sum_{x=11}^{43} p(x) \approx 4.105 \times 10^{-13}$$

This is extremely small, so we should reject the null hypothesis.

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  • $\begingroup$ Thank you for the answer Robert It was exactly what I was looking for. I was trying to calculate it as a series of multiplications (subtracting from one of the groups and the total for each multiplication) but I realized it was calculating a probability of a specific sequence of likes and I couldn't figure out how to calculate the probability for all permutations of sequences at once. As Nij mentioned in the comments above the sample is pretty small. Can you say anything about how accurate this result might be? $\endgroup$
    – JohannesH
    Sep 13, 2016 at 21:04
  • $\begingroup$ How accurate what result might be? The expression for the conditional probability I quoted is exactly correct: the number $4.105 \times 10^{-13}$ is accurate to $4$ significant figures, but I could give you as many significant figures as you want. Note that this is calculated under the null hypothesis. I have said nothing about probabilities under alternative hypotheses. $\endgroup$ Sep 13, 2016 at 21:57
  • $\begingroup$ If you want the exact rational number, it is $${\frac{ 2377803454610794730487137896550489466706819271496770498663431920862835976949843955105598051422355192713264126055898986861596756579090768624175204797367580061734076144478113319447093140169188867908343366347483 }{ 5791946037971906737681992131428274071800069549140781196145251405127344032207105977909432366286325603930745039636808646494775839826718734731359929631103532374936824348764886513768725638163599343741362681964815686400974118 }} $$ $\endgroup$ Sep 13, 2016 at 22:08
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  1. Populations of small size, very low density and low urbanization may have more reason to depend on Internet for social interaction, than larger populations living at higher density in cities.

  2. Social network connections and actions on the network are positively inter-correlated. Whether somebody will click "like" on a post is correlated with whether their "friends" in the social network liked it. A lot of things that occur on the network have a much higher probability of happening than in a model where the decisions on "likes" are made independently.

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  • $\begingroup$ Thank you for the insights. I agree that my problem is over-simplified and my sample size is way small. But in my case I think that people from Greenland have less fb connections on average than people from Denmark (I might be wrong). Also fb usage is lower for people from greenland than people from denmark. I can see this from the audience segments facebook shows me. $\endgroup$
    – JohannesH
    Sep 13, 2016 at 21:16
  • $\begingroup$ They may have fewer fb connections and less usage, but apparently they use what they do have to like your page. Or maybe it's just that the Danes hate it... $\endgroup$ Sep 13, 2016 at 22:10

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