A Question Most People Get Wrong

Edit: The poll is now over. A bunch of people voted after I wrote this article, but I decided not to redo all calculations because I am lazy.

A few days ago I created a Twitter thing called Geeky Polls. The first poll I posted was this one:

At first glance, this question may look like an obfuscated version of the liar paradox. However, it is not paradoxical, and you can answer correctly by choosing whichever option fewer people preferred.

When I made this poll, I expected that both answer choices will get roughly equal number of votes. Surprisingly, this turned out not to be the case: one of the options received 71% of votes. In case you haven’t voted yet, I will not disclose which option was more popular, since it would allow you to cheat.

But is this difference significant? Let’s find out. The poll attracted the attention of 70 people. The last time I checked, 71% of 70 was 50, so that’s how many people preferred the Popular Option. 20 people chose the Unpopular Option – what nice round numbers!

We can now start some kind of analysis. I will not use Bayesian approach because it is not suitable in this situation I know nothing about it, so let’s stick to good old frequentism. In this case, the p-value is the probability that 70 coin tosses will result in some side showing up at least 50 times. There are two ways to calculate it: smart one (do the math) and silly one (write a program). I will use the second approach.

Since you are probably tired of me posting Delphi apps, here is some nice and clean JS code:

var numTries = 100000;

var i, j, heads, skewed;

skewed = 0;
for (i = 0; i < numTries; i++) {
    heads = 0;
    for (j = 0; j  0.5) {
            heads++;
        }
    }
    if (heads = 50) {
        skewed++;
    }
}

// As you can see, I multiplied the p-value by 10000
// Hence, any output less than 500 will indicate significance
println(10000 * skewed / numTries);

This code gives the result of 4.3, which is much less than 500. Significance proven! Well, technically, it’s not. Since the program is randomized, there is always a possibility that it will take an insignificant result and mark it as significant. It means that we should conduct a statistical analysis to determine the significance of the result coming out of significance-testing program… My head hurts. That’s why serious people always choose the mathematical approach.

Anyway, goodbye for now. If you would like to know what the correct answer to the question is, just vote yourself!

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