Solved – Which test to use for a simple multiple-choice question

hypothesis testing

I asked 200 people a multiple-choice question with four possible answer options and received the following responses:

Option A: 2
Option B: 148
Option C: 50
Option D: 0

Each respondent could select only one option. What's the correct statistical test to use to show that these responses weren't random?

(My first thought was to carry out a binomial test for each option, but that would of course make the responses for Option C appear random.)

Best Answer

Chi-squared test where your null hypothesis is that they were chosen at random i.e. A=B=C=D and the alternative hypothesis is that the mean for at least one of the answers is not equal to the others.

The test statistic is $$T=\sum_i^N \frac{(observed_i-expected_i)^2}{expected_i}$$

Here $T$ follows approximately a chi-squared distribution. In the above formula, if everyone guessed you would expect each answer to get selected 50 times so that would be the "expected" value for each.

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