Significance of zero people out of 80 choosing something as their favourite from a selection of 42 things

Question

80 people were asked to choose their favourite object from 42 objects. Not one person chose a particular object.

The criteria for people choosing objects might be random or there might be a pattern so that people like and dislike the same things.

I know that the chances that zero people choose a particular object are 14% - so this to me suggests there is something unfavourable about the object. That doesn't mean though that there is an 86% chance that this hasn't happened by random and there is a bias/pattern in the selections?

How do I determine the existence of whether there is a pattern or not and quantify the faith in which the fact that no person chose an object suggest it is unfavourable?

To me it seems pretty obvious that if no people out of 80 chose something as their favourite then we can say with reasonable certainty that it is not favoured. But somebody else is telling me that no it is not enough of a sample size.

I also have the results of all the 80 peoples selections if that helps.

Answer 1

Maybe you have in mind to test a hypothesis:

If your null hypothesis is that each of the objects has an equal chance of being chosen, then the number of people out of 80 choosing your specific object, identified in advance is $X \sim \mathsf{Binom}(80, 1/42)$ and $P(X = 0) = 0.1455,$ as you say. In R:

dbinom(0, 80, 1/42)
[1] 0.1454673

Then you don't have evidence at the 5% level of significance that choices are other than random.

Reasoning similarly: if you had 130 people all avoiding this particular object, then you would have significant evidence to reject the null hypothesis at the 5% level:

dbinom(0,130,1/42)
[1] 0.04360072

There is nothing sacred about the 5% level, but it is often used. However, few statisticians would judge something that can occur with probability 0.14 as being especially rare.

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If, after all 80 have made their choices, you discover that one or more items were not chosen by anyone, that is not at all surprising. The probability is almost 1 that fewer than 42 items are chosen under random sampling. On average, only about 36 of the 42 items will be chosen. (Illustrated by simulation of 100,000 iterations in R below.)

set.seed(1012)
pop = 1:42
nr.ch = replicate(10^5, length(unique(sample(pop, 80, rep=T))))
mean(nr.ch < 42)
[1] 0.99968
mean(nr.ch)
[1] 35.88572

Even with 130 people there is still probability about 0.87 that fewer than 42 items are chosen under random sampling. (Variant of above simulation not shown.)

Answer 2

It seems to me that the relevant statistical question is "How likely is it that at least one of the 42 objects is not chosen, assuming that each item is equally likely to be chosen?". If this probability is moderately high then it seems unfair to criticize an item which was not chosen, because we should expect some item(s) to be un-chosen just by chance.

So let's suppose that when a person chooses an item, each item is equally likely to be chosen, with a probability of $1/42$, and that each person makes his choice independently of other people's choices. By inclusion/exclusion, the probability that at least one item is not chosen by anyone is $$\sum_{m=1}^{42} (-1)^{m-1} \binom{42}{m} \left( 1 - \frac{m}{42} \right)^{80} = 0.99973$$ which is quite high, verging on certainty. So we should not be surprised to find that some item is not chosen.