Hypothesis Testing: conditional vs unconditional probabilities

Question

Let's assume that we have a population of people and we study the percentage of them who are interested in sports. In that case by sampling some of the users and we can estimate the percentage.

Now let's assume that someone gives us the additional information of whether a particular person is subscribed to a sport magazine. Again we can estimate the percentage of people who interested in sports given they have subscribed to a sports magazine.

I want to perform hypothesis testing and check if the following is true:

$$P(\text{Interested in Sports}|\text{Subscriber}) > P(\text{Interested in Sports}) $$

I could perform a chi-square test nevertheless this will tell me if the "Subscriber" event is independent from the "Interested in Sports" event. I would like to focus only on the positive case and test if by knowing this extra information we can spot easier more athletic people.

On the top of my head, I am thinking that performing a T-test on independent samples for the percentages would work. For the unconditional case, I will use the entire sample (including subscribers and non-subscribers) and estimate the p of being interested in sports. For the conditional case, I will use only the portion of the sample who are subscribed to a sports magazine and I will perform the same. Using the percentages and the sample sizes, I will estimate the variance and I will conduct the T-test.

Do you see any problem with this approach?

Answer 1

A T-Test really shouldn't be used when comparing non-distinct samples (in this case the overlap would be all those who subscribed to a sports magazine).

It sounds like you’re looking for a method to glean some kind of relationship between the variables Is a Subscriber and Is Interested in Sports. Unfortunately neither a T-test nor a chi-squared test will help you there because both only determine independence between the two variables and won’t tell you anything about the relationship between them beyond whether they have a relationship or not (whether they are dependent or independent).

What you could do is:

1. Prove that the variable Is a Subscriber and Is Interested in Sports are not independent by performing a T-test with “A person’s subscriber status is independent of whether they are interested in sports” as your null hypothesis and only using persons for which you have an Is a Subscriber response for.
2. Once you’ve proved they are not independent, perform some form of catagorical data analysis to measure the assosiation between Is a Subscriber and Is Interested in Sports. A list of analyses for catagorial data is here.
3. Use the result to "test if by knowing this extra information we can more easily spot more athletic people".