Please see the embolded phrase below. When I read this for the first time, I didn't see this problem at all, and this problem didn't present itself immediately to me. After rereading this four times, I still don't understand this immediate problem!
> If you multiply both sides of each inequality by the common denominator
(all people) × (all smokers) you can see that the two statements are different
ways of saying the same thing:
>(married smokers) × (all people) < (all smokers) × (all
married people) Why doesn't `$\tag{3}$` work?
> In the same way, if smoking and marriage were positively correlated, it
would mean that married people were more likely than average to smoke and
smokers more likely than average to be married.
> One problem presents itself immediately. **Surely the chance is very small
that the proportion of smokers among married people is _exactly the same_ as
the proportion of smokers in the whole population.** So, absent a crazy
coincidence, marriage and smoking will be correlated, either positively or
negatively. And so will sexual orientation and smoking, U.S. citizenship and
smoking, first-initial-in-the-last-half-of-the-alphabet and smoking, and so on.
Everything will be correlated with smoking, in one direction or the other. It’s
the same issue we encountered in chapter 7; the null hypothesis, strictly
speaking, is just about always false.
Ellenberg, *How Not to Be Wrong* (2014), page 348.
Chgg Clou
·
2021-06-04T04:49:56Z (almost 3 years ago)