Good example and well explained. We should team up on a book on science for lay people!

Your point about specifying the null hypothesis and the p value is very important. Another way studies can fail is if you pick 20 different variables, like you mentioned, and then look to see if any of them give you p<0.05. So in your example, we measure smiling and 19 other factors besides being told jokes. Let’s say the weather, the day of the week, what color clothes the person is wearing, what they had for breakfast, etc. Again, due to statistics, one of those 20 is going to appear relevant by chance. You’re essentially doing 20 experiments in one so again you’ll get one spurious result that you can report as “success”.

Experimental design is tough and it’s hard to grok until you’ve had to design and run your own experiment including the math. That makes it easy for people to pass off bad science as legitimate, whether accidentally or on purpose. And it’s why peer review is important, where your study gets sent to another researcher in your field for critique before publication.

There’s other things besides bad math that can trip you up like correlation vs causation, and how the data is gathered. In the above example, you might try to save money by asking subjects to self report on their smiling. But people are bad at doing that due to fallible memory and bias (did that really count as a full smile?). Ideally you want to follow them around and count yourself, with a clear definition of what counts as a smile. Or make them wear a camera that does facial recognition. But both of those cost more money than just handing someone a piece of paper and a pencil and hoping for the best. That’s why you should always be extra suspicious of studies that use self reporting. As my social psych prof said, surveys are the worst form of data collection. It’s what makes polling hard because what people say and what they do are often entirely different things.