First question:
The reason why the title of the article is wrong can be summarized with the phrase "correlation is not causation."
Causation implies that one variable is the reason for the other one to occur. However, in this case, we lack information to conclude that. All the information we have is that both phenomena (the rise in the use of vitamins and the increasing crime rates) are happening at the same time. This mutual relationship is called correlation.
This correlation could be caused by a third variable (for example, demographics or income levels).
Another example could be: As sales of ice cream increase, drownings increase as well.
We would be tempted to conclude that ice cream causes drownings. However, the third variable of "hot weather" could provide a better explanation. As the weather gets warmer, ice cream sales go up, and the number of people in swimming pools goes up as well. With more people swimming, you get a higher number of drownings. This is called a "spurious relationship".
Second question:
In your study, you would need two groups: a control group and an experimental group. This is done so that you can control all other variables except the use of vitamins. If all other factors are controlled, you can confidently attribute any effects to the use of vitamins exclusively.
A double-blind study is when neither the participants nor the experimenters know who is in the control group and who is in the experimental group. This is to prevent any bias from altering the results of the experiment.
Other things to be careful about are the size of the sample (number of people should be large enough to make the results representative of the general population) and the timeframe.
