Are crazy sports fans just lonely people trying to fill a personal void? Or are they actually happier than non-sports fans?  An article in the New York Times profiles two extreme sports fans, the "Dome Ranger," a Syracuse University basketball fan, and "Wild Bill," a shirtless fan of Utah State basketball.

Inserted into the text of the article is a nod to psychological research:

In the 1990s, researchers began finding that “highly identified” fans
experienced higher levels of arousal — measured by heart rate, brain
waves and perspiration — and had fewer bouts of depression and
alienation than nonfans. Contrary to popular belief, this research
suggests, hard-core fans even have higher self-esteem.

a) This paragraph is describing some correlational research–several bivariate correlations. What are the variables in each association?  Sketch a scatterplot of the associations they are describing.

b) One of the associations here suggests that highly identified fans had fewer bouts of depression. The journalist seems to draw his headline from this correlational research, saying that "one way to cheer up is to cheer harder."  Can this causal claim be supported?  (Work through all three rules for a causal claim, starting with covariance between depression and sports-team identification.)

c) This story is mainly about the Dome Ranger and Wild Bill, two VERY enthusiastic sports fans who seem well adjusted and happy. Certainly, these two superfans were not in the original research. But what if they were? Revisit your scatterplot that you drew in question a), and add two points to it for these superfans. What might these outliers do to the correlation you'd observe between the two variables–would the outliers make the correlation stronger? Weaker? or have no effect?