Q:

Best way to convince managers importance of typography

Most designers know that typography has a big influence on design and, more importantly, readability (which is a big part of ux).  But consumers (as well as managers and business leads) don't know this unless they have been told so or compare good typography with bad typography. 
How do you convince your manager/coworker developers as a user experience specialists? 

A:

The best way is to show them. 
Take a screenshot of the tool itself and redesign it using properly treated Typography so they can see the difference.  You can present your case with "Before" and "After" slides.  In my experience because I like to advocate good design, sometimes will do this just for the heck of it because some bad designs look so gawd-awful. 
You can also ask if you can run an A/B test or do a quick hallway test to ask users which version provides a more positive experience for them.  You will end up with more measurable results that you can bring forward to your team. 
You can also show them other successful companies who maintain great Typography.  Providing case studies that show numbers can further your cause. 
Managers still not convinced?  Here are some more articles:

When done right, it can trigger the right emotion, mood, and even action required to get a visitor to convert. 

Read More: https://blog.kissmetrics.com/how-typography-affects-conversions/
This Kissmetrics article displays a case study of an experiment done by the New York Times:
http://opinionator.blogs.nytimes.com/2012/08/08/hear-all-ye-people-hearken-o-earth/?_php=true&_type=blogs&_r=0

Typography plays an important role in influencing people’s decisions. 
  Read More: http://blog.crowdspring.com/2014/05/how-typography-affects-conversions-and-sales/
Can Typography Impact Our Judgement?  Yes. 

Read More: http://conversionxl.com/the-effects-of-typography-on-user-experience-conversions/

