First, I invite you to join Skwinty ... download the paper (NOT the PR!), read it, and then ask questions (I'll be happy to try to help you understand the paper, and also show you how misleading the PR is).
Second, every conclusion drawn from astronomical observations comes with estimates of uncertainty. Further, in nearly every case, how the 'error budget' is determined is at least described (and in some cases, it is the sole topic of a string of very long papers!). Then, again in nearly every case, a clear distinction is made between estimated uncertainty due to known, essentially random factors, and estimated uncertainty due to systematic effects.
Systematic effects are the bane of astronomers' lives (or, to some, their primary research topic), and vast amounts of time and effort go into studying them. However, it is a fact of modern life that this 'back story' sells no papers, so you rarely see it (in PRs, popsci articles, etc, etc, etc). One unfortunate consequence is that all kinds of weird ideas start to float around the internet, based on little more than ignorance of estimates of uncertainty in astronomical results (and some folk apparently seek out this sort of thing to the exclusion of everything else).
Bottom line: if you are interested in the degree of confidence that can reasonably be given to any particular astronomical result, I think you have no choice but to roll up your sleeves and learn how the result was obtained, and what those engaged in doing the research have (and have not) already painstakingly taken account of.
OK, I'll climb down now.
Happy to oblige!
First, though, the sources.
This LAMBDA ("Legacy Archive for Microwave Background Data Analysis") webpage is by far the best single source ("Bibliography of WMAP Science Team Publications Five Year Data Scientific Papers"). The single paper you should read is "Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Likelihoods and Parameters from WMAP Data" (third from the bottom). If you haven't already done so, familiarise yourself with
the cosmology tutorial on Ned Wright's website; the FAQs include one called "
Age of the Universe"; I think that should answer your question.
The number you quoted earlier comes from a newspaper (!), but its likely source is a PR from the WMAP team, probably from their Three-Year results; you can learn about how observations of the CMB can be used to estimate the age of the universe from the LAMBDA webpage (above).