Your question got me curious about the date. I had always seen this poster referred to as being from the 50s, but I started to wonder if I could definitively date it.
The poster is has a clear signature; it is by Fred Ludenken. However, I couldn't find anything about the artist except in relation to this poster.
Then it hit me, the logo! Sure enough, the logo on the poster was used by American Airlines from 1945 to 1962, so we can at least definitively narrow it to that period.
FWIW I saw one page specifically state a date of 1955 for the poster, but they didn't mention how they arrived at that date.