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If you're wondering how this poll could so completely disagree with this poll:
It's because they're both fucking useless and you should ignore (and remember to always ignore in the future) anyone dishonest enough to pretend that you can draw conclusions about voters under 30 from either one of them.
The NY Times/Sienna poll had 179 18-29 year old respondents while the Economist/YouGov poll had 168 18-29 year old respondents; those sample sizes are of zero scientific value.
The main issue is that no one can generate random samples anymore. Not via landlines anyways. The problem is that scientific polling is nearly impossible now.
Neither 179 nor 168 is enough of a sample size to draw conclusions; if you don't understand that then your statistics professor completely failed.
The total sample doesn't matter when you're only pulling out a small part of it. If I polled 10,000 people but only 20 of them were over 100 I can't go around saying shit like "A majority of people over 100 say" this or that.
The smallest number I have heard for a statistically significant sample size is 30, from numerous statistics professors at different schools. A quick google search shows that 100 is generally agreed upon to be sufficient to get meaningful results. 179 and 168 should be plenty big enough.
However, there are other areas where these surveys could be erring: namely collection method or location, which could be skewing results one way or another. These factors are not reported on in the articles.
Personally, none of my friends who are in the target demographic are planning to vote for Trump.